Subsystems of Set Theory and Second Order Number Theory

CHAPTER IV

Subsystems of Set Theory and Second Order Number Theory Wolfram Pohlers Institut fiir mathematische Logik und Grundlagenforschung West f~lische Wilhelms-Universit~t, D-~81~9 Miinster, Germany

Contents 1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. P a r t i a l models for axiom systems of set theory . . . . . . . . . . . . . . . . 1.3. Connections to s u b s y s t e m s of second order n u m b e r theory . . . . . . . . . 1.4. M e t h o d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. First order n u m b e r theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Peano a r i t h m e t i c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Peano a r i t h m e t i c with additional transfinite induction . . . . . . . . . . . . 3. Impredicative systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Some remarks on predicativity and impredicativity . . . . . . . . . . . . . 3.2. A x i o m systems for n u m b e r theory . . . . . . . . . . . . . . . . . . . . . . 3.3. A x i o m systems for set theory . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Ordinal analysis for set-theoretic axioms systems . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

HANDBOOK OF PROOF THEORY Edited by S. R. Buss 9 1998 Elsevier Science B.V. All rights reserved

210 210 215 219 230 231 231 261 266 267 268 279 294 333

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w. Pohlers

1. P r e l i m i n a r i e s The aim of the following contribution is to present a sample of ordinal analyses of subsystems of Set Theory and Second Order Number Theory. 1 But before we start the presentation of results we think we should enter a general discussion about the type of results we are going to obtain. We want to keep close to Hilbert's program and try to give consistency proofs for axiom systems as constructively as possible - being well aware of all the obstacles to this enterprise resulting from Ghdel's Theorems. The emphasis will be on impredicative subsystems of Set Theory and Second Order Number Theory. Here we opt for a seemingly unconventional approach. We will try to construct partial models of theories within the constructible sets. The hierarchy L of constructible sets is determined by the ordinal line. Therefore special care will be given to notations for ordinals. In the Preliminaries we will introduce our concept of ordinal analysis. First we introduce some basic facts on ordinals, then introduce our concept of partial models and finally show how this is connected to the more conventional approach to ordinal analysis. 1.1. O r d i n a l s Ordinals will play the crucial role for all what follows. Therefore we start with a short introduction to ordinals as we will use them in the following contribution. Ordinals are regarded in their set theoretic sense, i.e., an ordinal is a hereditarily transitive set. Assuming that the membership relation E is a well-founded relation, this entails that every ordinal is well-founded with respect to the membership relation and every ordinal is the set of its E-predecessors. For the reader who is not so familiar with set theory we give a brief sketch of the theory of ordinals as far as it will be needed for this paper. Besides transfinite recursion (which may be regarded as a generalization of primitive recursion) all we need from Set Theory are the facts (On1) - (One) below. They may be viewed as axioms for the theory. To make the article not too long we will not give proofs here. Detailed information how to prove the results of this section from (On1) - (One) can be found in Pohlers [1989]. A linear order relation -~ well-orders its field iff it does not contain infinite -~-descending sequences. A class M is transitive iff a E M =~ a C_ M.

(On1) The class O n of ordinals is a non void transitive class, which is well-ordered by the membership relation E. We define ~ < ~ as c~ E O n A ~ E O n A c~ E ~. In general we use lower case Greek letters as syntactical variables for ordinals. The well-foundedness of E on the class O n implies the principle of transfinite induction

(V~ E On)[(Vr/< ~)F(~)

=~ F(~)] =~

(V~ E O n ) F ( ~ )

1i am indebted to Dr. Arnold Beckmann for proofreading. He not only detected a series of errors in the first versions but also made many valuable suggestions.

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211

and transfinite recursion which, for a given function g, allows the definition of a function f satisfying the recursion equation f(r/) -- g({f(~)[ ~ < ~}). (On2) The class O n of ordinals is unbounded, i.e., (V~ E On)(3r] E On))[~ < r/]. The cardinality IMI of a set M is the least ordinal ~ such that M can be mapped bijectively onto ~. An ordinal ~ is a cardinal if Ic~l - c~. (On3) If M C O n and IMI E On then M is bounded in On, i.e., there is an a E On such that M C ~. For every ordinal a we have by (On1) and (On2) a least ordinal c~' which is bigger than c~. We call a' the successor of c~. There are three types of ordinals: 9 the least ordinal O, 9 successor ordinals, i.e., ordinals of the form c~', 9 ordinals which are neither 0 nor successor ordinals. Such ordinals are called limit ordinals. We denote the class of limit ordinals by Lira. Considering these three types of ordinals we reformulate transfinite induction and recursion as follows: Transfinite induction: If F(O) and (Vc~EOn)[F(c~) (V~ < ,~)F(~) => F(A) for ,~ E Lira then (V~ E On)F(~).

=V F(c~')]

as well as

Transfinite recursion: For given c~ E On and functions g, h there is a function f satisfying the recursion equations f (O) - e~

f (~') = g(:(~)) f ( A ) - h({f(r/) I rl < An ordinal ~ satisfying

(R1) (R2)

,~})for

A e Lim.

~ e Lira

If M C_ ~ and [M] < ~ then M is bounded in ~, i.e., there is an a E ~ such that M C a

is called regular. The class of regular ordinals is denoted by JR. (On4)

The class R is unbounded, i.e., (V~ E On)(3r/E R)[~ _ r/].

We define sup M "= min {~ E On] (Mr/E M ) ( r / < ~)} as the least upper bound for a set M C_ On. In set theoretic terms it is sup M - U M. It follows that sup M is either the biggest ordinal in M, i.e., sup M - max M, or sup M E Lira. By w we denote the least limit ordinal. It exists according to (On4) and (On1). The ordinal Wl denotes the first uncountable ordinal, i.e., the first ordinal whose cardinality is bigger than that of w. It exists by (On3).

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212

For every class M C_ O n there is a uniquely determined transitive class otyp(M) C_ On and an order preserving function enM: otyp(M) onto>M. The function enM enumerates the elements of M in increasing order. Since otyp(M) is transitive it is either otyp(M) = On or otyp(M) E On. We call otyp(M) the order type of M. In fact otyp(M) is the Mostowski collapse of M and enM the inverse of the collapsing function (usually denoted by ~r). By (On3)we have otyp(M) E On iff M is bounded in On. Unbounded, i.e., proper classes of ordinals have order type On. If M is a set of ordinals then otyp(M) E On. If M is a transitive class and f: M ~ O n an order preserving function then we have a <_ f (a) for all a E M. A class M is closed (in a regular ordinal a) iff sup N E M holds for every class N C_ M such that IN I E On (IN[ < a). We call M club (in a ) i f f M is closed and unbounded (in a). We call an order preserving function f" M ~ O n (a-) continuous iff M is (a-) closed and f preserves suprema, i.e., sup {f(~)[ ~ E N} - f(sup(N)) for any N C_ M such that INI e On (INI < a). A normal (R-normal) function is an order-preserving continuous function f: O n

> O n or f: a

~ a respectively.

For M c_ O n (M C_ a) the enumerating function enM is a (a-)normal function iff M is club (in a). Extending their primitive recursive definitions continuously into the transfinite we obtain the basic arithmetical functions 4-, 9and exponentiation for all ordinals. The ordinal sum, for example, satisfies the recursion equations

a+O=a ~ + / ~ ' = (~+/~)' a + A = sup~ < ~(a + ~) :for A e Lira. It is easy to see that the function A(. a 4- ( is the enumerating function of the class {( E On] a < (} which is club in all regular a > a. Hence A(. a 4- ( is a R-normal function for all regular a > a. We define H := {~ e On I o~# 0 ^ (V~ < oL)(Vrl < o0[~ + r/< ~]}

and call the ordinals in H additively indecomposable. Then H is club (in any regular ordinal > w), 1 := 0' E H, w E H and w r q H = {1}. Hence end(0) = 1 and en~(1) = w which are the first two examples of the fact that (V~c E On)[en~(~) = w~]. Thus A(. w~ is a (a-)normal function (for all a E R bigger than w). We have

][-][ c_ Lira U {1} and obtain e M/i/(v~

< o0[,~ + o~ = 04.

(1)

213

Set Theory and Second Order Number Theory Thus for a finite set { a l , . . . , an} C_ H we get al +...

+ an = akl + "'" + akin

for { k l , . . . , kin} C_ { 1 , . . . , n} such t h a t ki < ki+l and ak~ _> ak~+l. By induction on a we obtain thus ordinals { a l , . . . , an } C_ H such t h a t for a r 0 we have (2)

a = al + " " + an and al > _ ' " >_ an.

This is obvious for a E H and immediate from the induction hypothesis and the above r e m a r k if a = ~ + 77 for ~, 77 < a. It follows by induction on n t h a t the ordinals a l , . . . , an in (2) are uniquely determined. We therefore define an additive normal form a=NFal+...+an'r

a=al+...+an,

{at,...,an}C_H

and al >_ " " >_ an.

We call { a l , . . . , an } the set of additive components of a if a =NF a l + ' ' ' + an. We use the additive components to define the symmetric sum of ordinals a =NF a l + ' ' ' + an and fl =NF an+l + " " + am by a ~ fl "= a~(1) + ' "

+ a~Cm)

where lr is a p e r m u t a t i o n of the numbers { 1 , . . . , m} such t h a t 1 _< i < j _< m =~ a~( 0 ___ a~(j). In contrast to the "ordinary ordinal sum" the symmetric sum does not cancel additive components. By definition we have

It is easy to check t h a t the symmetric sum is order preserving in its b o t h arguments. As another consequence of (2) we obtain the Cantor normal f o r m for ordinals for the basis w, which says t h a t for every ordinal a r 0 there are ordinals ~ 1 , . . . , ~n such that a = N F w ~'~ + " ' " + w ~".

Since ) ~ . w r is a normal function we have a _< w ~ for all ordinals a. We call a an e - n u m b e r if w ~ = a and define r

" - min { a I w" = a } .

more generally let ) ~ . Q enumerate the fixed points of ) ~ . w r . If we put :=

Z) :=

we obtain ~o "= sup expn(w, 0). n(og

For 0 < a < Co we have a < w ~ and obtain by the Cantor Normal Form T h e o r e m uniquely determined ordinals a l , . . . , an < a such t h a t a =NF w al + "'" + w an.

214

W. Pohlers

For a class M c On we define its derivative M' := {~ e On[ enM(~) = ~}. The derivative f ' of a function f is defined by f ' := enFix(/), where Fix(f) : - {r

f(r

= r

Thus f ' enumerates the fixed-points of f. If M is club (in some regular ~) then M ' is also club (in ~). Thus if f is a normal function f' is a normal function, too. If {M, I t e I} is a collections of classes club (in some regular ~) and III e O n ([I[ e ~) then ~,e, M~ is also club (in ~). These facts give raise to a hierarchy of club classes. We define c

(o) : = H := :=

cr(r

for

e Lira.

If we put ~ : - encr(~), then all 9~a are normal functions and we have by definition

a < fl => ~oa(~P~7) = ~ZT.

(3)

The function ~ is commonly called Veblen function. From (3) we obtain immediately

(PC~I~I ~-~ (~cx2~ ig

O/1 < C~2 and ~1 ~_~(~Ot2~ or al = a2 and 131 < ~2 or a2 < al and r < f12.

(4)

We define the Veblen normal form for ordinals ~ d / b y a =NF ~

: r

a = ~

and 7"1< a.

Then a = N F (P~IT]I and a = N F (P~2~2 ~ r "-- r and 771 = ?72. Since ~ < a and < ~ E Cr(a) implies ~ r we call Cr(a) the class of a-critical ordinals.. If a is itself a-critical then ~, r / < a =v qoer/< a. Therefore we define the class SC of strongly critical ordinals by SC := {a e On I ~ e C~(a) }. The class SC is club (in all regular ordinals ~ > w). Its enumerating function is denoted by A~. F~. Regarding that by (4) A~. ~,0 is order preserving one easily proves sc =

=

If we define 7o := 0 and %+1 := 9~. 0 then we obtain F0 = sup 7-. n
Set Theory and Second Order Number Theory

215

For every a < Fo there are uniquely determined ordinals ~1,... ,~n < a and ~h,..., r/n < a such that

--,F (P~lrh + . . . + cP~nrl,~ and rii < ~a~,rli for i E { 1 , . . . , n}.

(5)

This is all we need to know about ordinals for the moment. We will have to come back to the theory later. 1.2. P a r t i a l m o d e l s for a x i o m s y s t e m s of set t h e o r y Let /::(E) denote a language of Set Theory, i.e., a first order language which contains a symbol E for the membership relation. Later we will add a unary relation symbol Ad. We will distinguish between restricted quantifiers (Vx E a) and (3x E a) and unrestricted quantifiers of the form (Vx) and (3x). Recall the Levy hierarchy of/::-formulas. We say that a formula is A0 if it contains only restricted quantifiers. A formula F of/:: is a Hn-formula, if F = (VXl)(3x2)... (Q~xn)a(x~,... ,x~), where G(xl,... ,xn) is a A0- formula and V3..-(Qn) an alternating string of unrestricted quantifiers. Dually, a formula is En if -~F is logically equivalent to a 1-In-formula. A formula F ( x l , . . . ,xn) is a An-formula of an s 92 iff there are a 1-In-formula FH(Xl,... ,xn) and a En-formula Fr.(Xl,... ,x~) such that 91 I= (VZ)(F(Z) +4 .Fn(.~)) A (VZ)(F(Z) +4 Fs(.~)).

(6)

We call F a An-formula of a theory A x iff A x proves the formula in (6). The class of E-formulas is the smallest class which contains the A0-formulas and is closed under the positive boolean operations V and A, restricted quantification and unrestricted existential quantification. The class of II formulas is the dual of the class of E-formulas. A formula F ( x l , . . . , xn) is A in a structure 92 if there is a E-formula FE(Xl,... ,xn) and a H-formula Fii(Xl,... ,Xn) such that 92 ~ (VZ)[F(Z) ++ Fr~(Z)] and 91 ~ (VZ)[F(Z) ++ Fn(Z)]. It is A for a theory A x iff it is A for all models of Ax. Recall that the constructible hierarchy L is the union of its stages L= given by the following definition. 1.2.1.

Definition.

L0 - @

La+1 = Def(L~)

L~ = U{L~I f < ,~} for A E Lim, where a E Def(La) iff there is an /~(E)-formula F(x, yl,...,yn) with no other free variables and a tuple (bl,...,bn) of elements of L~ such that

a - {seL=l L, k

F(s,b~,...,bn)}.

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W. Pohlers

The formula F z is obtained from F by restricting all unrestricted quantifiers in F to z, i.e., replacing (Vx) by (Vx e z) and (3x) by (3x e z), respectively. The formula F z is obviously A0. We say that a formula G is II~ or E,~ if G-

FL~

for a YIn-formula F or En-formula F, respectively. Let Ax be an axiom system formulated in the language s

We define

[IAx[Ioo := min {hi L~ ~ Ax}

(7)

and call I[Ax[Ioo the c~-norm of Ax. By Lbwenheim-Skolem downwards and the Condensation Lemma for the constructible hierarchy we know that [[Ax[Ioo is a countable ordinal which may be quite big. Much too big for our purpose as we will see in a moment. To obtain smaller ordinals which may be characteristic for Ax we regard partial models for Ax. For any complexity class $" of/:-sentences we define ConT(Ax) := {FI F e $" A A x e - F }

(8)

I[Axl[~ := min {a[ L~ ~ Con~(Ax)}.

(9)

and

The complexity of the sentences in the axiom systems which we are going to study will never exceed 113. Therefore we always have IIAxll

= IIAxlloo.

We will only consider axiom systems which comprise the following basis theory K P whose axioms are: The ontological axioms of Extensionality and Foundation (Ext)

(Vu)(Vv)[u = v ~ (Vx e u)(x e v) A (Vx e v)(x e u)]

(Found)

(Vu)[(3x e u)(x e u) --+ (3x e u)(Vz e x ) ~ ( z e u)].

The closure axioms of Pairing and Union (Pair)

(Vu)(Vv)(3w)(Vx)[x e w ~ x = u V x = v]

(Union)

(Vu)(3w)(Vx)[x e w ~ (3y e u)(x e y)]

The set existence axiom schemes of absolute Separation and Collection (A0-Sep) (Vg)(Va)(3z)(Vx)[x e z ~ x e a A F ( x , ~*)] (A0-Col) (WT)(Vu)[(Vx e u ) ( 3 y ) F ( x , y, g) --+ (3z)(Vx e u)(3y e z ) F ( x , y, ~7)]. In both schemes we allow only A0-formulas F(x, g) or F ( x , y, g) of the language s respectively. This requirement guarantees the absoluteness of the sets whose existence is postulated. We use the abbreviations which are common in Set Theory. E.g. a = {x e u[ F(x)} stands for (Vx e a)[x e u A F(x)] A (Vx e u)[F(x) ~ x e a];

Set Theory and Second Order Number Theory

217

{ x 9 I F(x)} 9 b s t a n d s f o r (3y)[y = { x 9 I F(x)} A y 9 b], etc. Lower case Greek letters are supposed to range over ordinals. Adding the ontological axiom of Infinity (Inf)

(3u)[0 e

^ (w e

u

{z} e u)]

we obtain the theory K P w - and adding the Foundation Scheme (FOUND) (3x)F(x) ~ (3x)[F(x) A (Vy e x)-~F(y)] we obtain the theories K P or K P w respectively. If we restrict the formula F(x) in (FOUND) to a complexity class $" we talk about S'-Foundation, denoted by (gv-FOUND). An ordinal a is called admissible if L~ ~ K P . The theories K P and K P w are profoundly studied in Barwise [1975]. They prove 2-recursion- a very important t h e o r e m - and E-reflection, i.e., the scheme

F -+ (3x)F ~ for E-formulas F. As a consequence both theories prove the equivalence of any E-formula to a El-formula. Recall that a partial function f: L~ >p L~ is called a-partial recursive if its graph is E-definable over L~ i.e., if we have a E-formula F(x, y, ~ without further free variables and a tuple g of elements of L~ such that f (a) "~ b iff L,~ ~ F[a, b, c-] for all a, b E L~. Admissible ordinals are important for generalized recursion theory because L~ is closed under all a-recursive functions if a is admissible. Obviously w is an admissible ordinal and it is a folklore result that w~K, i.e., the least countable ordinal which cannot be represented by a recursive well-ordering on the natural numbers, is the next admissible ordinal. Therefore L~cK ~ K P w and, since there are no further admissibles between w and w~K, even IIKP~I[o~ w~K. Hence IIgPwtlH2 < w~K as well as iigPwll~l < w~K. For most impredicative theories Ax, however, we have w~K < [[Ax[l~,. Therefore we need to introduce the following ordinals. 1.2.2. Definition.

Let A x be a theory which contains K P - . Then we define

iiAxilr.~ := min {a I (VF)[F is a El-sentence A A x e - F T'~ =~

L~ ~ F]}

and analogously IHAxlHH~ "= min {at (VF)[F is a II2-sentence A A x e - F L~ =~

L~ ~ El}

The notation A x ~- F ~'~ has to be read with the necessary care. It anticipates that L~ can be defined in some way from the axioms in Ax. We need E-recursion to

218

W. Pohlers

define the function a ~ L~. To prove the E-recursion theorem it suffices to have K P - + El-FOUND. So for theories extending K P - + El-FOUND we need only a description of a in order to characterize La. We say that A x believes that ~ is admissible iff L~ is definable in A x and

Ax

(10)

holds for all sentences G E K P - . rule for a complexity class ~" iff A x ~- a

~

We say that A x proves the L- or L~-reflection

A x ~- (3~)(3u)[u = L~ ^ a ~]

or

A x e - a '-

Axe-

=

^

respectively, for all formulas G E .T. The following lemma is a first easy observation about partial models. 1.2.3. L e m m a . Let a E (w, IIAxII~] be an ordinal such that A x believes that a is admissible and A x proves the L~-reflection rule for El-formulas. Then Ilhxll,~ -

IIAxll r. P r o o f . We obviously have IIAxII~ ~ IIAxllr~ and need only to show the converse inequality. Thus put a : - []hxIl~.?, let (Vx)(3y)F(x, y) be a II2-sentence such that h x ~ (Vx E L~)(3y E L~)F(x,y) and choose some a E L,. We have to show that L~ ~ (3y)F(a, y). Since a is obviously a limit ordinal there is a ~ < a such that a E L~. By definition of a there is a El-sentence G such that A x ~- G L~ but LZ ~: G. Since A x proves L~-reflection we obtain A x ~-(3-y E L~)(3u E L ~ ) ( u - L~ A G u) and since A x believes that a is admissible by A0-collection relativized to L~ also h x ~ v E L~ --+ (3z E L~)(Vx E v)(3y E z)F(x, y). Choosing v - u we thus get A x ~- (3~/E L~)(3u E L~)(3z E L~)[u = L~ A G ~ A (Vx E u)(3y E z)F(x, y)]. Since A x believes that ~ is admissible this is equivalent to a El-sentence relativized to L~. Hence L~ ~ (37)(3u)(3z)[u = L~ A G ~ A (Vx e u)(3y e z)F(x, y)]. Because u = L~ is absolute for L~ we finally get G L~ and (Vx E L~)(3y E L ~ ) f ( x , y) for some 7 < a. Because ofL~ ~ G w e h a v e f l < 7. Hence a E L~ C_ L~ and it follows L~ ~ (3y)f(a, y)as desired. O If K P w - + El-FOUND c_ A x then w < [[Ax[l~ =" a and A x proves the L-reflection rule for E-formulas. Interpreting the provable sentences of A x it makes no difference if we think that every unrestricted quantifier is restricted by L~. Since K P - c_ A x this has the same effect as if A x believes that a is admissible. Therefore we obtain as a corollary of Lemma 1.2.3 1.2.4. Corollary.

If K P w - + E,-FOUND C_ A x then [[Ax[Ig 1 = [[Ax[[n2.

Another observation is that adding true II~-sentences does not increase the E~: ordinal of an axiom system.

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Set Theory and Second Order Number Theory

1.2.5. T h e o r e m .

Let G be a true H~ sentence. Then IIAx + GII~ = I I A x I I ~

Proof. Let G - H L~ for a Hi-sentence H. Assume that Ax + G ~ F L~ for a E1 sentence F. Then A x e - ( H --+ F) L~ and H -+ F is El. For c~ " - I I A x [ ] ~ we thus have L~ ~ H --+ F. From ~ _ a, L~ ~ H and the downwards persistency of Hi-sentences we get L , ~ H which in turn entails L~ ~ F. Hence [lAx + GII~ ~ _ I[Axll~?. But the converse inequality holds trivially and we have [lAx + GII~ ~ =

ItAxII~r

D

We introduce the following notation.

1.2.6. Definition.

IIAxll~ = [IAxl[~

Because of Lemma 1.2.3 we get IIAxII~ - IIAx[Inu for theories Ax satisfying the hypotheses of the lemma. We call the computation of the ordinal [IAxll~ a a-ordinal analysis for Ax. It will turn out that IIAx[[~CK will be the most important ordinal. In Section 2.1.4 we will see that there is also something as an w-ordinal which gives a characterization of the Skolem functions of the provable H~'-sentences of an axiom system Ax in terms of a sub-recursive hierarchy. 1.3. C o n n e c t i o n s to s u b s y s t e m s of second o r d e r n u m b e r

theory

Let L~ be the language of Second Order Arithmetic. We assume that L~ contains a constant 0 for 0 and constants for all primitive recursive functions and predicates. We restrict the language to unary predicate variables and talk about set variables. This means no real restriction since we have a primitive recursive coding machinery. We use capital Latin letters as syntactical variables for sets and write t E X instead of X(t). We assume familiarity with the complexity classes in the arithmetical and analytical hierarchy. Since all primitive recursive functions and predicates have A0-definitions in K P w - + E1 - FOUND, we may regard L~ as a sublanguage of L(E) by restricting all first order quantifiers to w and replacing all second order quantifiers (VX) and (3X) by (VX C w) and (3X C_w), respectively. We may therefore transfer the notions of the arithmetical and analytical hierarchy to the language of/:(E). Whenever we talk of a H ~ E ~ 1-I~, . . . - sentences in the language of Set Theory without further comments we think of a translation of the corresponding s 2 One of the basic facts for the things to come is the w-Completeness Theorem for H~-sentences. We will use the w-Completeness Theorem to introduce the notion of truth complexity for II~-sentences. The value t N of a closed term t and the truth value of an atomic sentence in the standard structure N are primitive recur-sively computable. Since there are symbols for all primitive recursive functions and predicates we obtain the diagram of N D(N) "= {A I A is an atomic sentence and N ~ A }

W. Pohlers

220

as a recursive set. For arithmetical sentences which are not atomic the truth definition is given inductively by

N ~ A1 and N ~ A2=v N ~ A 1 A B1 N~A,.forsomeie{1,2} = ~ N ~ A 1 VA2 N ~ A(n) for all n e N =~ N ~ (Vx)A(x) N p A(~) yor ~om~ ~ e N ~ N p (3~)A(~). To extend this truth definition to Hi-sentences we introduce an infinitary calculus. For technical reasons we opt for a one sided sequent calculus g la Tait. First we fix the language of the Tait calculus. The non logical symbols for the Tait-language of s are:

9 The constant 0 as well as constants for all primitive recursive functions and relations. The logical symbols comprise: 9 Bounded number-variables, denoted by x, y, z, Xl,... and set variables, denoted by X , Y, Z, X1,... 9 The logical connectives A, V and the quantifiers V, 3. 9 The membership symbol E and its negation ~. Terms are built up from 0 and function symbols in the familiar way. We use S as a symbol for the successor function. Terms of the shape ( S ~ ~ 0 ) are numerals and n-times

will be denoted by n_. Atomic formulas are t E X, t ~ X and R ( t l , . . . , t ~ ) , where t, t l , . . . , t ~ are terms, X is a set variable and R is a symbol for an n-ary relation symbol. From the atomic formulas we obtain the formulas of s in the familiar way. Notice that we do not have free number variables in the language. The negation symbol is not a basic symbol of the Tait-language. We define the negation of a formula by de Morgan's laws.

~(t e x ) . - (t r x); ~(t r x ) . = (t e x) ~ ( R t l . . . t ~ ) "=_ ( R t l . . . t n ) where R is a symbol for the complement of R ~(A A B)"_= (-~A V -~S); -~(A V B ) : - ( - ~ A A -,S) -,(Vx)A(x) "=_ (3x)-~A(x); -,(3x)A(x) :- (Vx)-~A(x). It is obvious that we have

-,-,A=A.

(11)

The semantics for the Tait-language is straightforward. We easily check N ~ ( ~ A ) [ S l , . . . , Sn] iff N ~s A[S1,..., S~] for any assignment of sets S 1 , . . . , S~ to the set variables occurring in A. We use capital Greek letters A, F, A, A1, ... as syntactical variables for finite sets of/:~-formulas.

221

Set Theory and Second Order Number Theory

1.3.1. Definition.

We define ~ A inductively by the following clauses:

(AxM) I f A N D(N) ~ q} then ~ A for all ordinals (~. (AxL) If t N = s N then ~ A, s r X, t e X for all ordinals ~. (A)

If ~

A,A,

and~i < a f o r i =

1,2 then ~ A, A1 A A2 .

(v)

If ~ A , A , a~d ~ < ~ 1o~ ~om~ i e

(V)

If ~

(3)

y ~ A, A(i) a~d ~o < ~ 1o~ ~om~ i e N t h ~ ~ A,

{1,2} t ~

~ A,A~ V g~.

A, A(i) and ai < a for all i e N then ~ A, (Vx)A(x) .

(3z)A(~).

The relations ~ A is to be read that there is an infinite proof tree of V A whose depth is bounded by the ordinal a. It is obvious from the definition that we have A,

a _< Z,

A C_ F

=~

~ F

(12)

and a simple induction on a shows F1, . . . , Fn =~ N ~ (F1 V . . . V Fn)[S1, . . . , Sm]

for all assignments $ 1 , . . . , Sm to the set parameters occurring in (F1 V ... V Fn). We thus have F (9~) =~ IN ~ (V)~)F()~)

(13)

for all H~-sentences (V)~)F(s If F is a true arithmetical sentence, i.e., if F does not contain set parameters, we obtain by a simple induction on the complexity rk(F) of the formula F (which can be taken to be the number of logical symbols occurring in F) [rk(F)F.

(14)

This shows that ~ F is indeed an extension of the truth definition for arithmetical sentences. However, to prove that ~ F is also complete for II]-sentences, i.e., the opposite direction in (13), needs harder work. We follow Schiitte's proof via search trees. Let Seq be the set of (codes for) finite sequences of natural numbers. For s, t C Seq we denote by s C_ t that s is an initial segment of t. A tree is a set of finite number-sequences which is closed under initial segments. The elements of a tree are called nodes. A set of nodes in a tree is a thread, if it is linearly ordered by C_. A maximal thread in a tree is a path. A binary relation -~c_ w x w is well-founded iff there are no infinite -~-descending chains. For well-founded -~C w x w we define otyp.~ ( n ) : and

sup (otyp.~(m) + 11 m -~ n} wl

if n e field(-~) otherwise

W. Pohlers

222

otyp(- ) :=

I

e field(-<)}.

For a tree T we define the tree-relation -
s-'
otyp(T) -- otypT(()). Recall that the first non recursive ordinal is defined by w~K "= sup {otyp(-<)l

-<

is a recursive well-ordering}.

We are going to define search trees for finite sequences of formulas. Such a sequence is called reducible if it contains at least one non atomic formula. The left most non atomic formula in a reducible sequence is called distinguished. The reduced sequence A r of a reducible sequence A is obtained by removing the distinguished formula from the sequence. The search tree for a finite sequence A of s is a tree Sz~ together with a label function which assigns a finite sequence 6(s) of s to each node s 6 Sz~. It is inductively defined by the following clauses:

(S()) <>6S~ and6(<>)=A. (S~.) If s 9 Sa and 6(s) is an axiom according to (AxM) or (AxL), then s-
If 5(s) is not reducible then s~(O) 6 S~ and 6(s~(O)) = 6(s).

(S^)

If Fo A F~ is the distinguished formula in 6(s) then s~(i) 6 S~ for i = O, 1 and 6(s~ (i)) := 6(s) r, F~.

(Sv)

Let Fo V F~ be the distinguished formula in 6(s). 6(s~(io)) := 6(s) ~, F0, F~.

(Sv)

/f the distinguished formula in 6(s) is (Vx)F(x), then s~ (i) 6 S~ for all i 6 N

Then s~
and 6(s~(i)) = 6(s)~,F(i). ($3)

If the distinguished formula in 6(s) is (3x)F(x), then s~(O) 6 Sz~ and at rat umb r that n # t N for all formulas F(t) 6 Usocs 6(s0). =

Observe that we introduced clause (SAx) only for better readability. It follows from the other clauses and the fact that Sz~ is inductively defined. There are two main lemmas.

Set Theory and Second Order Number Theory

1.3.2. Syntactical Main Lemma. and jowp(S~) A .

223

If Sa is well-founded then otyp(Szx) < w~K

P r o o f . Let SLX be well-founded. The tree relation "
(1) (2)

If A is an atomic formula occurring in s E SLX then A occurs in all t such that sC_teSzx. If a non atomic formula F occurs in some f[n] then there is an m > n such that F is distinguished in f[m].

The proof of (2) is an easy induction on the number of non atomic formulas occurring left of F in 6(f[n]). Using (2) the proofs of the following observations are almost immediate from the definition of SLX.

(3)

If a formula A A B occurs in f[n] then there is an m such that either A or B occurs in f[m].

(4)

If a formula A V B occurs in f[n] then there are mA and m s such that A occurs in f[mA] and B occurs in f[ms].

(5)

If a formula (Vx)F(x) occurs in f[n] then there are numbers i and m such that

F(~) o ~ r ~ ~ f[m]. (6)

If a formula (3x)F(x) occurs in f[n] then for every number i there is a term t and an mi such that t TM = i and F(t) occurs in f[mi].

To prove fact (6) we assume that (3x)F(x) is distinguished in 6(f[n]). By (2) this means no loss of generality. Then 6(f[n + 1]) = 6(f[n]) r, F(j), (3x)F(x) and, if i < j we have F(t) in f[m] for some t and m _ n such that t TM - i. If j < i then F(t) will occur in f[m + 1] for some m _> n and t with t N = i as soon as ( 3 x ) f ( x ) becomes distinguished in 6(f[m]) and F(t) has occurred for all t N < i. We define an assignment r

: - {t ~ I (t r X ) occurs in f }.

Here F occurs in f means that F occurs in f[n] for some n. An easy induction on the length of a formula G, using observations (3) - (6) and the fact that f must not contain an axiom, shows iN ~= G[(I)] for all formulas G occurring in f . Since all formulas of A occur in f[0] this yields iN ~ V { F I r e A } [(I)]. D

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W. Pohlers

The Syntactical Main Lemma together with the Semantical Main Lemma prove the following theorem. 1.3.4. w-Completeness Theorem. Let (V)~)F(X) be a H~-sentence. Then we have N ~ (V)~)F()() iff there is an ~ < w~K such that ~ F (X). Proof. The soundness is already stated in (13). For the completeness direction we assume ~ F()~) for all ordinals ~ < w~K. Then, by the Syntactical MainLemma, the search tree for F()~) cannot be well-founded. Applying the Semantical Main-Lemma we obtain an assignment (I) over N such that N ~ f(s Hence N ~= (V)~)F()~). D In view of Theorem 1.3.4 we call formulas which contain at most free set variables sloppily II~-sentences. Semantically we treat these pseudo II~-sentences as their universal closure, i.e., N ~ F()~)

:r

N ~ (V)~)F()~).

We use Theorem 1.3.4 to define the truth complexity of II~-sentences. 1.3.5. Definition. (Truth Complexity) For a II~-sentence G := (V)~)F()~) or G : - F()~) we define tc(G) "= { Wl min (c~ I ~ f ()~)}

if N ~ G otherwise.

Using truth complexities we may restate Theorem 1.3.4 for H~-sentences F as N ~ f

iff t c ( f ) < w~K.

(15)

As we have seen in (14) we have tc(F) < w

(16)

for all true arithmetical sentences F. But in contrast to that we have by Corollary 1.3.9 and Theorem 1.3.10 below sup {tc(F) ] (V.7)F(,7) is a H~-sentenee and r~ ~ F(X)} = w~~ which shows that for "real" II~-sentences truth complexity is a non-trivial notion. We call a binary relation -< arithmetical if there is an arithmetical formula F(x, y) such that m -< n iff N ~ F(m, n_).

The following Boundedness Theorem is one of the most important theorems for this contribution and will return in different variations.

1.3.6. Boundedness Theorem. T! (-<, X) be the formula

Let ~ be an arithmetical definable relation and

Set Theory and Second Order Number Theory

225

(Vx E field(-~))[(Vy)(y -~ x --+ y e X) --+ x e X] --+ (Vx e field(-~))[x e X]

~=p~,,i.g i.du~tio. .lo.g -<. Th~.

otyp(-<) < tc((VX)TI(-<.X)).

To obtain the Boundedness Theorem we prove the more general Boundedness Lemma. We prepare the Boundedness Lemma by a few notions and observations. An /:~-formula is X-positive if it does not contain occurrences of t ~ X. An obvious property of X-positive formulas is stated in the next lemma. 1.3.7. M o n o t o n i c i t y L e m m a . Let F be an X-positive formula not containing further set variables and M C g C_ N. Then N ~ F[M] entails N ~ F[g]. The proof is straightforward by induction on the length of the sentence F.

[3

By induction on a we obtain the following inversion properties:

A, A1 V A2 =~ ~ A, A1, A2 ,

(17)

A, A1 A A2 =~ ~ A,A, for i = 1,2

(18)

A , (Vx)F(x) =~ ~ F (/) for all numbers i.

(19)

and

Let Prog(-.<,X) denote the premise (Vx e field(-~))[(Vy)(y -< x --+ y e X) --+ x e X] of the II~-sentence TI(-<,X). For a fixed well-founded binary relation -< let e--ff~z1.....z.} denote the enumerating function of the complement of the set {otyp.<(zl),..., otyp~(zn)}, i.e. of On \ {otyp.<(zl),..., otyp.<(zn)}. Define

M ~ , .....z.}(a) := { m l otyp.~(m) _< e-ff~z,.....z,,}(o~)} u { Z l , . . . , Z n } . 1.3.8. B o u n d e d n e s s L e m m a . and assume that

Let -< be a well-founded transitive binary relation

-~Prog(-~,X),z 1 r X , . . . , z , r X , A for a set of X-positive II~-sentences A = {F1,...,Fn} not containing further setvariables. Then .-9

[M{,, .....,,,} (a)].

The Boundedness Theorem follows from the Boundedness Lemma. tr

_ ~.

-~Prog(~,X),n q~field(-<),n E X . Because of we obtain by the Boundedness Lemma (Vn e field(-~))(3a0 < a)[otyp(n) _ a0], hence otyp(-~) _ c~.

Assume

Th~n the~e is fo~ every ~ e field(-<) an '~o < ~ such that

Mo(ao) =

{m] otyp(m) _< a0}

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W. Pohlers

We prove the Boundedness Lemma by induction on a. Assume -~Prog(-<, X), z, r X , . . . , zn r X, A.

(i)

The claim is trivial if D(N) n A r 0. If (i) holds by (AxL) then there is a formula tEXinAsuchthatt N = z i f o r s o m e i E { 1 , .. .,n}. SinceziE M{~, "~ .....z.}(a) this yields the claim. If (i) is obtained by the premises -,Prog(-<, X ) , z__1 r X , . . . , z_, r X , Ai

then we obtain the claim by the induction hypothesis together with the Monotonicity Lemma and the soundness of ~ . The really interesting case is that (i) follows from the premise -~ P ro g ( - < ,X ), z E field(-<) A (Vy)[-~y -< z V y E X] A z r X,

z~ r x,...,z, r x, zx

(ii)

By inversion we get from (ii) ~ P r o g ( - < , X ) , z E field(-<) A ( V y ) [ W -< z V y E X],z 1 r X , . . . ,z n r X, A (iii)

and

--,Prog(-~,X),z r X,z~ r X,...,z__, r X,/',.

(iv)

If

N ~ V ~[M~, .....~.~(~o)]

(v)

N ~ V ~X[M~..... ~.~(~)]

(vi)

then

by Monotonicity. If N ~= V AIMS1 .....z.} (co)]

(vii)

then the induction hypothesis applied to (iii) gives z E field(-<) A (Vy-< z)[y E M ~ , .....z.}(ao)].

(viii)

From (iv) we get by the induction hypothesis N ~ V AIMS, .....~.,~}(ao)].

(ix)

We claim otyp.~(z) < ~-~, .....~.}(ao + 1) < e-rib, .....z.}(a).

(x)

To prove (x) recall that otyp.~(z) := sup {otyp~(y) + 1[ y -< z} and observe that for y -< z we obtain otyp.~(y) _ e-n~l .....~.}(ao) or y E {Zl,... ,zn} by (viii). Hence

:= sup {otyp~(y) + II y -~ z ^ y r {z~,... ,z~}} < ~ ,

.....~.}(ao + 1)

Set Theory and Second Order Number Theory

227

and for y 9 z such that y E { z l , . . . ,z,} we get otyp.~(y) < e--ff~z,..... z.}(ce0 + 1) since otyp.~(y) is omitted in the enumeration. Because of

~-ff'~.,.....z.,.)(ao) _< ~-fi~,~,.....,,.}(ao + 1) _< e-ffS, .....z.}(a)

(xi)

we get by (x) and (xi) M(z"~1.....~.,~}(ao) C_ M {~1 "~ .....~.}(a) . This fnally yields N V A [ M ~ .....z.} (a)] by (ix) and Monotonicity. [3 As a consequence of the Boundedness and the w-Completeness Theorem we obtain 1.3.9. Corollary.

If 9 is an arithmetical definable well-ordering then otyp( 9 <

0.) ~ K .

It should be mentioned that Corollary 1.3.9 can - with just a little effort - be extended to ~-definable well-orderings 9 and thus comprises the well-known Boundedness Theorem of recursion theory without referring to the Analytical Hierarchy Theorem (cf. Beckmann and Pohlers [1997]). Conversely to the Boundedness Theorem we obtain otyp( 9 also as an upper bound for tc( T l ( 9 X ) ) . We show I~(~

~Prog( 9 X ) , n_ E X

(20)

by induction on otyp.~(n) where - for simplicity- we assume that 9 is a primitive recursively definable relation whose field is all of N. We have 15"(~

-~Prog( 9

9 n_,m E X

(i)

either as an instance of (AxM) or by induction hypothesis. Hence 15"~

-~Prog( 9

(Vy)[y 9 n_ -+ y E X]

(ii)

by two (V) and one (V) "inference". By (AxL) we have -~Prog( 9

n_ q~ X, n_ E X .

(iii)

By (ii) and (iii) we obtain 15Otyp'~(n)+4 ~ P r o g ( 9

(Vy)[y 9 n_ -+ y E X] A n_ ~ X,_n E X .

(iv)

One additional "inference" (3) leads to [5.(otyp~(n)+l) ~Prog( 9 X ) , n E X .

[3

From (20) we obtain by a clause (V) and two clauses (V) the following theorem. 1.3.10. T h e o r e m . If 9 is a primitive recursive well-founded relation whose order type is a limit ordinal then otyp( 9 <_ t c ( T l ( 9 <_ otyp( 9 + 2.

W. Pohlers

228

Of course we should read Theorem 1.3.10 as otyp(-<) = tc(Tl(-<,X)), since the "§ is just due to the syntactical peculiarities in the definition of ~ A . But, since all important ordinals will be limits, this is of no importance. We define specni(N) := {to(F) [ F is a II~-sentence and N ~ F} and call specs] (N) the II~-spectrum of N. Due to Theorems 1.3.4 and 1.3.10 we then have specrq (N) = w?K.

(21)

More generally we define the II~-spectrum of a theory Ax in the language of second order arithmetic by specni(Ax) : - (tc((V)~)F()()) [ (V)~)F()~) is a II~-sentence and A x ~ F()~)}.

(22)

IrAxllni : - sup(specnl (Ax)).

For a recursively enumerable theory Ax the set {F[ Ax ~ F} is recursively enumerable, too. Since w~K is recursively regular we get by Theorem 1.3.4 and the fact that for true IIl-sentences F an upper bound for tc(F) can be effectively computed from F via the depth of its search tree

IIAxll,: < w~ ~ for all recursively enumerable theories Ax. The Ax is commonly defined as

proof theoretic ordinal of a

][Ax[] : - sup {otyp(-<)] -< is primitive recursively definable and A x ~

theory

TI(-<,X)}.

From the Boundedness Theorem we get immediately

IIAxII _< IIAxII.:
(23)

for all recursively enumerable theories Ax. By showing that for every ordinal a < []Ax[]n~ there is a primitive recursive order relation -< such that a _< otyp(-<) and A x e - TI(-~,X) we get from Theorem 1.3.10 together with (23)

IIAxII = liAxJlnl

(24)

for all theories which will be analyzed. There is, however, also a more general argument for (24) which we are going to sketch roughly. Assume that Ax is a theory comprising PA and let (VI~)F(]~) be a II~-sentence. Denote by -
229

Set Theory and Second Order Number Theory

formulas with parameter T. Since we have induction in ffJ~ for first order formulas we obtain ~ ~ F(]~) [(I)] as in the proof of the Semantical Main Lemma using a local truth predicate. Hence Ax ~ F(] ~) and we have shown

hx ~ F(?) ~ Ax ~- TI(-~(~>, X). Since -~sv(~) is primitive recursively definable and we have tc(F(]?))

___

otyp(-~sF(~)) _< [[Ax[[ if Ax ~ - F ( ] ?) this implies [lAX[[HI ___ [tAxi]. Summarizing we get 1.3.11. T h e o r e m .

Let P A C_ Ax then

IIAxll- IIAxll,].

The computation of the ordinal IIAxil is commonly called the ordinal analysis of Ax. In view of Theorem 1.3.11 we also talk about a II~-analysis of Ax. To explain the connection between liAxi[n] and ]lAx[] ~cK we use again Theorem 1.3.4 which 1Ell

says

N ~ (V)~)F()()

r

(3a < w~K)[B

F(2)].

(25)

Assume that we have coded the language/::~ within Set Theory (c.f. Barwise [1975]). The fact that "z is an infinitary proof tree for F()~) of length _< a" can be expressed by a A0 formula, say G(a, z, rF(X)7). It is easy to check that for a e On and z e L such that L ~ G(a,z, rF()~)7) we have z e La+n for some n < w. If we take into account that F()~) may contain additional number parameters, say g, (25) turns into

[N k

**

<

z,

This is the well known Hyperarithmetical Quantifier Theorem telling that every II]-formula is equivalent to a El-formula over LucK. If we put [His1 := min {c~ I L~ ~ H}

(26)

for El-sentences H we get

tc:((V)()F()~)) < I(:t[)(:tz)G([, z, rF()~;)[:s I _ tc((V)~)F()~))+ n for some n < w. Defining

Ilnxll

~o~ : =

sup {l(3~)(3z)G(~,z, rF(X)')l=, I nxl- F(X)}

for a / ~ - t h e o r y Ax we get IIAxli ,c~ = IIAxllHi = IIAxil . E1

(27)

It is evident that the ordinal ilAx][ ~cK for L~-theories is the exact counterpart of ]Ell the ordinal ]]Axl[~cK which we defined in the previous section for theories in the language/::(E) of Set Theory.

230

W. Pohlers

Analogously to Theorem 1.2.5 we get 2 IIAx + Fllnl = IIAxII,I

(28)

for all true ~-sentences F with the same proof. We mentioned already in the beginning of the section that s can be regarded as a sublanguage of s Gall F a H~-sentence of s if it is obtained as a translation of a II~-sentence of/2~, i.e., if it has the form (Vx)[x C_ w --+ G(x)] where G ( x ) is a A0-formula whose quantifiers are all restricted to w or to natural numbers. If A x is a theory extending K P then we can use the familiar unsecured sequences argument to show that A x proves that for every H~-formula F(Z) there is a A0-definable order relation -~f(~) such that Axe-(VZ)[F(Z) ++ Wf(-4F(~))]. We will see later (cf. Section 3.3.3) that for a theory A x which proves Axiom /3 - which says that every well-ordering can be order isomorphically mapped onto an ordinal- we have directly A x e - (VZ)[Wf(-4F(~)) ++ (3~ < L~cK)IF(~, Z)] for a ~l-formula IF(~,~,). It is easy to check that otyp(-4F(~)) ~ [(3~)IF(~,g)iSl. Thus we have Ilhxll,l = IIAxll = IIAxll~Tx~ for sufficiently strong theories in the language of Set Theory. This rough sketch should suffice to explain that the ordinals I[Axll and

(29)

IIAxl[

~o~

carry the same information. So 1-I~-analysis and v~X-ordinal analysis are the same things. We will see that indeed a good deal of information is contained in IIAx[I. 1.4. M e t h o d s Before we come to examples of analyzed theories we want to outline the methods used in ordinal analyses. There are two main steps. The first is to compute upper bounds for the ordinals in specnl (Ax), the second to show that these bounds are the best possible ones. We explain the general pattern on the example of an s Ax. If A x ~- F then there are formulas A 1 , . . . , An E A x such that A1,...,An~F

(i)

in pure predicate logic. By Gentzen's Hauptsatz we may assume that this derivation is cut-free. It will be quite easy to transform (i) into a truth definition ~A1, . . . , - , A n , F

2Defining Ax + (3X)H(X) ~- G :r notion of proof for Second Order Logic.

(ii) Ax ~ -~H(X) V G this makes sense even without a

Set Theory and Second Order Number Theory

231

where a will depend mainly on the complexity of the formulas A1,..., AN and F. Then we have to compute upper bounds for tc(A) for all formulas in Ax. This gives Ai

(iii)

if tc(Ai) _ a, for i e {1,..., n}. The problem is now to link (iii) and (ii). This will be achieved by extending the truth definition ~ A into a semi-formal calculus ~ A by adding the cut rule (cut)

~A,F;

~A,--F;

/31,/32
=:~ ~ A .

This will of course destroy its meaning as a truth definition. But we will still have

F

F

(iv)

By (ii) and (iii) we obtain

Y

(v)

where/3 and p are computable from a l , . . . , an and a and the problem reduces to the elimination of cut in the semi-formal system. Since the semi-formal system is obviously sound we get F

=~ N ~ F

=~ (35
=V (35
showing that the Cut-Elimination Theorem holds for the semi-formal calculus. But this is of little help since we do not know how to compute 5 from /3 and p. In a moment, however, we will see that for predicative theories we can sharpen CutElimination to

A.,o

(vi)

By (v), (vi) and (iv) we get tc(F) < ~p/3

(vii)

for Ax ~ F. Since/3 and p only depend on the formulas in Ax this will give an upper bound, say r for IIAxlln~. To show that r is the best possible bound it suffices to prove that for every c~ < r there is a primitive recursive well-ordering -< such that c~ < otyp(-<) and A x ~ T I ( ~ , X ) . It will become clear from the following text how this concept has to be modified as to serve also for theories in the language L:(~) of Set Theory and for impredicative theories. But before that we demonstrate some details on the example on Gentzen's result. 2. F i r s t o r d e r n u m b e r

theory

2.1. P e a n o a r i t h m e t i c The paradigm for ordinal analysis is still Gentzen's result on Peano Arithmetic. So we opt for it as our first and simplest example. However, rather than to analyze

232

W. Pohlers

PA itself we will analyze a conservative extension N T of PA which allows constants for all primitive recursive functions. We start with an introduction of the theory NT. 2.1.1. T h e axiom s y s t e m N T The language s is a first order language which contains set parameters denoted by capital Latin letters X, Y, Z, X1, ... and constants for 0 and all primitive recursive functions and relations. We assume that the symbols for primitive recursive functions are built up from the symbols C~ for the constant function, P~ for the projection on the n-th component, S for the successor function by a substitution operator Sub and the recursion operator Rec. The theory N T comprises the following sentences: The successor axioms = = s(y)

9 = y]

The defining axioms for function and relation symbols which are the universal closures of the following formulas (z,,

. . . , z,)

=

k_

P~ (Xl,..., x.) = x~ Sub(g, hl, . . . , h m ) ( X l , . . . , x , )

= g ( h l (Xl, . . . , x , ) ) " " ( h m ( X l , . . . , x , ) )

R e c ( g , h)(O, x l , . . . , x , ) = g ( x l , . . . , x , ) R e c ( g , h ) ( S y , x l , . . . , x , ) = h ( y , Rec(g, h ) ( y , Xl, . . . , x , ) , x~, . . . , x , )

(Xl,...,Xn)

e R ~ )(.R(Xl,...,Xn) = 0

The scheme of Mathematical Induction F(O) A ( V x ) [ F ( x ) --+ F ( S ( x ) ) ] - +

for all s

(Vx)F(x)

F(u).

2.1.2. A n u p p e r b o u n d for specrq (NT) Following the general pattern as sketched in the previous section we have first to deal with the truth complexity of logically valid sentences. Therefore we have to fix a calculus for pure predicate logic and we opt for a cut free Tait calculus, i.e., one sided sequent calculus, which is given by the following rules: 2.1.2.1. Definition.

(AxL) ~ - A , A , - ~ A f o r a n y m , if A is an a t o m i c f o r m u l a (V)

I f ~2_ A , A i f o r s o m e i E {1, 2}, t h e n ~-- A , A1 V A2 f o r all m > mo

(A)

I f ~-- A , A i a n d m i < m f o r all i E (1,2}, t h e n ~ - A , A1 A A2

Set Theory and Second Order Number Theory

233

(B)

If ~

A, A(t), then ~- A, (Bx)A(x) for all m > mo

(V)

If ~2_ A, A(u) and u not free in A, (Vx)A(x), then ~- A, (Vx)A(x) for all m>mo.

The identity axioms are the following formulas ( w : ) [ x = ~]

( w ) ( v y ) [ ~ = y -+ y = ~] ( w ) ( v y ) ( V z ) [ ~ = y ^ y = z -+ 9 = z]

(V..~)(Vy~[xl = (v~)(v~[x,

y, A

= y, ^

(W:)(Vy)[~ = y

-+

...

A x n - - Yn

. . . ^ x,, = y,,

,Yn)]

"-'+

f(xt,...,xn)

-- f(y,,...

-+

(R(~,,...

,~,,) ~ R ( y , , . . .

,y,,))]

(x e X --+ y e X ) ]

Due to Gentzen's Hauptsatz we have the following theorem:

Let A be a finite set of formulas such that V A is valid in the sense of first order predicate logic. Then there are finitely many identity axioms I1,..., In and an m < w such that ~-- -~I1,.. .,-,In, A.

2.1.2.2. T h e o r e m .

Let t7 be a list containing all number variables which occur free in A. induction on m using the fact

A (~) ~,~d ~ = t ~ =. ~ A (t)

An easy

(30)

shows

r- A(~) ~

~ A (~)

(31)

for every tuple g of numerals. We have (w)(vy)[~ = y

-~

(~ e x -+ y e x ) ]

and all the other identity and defining axioms for primitive recursive functions and relations are true arithmetical sentences. Thus, using also (16), we have tc(F) < w

(32)

for all mathematical and identity axioms except induction. What really needs checking is the truth complexity of the scheme of Mathematical Induction. Here we need the following lemma. 2.1.2.3. T a u t o l o g y L e m m a .

For every s

we have [2.rk(F) A, ~F, F .

The proof is immediate by induction on rk(F). The truth complexity for all instances of Mathematical Induction follows from the Induction Lemma.

234

W. Pohlers

2.1.2.4. Induction Lemma.

For any natural number n and any s

F ( n ) we have

I

~F(0),-~(Vx)[F(x) -+ F(S(x))], F(n_.).

(33)

The proof by induction on n is very similar to that of (20). For n - 0 we get (33) as an instance of the Tautology Lemma. For the induction step we have 12"['k(F(~-))+"l ~F(0), ~ ( V x ) [ F ( x ) -+ F(S(x))], F ( n )

(i)

by the induction hypothesis and obtain j2.~k(F(~_)) ~F(0), ~(Vx)[F(x)-+ F ( S ( x ) ) ] , ~ F ( S ( n ) ) , F ( S ( n ) )

(ii)

by the Tautology Lemma. From (i) and (ii) we get 12"[~k(F(~-))+"]+l ~F(0),--,(Vx)[F(x) ~ F(S(x))], F ( n ) A ~ F ( S ( n ) ) , F ( S ( n ) ) . (iii) By a clause (S) we finally obtain

]2.[,k(F(n_))+n]+2~F(O), -~(Vx)[F(x) -+ F ( S ( x ) ) ] , F ( S ( n ) )

.

[3

By Lemma 2.1.2.4 we have tc(G) _~ w + 4 for all instances G of the Mathematical Induction Scheme. Together with (32) we get tc(F) g w + 4

(34)

for all identity and non-logical axioms of NT. If N T ~- F then there are s {F1,..., Fn} and a natural number m such that ~- -,F~, . . . , ~Fn, F

(35)

and for all i E {1,..., n} the formula Fi is either an axiom in N T or an identity axiom. For every s F such that N T ~ F we thus get by (35) and (31) LN-sentences F1,..., F~ such that -~F1, . . . , -~Fn, F

(36)

and by (34) I~+' F/

(37)

for all i E {1,..., n}. As sketched in Section 1.4 the problem of linking (37) and (36) will be solved by introducing a semi-formal calculus. 2.1.2.5. Definition. We define ~ A for a finite set A of s contain at most free set-variables inductively by the following clauses: (AxM) I f A N D(N) r q} then ~ A for all ordinals c~ and p.

which

Set Theory and Second Order Number Theory

235

(AxL) If t N = s N then ~ A, s r X, t E X for all ordinals a and p.

(A)

Is $ A, A, a . ~ . , < ~ So~ i = 1, 2 the. ~ A, A, ^ A~

(v)

If ~ t,,A, a , ~ . o < - io~ ~om~ i e {1,2} th~. ~ A,A~ V A~.

(V)

If ~ A, X(i) a,~ ~, < . fo~ aUi e N t ~ . ~ A, (W)A(~).

(3)

If ~- A, A(~) a . ~ . o < - fo~ ~om~ ~ z N the. ~ A, (3~)A(~).

(cut)

If ~ A , F , ~ A , ~ F , ai < a ]ori E {1,2} andrk(F) < p then ~ A.

We have to read ~ A as" "There is an infinitary derivation of A of height < a whose cut formulas are all of ranks < p." We call F the main formula of a clause in the definition of ~ A, F if F is responsible for A, F being an axiom in (AxM) and (AxL) or if the logical symbol introduced in the clause belongs to F. Thus (cut) possesses no main formula. From the definition we get immediately

~

~.

~ ~

(3s)

From (31) and (38) we have

for every tuple z7of numbers. Another immediate property is

~A,

o~<_~,

p<_a

and

ACF

=~

~F.

(40)

The calculus ~ is obviously sound. By induction on a one proves easily F I , . . . , Fn =~ N ~ (F1 V ... Y Fn)[(I)] for every assignment (I):set variables ness of ~ together with (38) show 2.1.2.6. Cut E l i m i n a t i o n T h e o r e m .

(41)

~ Pow(N). Soundness of ~ and Complete-

If ~ A then there is a 9/< w~x such that

~zx. But, as mentioned before, Theorem 2.1.2.6 does not help us in ordinal analysis. The bound for 7 is much too large. What we are looking for is a function which computes a value for 7 from the data a and p. The key here is the Reduction Lemma which tells us how to avoid a cut of rank p for the costs of an increasing length of the derivation. 2.1.2.7. R e d u c t i o n L e m m a .

Then ]~ p t~ A,F.

Assume ~ A, F and ~ F,-~F as well as r k ( F ) = p.

W. Pohlers

236

The proof is by induction on a ~ ft. Assume first that F is not the main formula of the last clause in the definition of ~ A, F. Then either ~ A, and hence also O~

[p ~ ~ A, F, holds by (AxM) or (AxL) or we have ~ Ai, F for ai < a. But then we get by induction hypothesis [p~ A /

, F , and obtain [p ~ ~ A , F by the same

clause.The case that -~F is not the main formula in ~ F, -~F is symmetrical. We may therefore assume that both, F and -~F, are main formulas. Let us first assume that p = 0, i.e., that F is atomic. Then both ~ A, F and ~ F,-~F are axioms whose main formulas are F and -~F, respectively. This excludes axioms according to (AxM) because we cannot have {F,-~F} C_ D(N). But then we have axioms according to (AxL) and we may assume that F is a formula t E X. But then -~F - t ~ X and we have a formula Sl ~ X in A and a formula 82 E X in F such that SlN - t N = s~. But then [p ~ ~ A, F holds by (AxL) Now assume 0 < p. We will only treat the more complicated case that F is a formula (Vx)A(x). The remaining cases are either simpler or symmetrical. Then -~F is the formula (3x)--,A(x) and we have the premises A, F, A(i)

(i)

for all i E N and F,-~F,-~A(/_o).

(ii)

From (i) and ~ F, F as well as from (ii) and ~ A, F we obtain by the induction hypothesis

[p~,o # ~ A, F, A(/o)

(iii)

and

~ 8o A,F -~A(/_o)

(iv)

Since rk(A(/_o)) < rk(F) = p and aio ~ fl < a ~/3 as well as a ~/30 < a # fl, we obtain [p ~ ~ A, F from (iii) and (iv) by cut.

[3

As a first consequence of the Reduction Lemma we obtain the Elimination Lemma. 2.1.2.8. E l i m i n a t i o n L e m m a .

If ~

A then ~-~ A.

The proof is by induction on a. If the last inference is not a cut of complexity p we obtain the claim immediately from the induction hypothesis and the fact that A~. 2~ is order preserving. The critical case is a cut ~+~ A , F ; r.-~ A , ~ F =~ ~.--~2-7"~ A with rk(F) - p. By the induction hypothesis and the Reduction Lemma we obtain [::! 12~1 ~ 2~2 A and we have 2~1 ~ 2~2 < 2max{~l,~2} 92 < 2~ The Elimination Lemma provides the first step in the proof of the following more general lemma.

Set Theory and Second Order Number Theory 2.1.2.9. P r e d i c a t i v e E l i m i n a t i o n L e m m a .

If

237

18+~. A then '8 ~

A.

The proof is by induction on p with side induction on a. For p = 0 we obtain ""2~-A by the first Elimination Lemma which, since 2a < w" = ~0a, entails the claim. Now assume p > 0. If the last clause was not a cut of rank >/3 we obtain the claim from the induction hypotheses and the fact that the functions qOp are order preserving. Therefore assume that the last inference is ~ A,F 18+wp

I8+wp ~: ~X,-~F

~

I8+wP ~ A

such that ~ _ rk(F) < ~+wP. But then there is an ordinal r such that rk(F) = ~ + r which, writing r in Cantor normal form, means rk(F) = ~ + w ~1 + . . . + w ~" < ~+wP. Hence al < p and, putting a := al, we get rk(F) < ~ + w~. (n + 1). By the side induction hypothesis we have ~ A F and ~ A, ~ F . By a cut it follows '8

'

'8

Ivp"i+~pa2 A Ifwe define ~ ) c ~ .__ c~ and ~^(n+l) c~ . = ~a(~(n)c~) then we obtain from 8_i_wO..(n+l)

9

a < p by n + 1-fold application of the main induction hypothesis 18

(~Opal+~opa2) A .

Finally we show ~(~)(~pal + ~pa2) < ~pa by induction on n. For n = 0 we have ~(a0) (qOpO/1 + (/9pO~2) = ~pO/1 -{- ~pO/2 < qOpO/ since ai < a and ~pO/ E C r ( 0 ) . n(n+l)

For the

induction step we have ~ (~pal + ~pC~) = ~ ( ~ ( n ) ( ~ p a l + ~pa2)) < ~pa since < p and ~(~)(~pal + ~pa~) < ~pa by the induction hypothesis. Hence ~ El ~8 A By the Predicative Elimination Lemma we obtain the function which computes an upper bound for the height of the cut free derivation. 2.1.2.10. E l i m i n a t i o n T h e o r e m .

Then

i i0

Let ~p A such that p ---NF 0)Pl -I-"'" "~-Wpr~ 9

A

Back to the axiom system N T . If N T ~- F then we obtain by (36), (37), and (38) I~+4+n F for m := max{rk(F1), rk(Fn)} + 1 < w By the Elimination Theorem "

9

9

(or even m-fold application of the Elimination Lemma) we obtain I~p~(~''~ + 4 + n) F i 0

9

Hence tc(F) _ expm(w, co + 4 + n) < e0 and we have 2.1.2.11. T h e o r e m .

specH~(NT) C_ e0.

2.1.3. Lower b o u n d s for specH~ (NT) We want to show that the bound given in Theorem 2.1.2.11 is the best possible one. By Theorem 1.3.10 it suffices to have Theorem 2.1.3.1 below. 2.1.3.1. T h e o r e m . For every ordinal ~ < Co there is a primitive recursive well-order -.( on the natural numbers of order type ~ such that N T ~ 71(-K, X ) .

W. Pohlers

238

The first step in proving Theorem 2.1.3.1 is to represent ordinals below 60 by primitive recursive well-orders. This is done by an arithmetization. We simultaneously define a set On c_ 1N and a relation a -< b for a, b E On together with an evaluation map 1" I" On ~ O n such that On and -~ become primitive recursive and a -< b r la] < ]b]. We put 9 0 E O n and]O]=O

9 If zl,...,zn C_ On and zl ~_'" ~" zn then (zl,...,zn) E On and ](zt,...,zn)] = wlzal + ... + wlz-I

and

9 a-
V (3i) < min{lh(a), lh(b)})(Vj) < i)((a)j = (b)j A (a)i ~ (b)i)] Observe that On and -< are defined by simultaneous course of values recursion and thence are primitive recursive. It is also easy to check that a -< b r ]a I < ]bI. The order (On,-<) is a well-order of order type r We may therefore represent every ordinal a < r by an initial segment -
9

.~

(v~)[~<~-~ex]

Prog(X) :r

(Vt~)[oLc_ X -+ oL E X]

7-t(~,x)

prog(X) ~ ~ c x

:~

Our aim is to show TI(a,X) for all a < Co. Since TI(O,X) holds trivially and eo = sup { ezp"@, 0)] n e w}, we are done as soon as we succeed in proving N T ~ - Tl(a,X) =~ N T ~ - TI(w",X)

(42)

because N T ~- TI(a,X) and fl < a obviously entails N T }- TI(fl, X). The first observation is N T ~- F(X) =~ N T ~- F({x I G(x) })

(43)

for all /:N-formulas G. The formula F ( { x l G(x)}) is obtained from F(X) by replacing all occurrences of t E X by G(t) and those of t r X by -,G(t). To prove (43) assume N T }- F(X) and let ~ be an arbitrary /:N-structure and (I):set-variables signment such that |

~ NT[(~].

(i) pow ( ~ ) an as-

(ii)

Set Theory and Second Order Number Theory

239

We have to show

r I= F({xl a(=)})[,I,].

(iii)

Define a new assignment

~ ( z ) :=

{@(Y) {n e ~1 ~ I= a(=)[n, ~]}

if Y =~X otherwise.

Then | ~ F(X)[@] iff | ~

F({zl a(z)})[,I,].

(iv)

We claim |

~ NT[~].

(v)

Then (v) together with (i) and (iv) prove (iii). To check (v) we have only to take care of formulas in N T which contain the set variable X. This can only happen in instances of the scheme of Mathematical Induction or in identity axioms. Let

I(X)

:r

H(X,O) A (Vx)[H(X,x) -+ H(X,S(x))]--+ (Vx)H(X,x)

be an instance of Mathematical Induction. We have

~/(x)[~]

~ff ~ I= 1({~1 C(x)))[~].

(vi)

The right formula in (vi), however, holds by (ii) since H({x] G(x)},x) is also a formula in NT. Instances of identity axioms are treated analogously. O The above proof shows the importance of formulating Mathematical Induction as a scheme. Let

s(x) := {o~1 (v,~)[,~ c x ~ ,~ + ~o c x ] } denote the jump of X. Then, if we assume N I l - - Prog(X) -+ Prog(J(X)),

(i)

we obtain N T ~ - Tl(a, J ( X ) ) --+ Tl(w",Z).

(ii)

To prove (ii) assume (working informally in NT) Tl(a, J ( X ) ) , i.e.

Prog(J(X)) -+ a C_ J ( X )

(iii)

which entails

Prog(J(X)) --+ a e J ( X ) .

(iv)

Choosing ~ - 0 in the definition of the jump turns (iv) into Prog(J ( X ) ) -+ w '~ C_ X,

which, together with (i), gives

(v)

W. Pohlers

240

Prog(X) --+ w ~ c_ X,

(vi)

which is TI(w~,X). Once we have (ii) we also get (42) because T l ( a , X ) implies Tl(a,,.7(Z)) by (43). It remains to prove (i). Again we work informally in N T . Assume

Prog(X).

(vii)

We want to prove Prog(,.7(X)) i.e. (Va)[a C_ J ( X ) -+ a E fl(X)]. Thus, assuming also a c_ 3"(X),

(viii)

we have to show a E 3"(X). i.e. (V{)[{ C_ X + { + w ~ C_ X]. That means that we have to prove 77 E X under the additional hypotheses C_ X

(ix)

and

n <

+

(x)

If 77 < ~ we obtain r/ E X by (ix). L e t ~ _ < r / < ~ + w ~. I f a = 0 t h e ~ = ~ a n d w e obtain r/E X by (ix) and (vii). If a > 0 then there is a a < a and a natural number (i.e. a numeral in N T ) , such that ~ < w ~ + . . . + w% =: w ~. n (c.f. the proof of the n-fold

Predicative Elimination Lemma). We show

a < a -+ , ~ + w " . n C_ X

(xi)

by induction on n. For n - 0 this is (ix). For n := m + 1 we have + w ~. m C_ X

(xii)

by the induction hypothesis. From a < a we obtain a E J ( X ) from (viii). This together with (xii) entails ~ + w ~. n = ( + w ~. m + w ~ E X. This finishes the proof of (i), hence also that of (42) which in turn implies Theorem 2.1.3.1. [3 Summing up we have shown 2.1.3.2. T h e o r e m .

(Ordinal Analysis o f N T ) IINTlln~

- co.

As a corollary of Theorem 2.1.3.2 and (24) we obtain 2.1.3.3. G e n t z e n ' s T h e o r e m . N T is ~o.

The proof theoretic ordinal of the axiom system

As another consequence of Theorem 2.1.2.11 and Theorem 2.1.3.1 is

There is a II~-sentence (VX)(Vx)F(X, x) which is true in the standard structure N such that N T ~- F ( X , n) for all n E N but N T ~ (Vx)F(Z, x) .

2.1.3.4. T h e o r e m .

Set Theory and Second Order Number Theory To prove the theorem choose F(X, x)

:ca Prog(X) --+ x E On --+ x E X.

241 [3

Theorem 2.1.3.4 is a weakened form of Ghdel's Theorem. The general form of Ghdel's Theorem says that Theorem 2.1.3.4 holds already for a H~

(Vx)F(x). 2.1.4. C o m p u t a t i o n a l c o m p l e x i t y of H ~ H e u r i s t i c s . From the point of view of minimal models, Gentzen's result appears surprisingly complicated. Shouldn't it be trivial that the minimal model for P A is obtained at stage w? Consequently there have been comments s t a t i n g - faintly j o k i n g - that Gentzen tried to secure induction up to w by transfinite induction up to c0. However, (27) tells us that this is not at all the case. Since we have tc(F) < w for all arithmetical sentences we get the least II~-model for P A at L~ for more or less trivial reasons. But it is not at all trivial where to locate the least model for wCK the E11 -sentences which correspond to the provable II~-sentences of P A according to the Hyperarithmetical Quantifier Theorem. All we know without proof theory is that it must be at some ordinal < w~~. Since Gentzen's Theorem tells us tc(F) < Co for all provable II~-sentences we know that this will be at L~o. Nevertheless it is an unsatisfactory situation that the II~-spectrum tells us nothing about the provable arithmetical sentences of Ax. By Ghdel's Theorem we know that there are arithmetical (even H ~ sentences which are not provable from A x while by (14) we have tc(A) < w for all true arithmetical sentences. Thus we are far from having tc(F) < IIAxll =~ A x ~ F. After all this work, however, one has the feeling to deserve a stronger result. The aim of this section is to show that we can obtain much more information with just a little more effort. This has been quite recently detected by Weiermann [1996] and it is appealing to include his result here because it is inspired by the collapsing techniques of impredicative proof theory on which we want to put the emphasis of this contribution. The basic idea is quite simple. Recapitulating the computation of the upper bound Co for specn~ (Ax) we see that we started with a derivation 0~ -~F1,..., -~Fn, F and ended up with an infinitary derivation ~ F for some a < s0. Call this derivation ~. If we assume that F is a E~ (qy)A(y) then ~r cannot contain applications of the (V)-rule. Hence ~ is finite and we may hope that the depth of r can tell something about the size of a witness for the existential quantifier in (qx)F(x). So there are two things to take care of:

1. Assure that the depth of a derivation majorizes witnesses for existential formulas 2. Find a method to extract a bound .for the length of the finite derivation of a E~ from its assigned ordinal. The realization of 2. requires the possibility to collapse ordinals bigger than w into finite ones. This technique, however, is known from impredicative proof theory where we have to collapse (recursively) uncountable ordinals into recursively countable ones. Once we have solved 2. is easy to realize also 1. All we have to require is that

W. Pohlers

242

the witnesses in (3)-clauses are majorized by the collapse of the derivation length. Later we will see that in more complex systems this becomes an even more natural requirement. While functions which collapse ordinals of transfinite number classes into lower transfinite number classes occur quite naturally in the development of notations for impredicative ordinals, i.e., ordinals above F0, it had for long been unclear which are the right finitary collapsing functions, i.e., functions which collapse transfinite ordinals into finite ones. It follows from work of Buchholz and Wainer that finitary collapsing functions must be connected to functions in the sub-recursive hierarchy. There is a result that for a certain choice of an infinitary calculus we have (roughly speaking)

~o (3x)F(n,x) =v N ~ (3x < H~(n))F(n,x) for quantifier free formulas F(x, y), where Ha is the a-th function in the Hardy hierarchy (c.f. the article by Fairtlough-Wainer in this volume). The infinitary system is tailored in such a way that it gives a nice characterization of the Skolem functions for II~ which are provable in Peano arithmetic. It turned out, however, that the generalization to stronger, especially to impredicative axiom systems is by far not straightforward. But there are finitary collapsing functions which work smoothly together with the impredicative collapsing functions. The basic idea in their definition is strikingly simple. As soon as we have term notations for ordinals we may define the norm N ( a ) of an ordinal c~ as the number of symbols in its term notation. The norm is thus a finite ordinal with the property that for every natural number k the set {c~ I g ( a ) < k} is finite. For any starting function G: 1N -~ IN the function r

:= sup {r

+ 11 ~ < ~

and g(~) <_G ( g ( a ) ) } U {0}

(44)

is then a finitary collapsing function for ordinals possessing a term notation. C o m p u t a t i o n a l c o m p l e x i t y . In order to obtain a more refined complexity for arithmetical sentences we refine the truth definition given in Definition 1.3.1. Here a crucial part is played by the function r To specify it we have to choose the starting function G. There are symbols for all primitive recursive functions and relations ins s The truth of any atomic sentence R(tl,..., tn) is thus decidable in just one step which means that the starting point in the measurement of computational complexity must be above the primitive recursive functions. Therefore we introduce a function P a variant of the Ackermann-Peter function - which majorizes all primitive recursive functions and choose P as the starting function in the definition of the finitary collapsing function r := CP. 2.1.4.1. Definition.

Po(x) := 2x P.+, (x) "=

We put

p(x+2)(x)

Set Theory and Second Order Number Theory

243

and finally

P(x) := Px(x). Then we have the obvious properties

x < Pn(x) for any n,

(45)

x < y =a P,(x) < P,(y) for all n

(46)

as well as

m < n =~ Pro(x) < Pn (x) for all x.

(47)

Every primitive recursive function f is eventually majorized by P. To see this we assume that f is represented by the constant f and show by induction on the definition of "f is a constant for primitive recursive function" the existence of a number e I such that

f ( x l , . . . , X # f ) < P~i(max{xl,... ,x#f })

(48)

holds for all #f-tuples (Xl,..., x#i). This is obvious for f E { C~, P~, S}. For f = Sub(g, hi,... ,hm) we put e := max{eg, eh,,... ,eh~} and obtain by the induction hypothesis and (45)- (47)

f(~) < Pe(Pe(max{x~,...,x,})) <__Pe+l(max{xl,...,xn}). Thus let e I := e + 1. If f = Rec(g, h) we put e := max{eg, eh} and obtain

f ( m , x ~ , . . . , x n ) < P(m+l)(max{x~,...,xn,m}) <_ P~+~(max{x~,...,xn,m}) by induction on m. Therefore let e I : - e + 1.

[]

We define N(0) : - 0 and for an ordinal a =BE ~P~lr/1 + ' ' " + ~p~ r/, such that n>l 71

N(a) := E ( N ( ~ , ) +

N(r/,)+ 1).

(49)

i=1

This defines a norm for all ordinals < Fo. For a < Fo the finitary collapsing function is thus defined by r i.e., r

:= max { r

+ 11 ~ < a and N(~) <_ P(N(a)) } U {0},

:= OR in the sense of (44). We introduce the relation a<<~"

r

a<~

andg(a)<_P(N(~)+g(~)).

We obtain the transitive closure <<~ of <<} by putting

and

(50)

W. Pohlers

244

<<~/~.

~

(3~)[~ <<~ Z].

Instead of a <<0 fl we write shortly a << fl and call a collapsibly less than ft. This is justified by ce << fl =~ Cw(a) < r

(51)

It is perhaps noteworthy that r

= max {k[ 0 <<0k a}.

We moreover have N(~) = ~ for ~ < w. Hence fl
=~ (aKKfl r

a
(52)

Using the collapsibly less relation we define a refinement [~ A of ~ A for ordinals a, p < F0 as follows: 2.1.4.2. Definition.

(AxM) i/ A fq D(N) # 0 then I~ A

for all ordinals a and p.

(AxL) If t N = s N then I~ A, s ~ X, t e X for all ordinals a and p.

(^)

if

(V)

If I~p~ A, Ai and ao << a for some i E {1,2} then I~ A, A1 V A2.

(V)

If [~-~ A, A(i) and ai <
(3)

If [~pOA, A(/) and ao << a and i + 1 << a for some i E N then [~ A, (3x)A(:r,).

(cut)

I: I~ a, F, I~ a, -F, ~, << ~ :or i ~ (1, 2} an,t ~k(t) < p th~n I~ a.

I~-p A, ai and a, << a for i = 1, 2 then I~ A, al A Z2.

As a word of warning we want to emphasize that the definition of the << relation and thus also the definition of the relation [~ A depends on the term notation for ordinals (actually only on the norm function N(a)) and thus is only defined for ordinals below F0. But it is obvious how to extend [~ A as soon as we have term notations for larger ordinals. From Definition 2.1.4.2 we obtain

I~ A,

a <__
p
and

A C F

=~

I~ F

(53)

immediately by induction on a. The next observations will help us to a better understanding of the relation I~ A. First we observe

I~A

:=> ~ A

~

In contrast to the calculus ~ property:

~A.

(54)

the refined calculus has the following collapsing

Set Theory and Second Order Number Theory I~~~ A and A C E ~ =~ II~

A.

245 (55)

Observation (55) is immediate by induction on a. The crucial point is that a derivation of a set of E~ which is cut free cannot contain applications of an (V)-rule. Hence all ordinal assignments ~ in this derivation increase in the sense of the collapsibly less relation and m a y - by (51) and (52) - thus be replaced by r On the other hand we also have l~o (3x)F(x) for an E~

(2x)F(x)

=:~ N ~ (3x < n)F(x).

(56)

=~ [~ A.

(57)

The proof of (56) is by induction on n and needs

I~ A, F and N ~ F .for F quantifier free

We prove (57) by induction on a. If F does not belong to the main formula of the last inference in I~ A, F the claim follows immediately from the induction hypothesis. If F is the main formula of the last inference it can neither be an axiom (AxL) (because F is a sentence) nor an axiom according to (AxM) (since F ~ D(N)). Thus F is either a conjunction Ft A F2 and we have the premises I~- A, F, Fi for i E { 1, 2} and 1N ~ Fi for some i E {1, 2} or F is a disjunction F1 V F2 and we have the premise I~2- A, F, Fi and 1N ~ Fi. In both cases we obtain ]~ A by two fold application of the induction hypothesis and (53). O It is a good exercise to try to generalize (57) to sentences F containing quantifiers. What goes wrong if F contains universal quantifiers? Back to the proof of (56). The only possibility to obtain ]~ (3x)F(x) is a clause (3) with the premise

[~o (3x)F(x), F(i)

(i)

for m < n a n d i + l < n. I f N ~ F(/) we are done because o f i < n, otherwise we obtain 10~ (3x)F(x) by (57) and then N ~ (3x < m)F(x) by the induction hypothesis. Since m < n this yields the claim. O By (55) and (56) we see how to obtain upper bounds for the witness of a E ~ sentence F once we succeed in getting I~ F. To obtain (55)it is important to replace as far as possible- the natural order relation on the ordinals by the collapsible less relation on the term notations. The crucial condition for (56) is to majorize the witnesses of (3)-clauses. What still needs explanation is the ordinal assignment to (V)-clauses. There are only finitely many ordinals/3 << a. So we cannot require the ordinals to increase in the sense of the <<-relation. We will come back to that point. In analogy to the definition of the truth complexity for II~-sentences we define the computational complexity of a sentence F by

-

cc (F) "= min {c~] I~ F}.

(58)

By (54) we then have tc(F) _ cc (F)

(59)

W. Pohlers

246

for all sentences F. One is tempted to define specro(Ax) := {cc (F) I F is a E~

and A x ~ F }

in analogy to spec~i(Ax). However, we easily get I[Lt-~,0 (3y)F(y) if N ~ F(n)_ which entails cc ((3y)r(y)) = min {n + 21 N ~ F(n)} for a E~ (3y)F(y). Therefore we have specro (Ax) = w for any axiom system A x which at least contains the successor axioms which shows that the E~ specro(Ax) carries no information about Ax. The analogy to specn~ (Ax) is apparently the wrong one. wCK

But recall that YI~ corresponds to E11 wOK

wOK

on the side of sub-systems of Set Theory and

E11 -models are the same as II2 ~ -models (cf. Lemma 1.2.3). The computational wCK

aspect of an axiom system is however better reflected in its YI2~ -model because it wCK says something about its provably total E1 ~ -functions. Pulling this down to w one should rather look at H~ and try to define something as the II~ of an axiom system. As a first observation in that direction we show

I~ A, (Vx)F(x) =~ II"+'+l,p A, F(/)

(60)

for all i 6 N by induction on a. If the main formula of the last inference is different from (Vx)F(x) then the claim follows directly from the induction hypothesis and the fact a~<
=~ a ~ + i + l < < j a + i + l

(61)

for all j and i. The crucial case is that of a clause (V) with main formula (Vx)F(x). There we have the premises

F(j) with a# <
(i)

To get the claim from (i) it suffices to check aj<
=~ a j + i + l < < c ~ + i + l .

(62)

From ~ <<~ ~ we get ~j + i < ~ + i and N(~j) _< P(N(~) + i) which entails also N(c~j + i + 1) = N(c~j) + i + 1 _< P(N(oL + i + 1)). Hence ~j + i + 1 <<1 c~ + i + 1. Iterating the procedure we get c~j + i + 1 <
2.1.4.3. T h e o r e m . If cc ((Vx)(3y)F(x,y)) = a then, for every i 6 N, N (3y < Cw(a + i + 1))F(i, y).

a

0 To prove the theorem let (Vx)(3y)F(x,y) be a II2-sentence and put := cc ((Vx)(3y)F(x, y)). Then I~ (Vx)(3y)F(x, y) and we get by (60)

Set Theory and Second Order Number Theory

Iioa + i + l

(3y)F(~_,y) for a l l / E N. Hence 1N ~ (=ty< r

247

+ 1 ) ) F ( i , y ) by (55)

and (56).

U

According to Theorem 2.1.4.3 we define W,~(i) := ~ ( a

+ i + 1).

(63)

By Theorem 2.1.4.3 it follows that W~ majorizes a Skolem function of a II~ F if cc (F) _< a. If we put

I(Yx)(3y)F(x, y)lno := min { a I (Vi)[N ~ (3y < W~(i))F(/, y)]} for a II~

lal o _< We define the II~

(Vx)(3y)F(z, y) then we obtain for II~

(a).

(64)

G

(65) of a theory A x as

specno(Ax) = {IFIno I F is a g~

and A x ~ F }

provided that IFIno is defined for all such F as well as

llAxlE,o :=

up(spec,o(Ax))

and obtain IIAxl[no _< sup { cc (F) I A x ~ F } .

(66)

We call the computation of IIAxlJno a n~-analysis of Ax. The definition of the H~ is admittedly less intrinsic than that of the H~spectrum. We want, however, give some reasons why we think that II~ do have an intrinsic meaning. The H~ depends on the function c P which in turn depends on the starting function P and on the term notation of the ordinals in the H~ Indeed the II~ is rather a set of ordinal terms than a set of ordinals. We have already argued that the dependence on a starting function P has natural reasons. P has to majorize all the functions for which there are function symbols in the language (c.f. also the proof of Lemma 2.1.4.6 below). With another choice of the language, e.g. Peano arithmetic with the only function symbols for addition and multiplication or even the language of Set Theory with no functions symbols, a weaker starting function, e.g. something like )~x. 3x , will do the same job. Also the hierarchy of functions WaR "- ,ki. CP (a + i + 1) does not depend too much on the starting function, i.e., for not too different starting functions P and G we will obtain a comparatively small ordinal a such that W P ~ WaG which means that W P is elementary in WaG and vice versa. More serious is the dependence on the term notations for the ordinals. However, it is hard to imagine an explicit term notation which could alter the II~ Weiermann has shown that the fast growing subrecursive hierarchies- and the hierarchy Wa is fast growing - are very stable against alterations of the term notations (which is not true for the so called slow growing hierarchies). We believe that at least every term notation usable in ordinal analysis will lead to the same finitary collapsing function r and thus to the same H~ - although we have to admit to see no way of proving this. The lack of a general and "natural" term notation system for all recursive ordinals (or equivalently

W. Pohlers

248

the lack of a natural subrecursive hierarchy for all recursive functions), hinders us from defining generally

{IFIno I F a n~

specno(N) :=

and N ~ F}

although we conjecture that any possible definition should lead to specno(N ) specrq (N) = w~K (which is motivated by the fact that we have specno(Ax ) = specn~ (Ax) for all "regular" axiom systems which are so far analyzed). Anyway, the II~ of an axiom system has pleasant properties. Every H~ (Vx)(3y)F(x, y) defines a partial recursive function fF :-- #Y. F(x, y) and we call f f provably recursive in A x iff A x ~ (Vx)(3y)F(x, y), i.e., iff A x proves that fF is total. If fF is provably recursive in A x then there is an a E specno(Ax) such that f f -- #Y < Wa(x). F(x, y). Therefore all provably recursive functions of A x are primitive recursive (even elementary) in Wa for some a e specno(Ax). By induction on a we obtain (Vx)(3y)[W~(x) -- y], i.e., we have

< IIAxll ~

A x ~ (W)(3y)[W~(x) - y]

(67)

for all axiom systems A x which allow the definition of the functions W~. (For this it certainly suffices that A x allows the definition of all primitive recursive functions. In the rest of the paper we tacitly assume that this is true for all axiom systems considered. Weaker systems need more subtle considerations which are outside the scope of this contribution). For axiom systems satisfying this assumption we obtain as a corollary of Theorem 2.1.4.3

2.1.4.4. Lemma. If cc ((Vx)(3y)F(x, y)) < IIAxll then there is a provably recursive function f of Ax such that N ~ (Vx)F(x, f(x)). Because of I(V~)(3y)[W~(~) - y]l.o = ~ + 1 we obtain from (67) also a < IlAxll.o.

(68)

IIAxllnl- IIAxII ~ IIAxllno.

(69)

a < IIAxII

~

Hence

In general the inequality in (69) is proper. For ~ : - [IAxllno, for instance, we get by (28) the inequalities IIAx + (w)(3y)[Wo(~) = y]lln~ = [[Axl[ul = [[Ax[[ _< a < [ [ A x + (Vx)(3y)[W~(x) = y][[no. In most cases, however, we obtain - spending a little more care on the ordinal assignment - sup {tc(F)] A x ~ F } = sup {cc (F) I A x e - F } . This then entails IIAxlln} = sup {tc(F) I A x e - F } = sup {cc ( F ) [ A x e - F } . Together with (69) we then obtain

IIAxII - IIAxllrxl _ IIAxllno ~ sup {cc (F) I Ax I~ F} - IlAxllrq,

(70)

i.e., IIAxlln, = IIAxllno. We are going to call axioms systems for which we have IIAxlln] = I]Axllno regular. Another consequence of (67)is the following theorem.

Set Theory and Second Order Number Theory

249

Let A x be a regular axiom system, i.e., let IIAxll = IIAxllno. Then the provably recursive functions of A x are exactly the functions which are primitive recursive (even elementary) in some W~ for a < IIAxll. 2.1.4.5. T h e o r e m .

Without further hint we just remark that the functions W P are closely connected to the Hardy-functions Ha. A detailed study is in Buchholz, Cichon and Weiermann [1994]. We want to close this section with the remark that there is also a YI~ ordinal for theories, whose intention is to express the order type of the shortest primitive recursive well-ordering which is needed to prove the consistency of the theory within a finitistic framework. Due to certain pathologies (cf. Remark 7.1.9. in Girard [1987] which exposes an example due to Kreisel) the definition of the II ~ ordinal is not completely straightforward. We omit a discussion since we believe that the known concepts are still too far from a final form and need further research.

Computational c o m p l e x i t y of N T . As an example we want to compute specno (NT). The first step consists in computing the computational complexities of the axioms of N T . We observe that

(VXl)""" (VXn)a(Xl,... ,Xn)

(71)

holds for all true sentences (VXl)""(VXn)e(Xl,...,Xn) where G(ul,...,un) is a quantifier free formula. The proofis simple. For every n-tuple (zl,..., zn) we have G(zl,..., zn) E D(N). Hence [~ (Vxl)... (Vx~)G(xl,... ,x,,) by n-fold application of a clause (V). r] All mathematical axioms of N T , except the induction scheme, are H-sentences of the form ( V x l ) . . . ( V x , ) G ( x l , . . . , x n ) with G ( u l , . . . , u , ) quantifier-free. Thus (71) gives us bounds for the computational complexity of all these axioms. To compute the computational complexity of the scheme of Mathematical Induction we first prove

I,olw'rk(F) /k, F, ~ F

(72)

by induction on rk(F). The proof is essentially that of the Tautology Lemma (Lemma 2.1.2.3). A bit more care is needed for the case that F is a formula (Vx)G(x). There we have

I,ol'''rkcac=-)) A, a(z_),--,a(z)

(i)

for all E N by the induction hypothesis and obtain

G(z_), (Sx)~G(x)

(ii)

for every z E N by a clause (3). But w. rk(G(z)) -t- z ((z w. (rk(G(z)) -i- 1) holds for all z E N and we obtain

~.rk((V=)C(=))

by a clause (V).

W. Pohlers

250

The computation of the computational complexity of instances of the scheme of Mathematical Induction is obtained as in Lemma 2.1.2.4 with some extra care on the ordinal assignment. We prove

~ F(Sx)],F(n)_

II0W'rk(F(~

(73)

by induction on n. For n = 0 this is (72). For the induction step we get by (72) and the induction hypothesis by an inference (A)

][0w'rkCf(0--))+2n+l--,F(O),~(Vx)[F(x) -+ f(Sx)],F(n_)

A--,F(Sn)_, F(Sn)._

(i)

We have n << w.rk(F(O))+2.(n+ 1) and w.rk(F(O_))+2.n+ 1 << w.rk(F(O))+2.(n+ 1) and obtain thus from (i) ]10~.rk(f(0))+2(n+l) -,f(O), -~(Vx)[f(x) --+ f(Sx)], f(Sn)

(ii)

by a clause (3).

D

Noticing that w.rk(F(O))+2n <
IIo

~F(0),--,(Vx)[F(x) ~

F(Sx)], (Vx)F(x).

(74)

This gives an estimate for the computational complexities of instances of Mathematical Induction. From (72) we obtain also

iw'rk(F)_z1 r

I,0

k_l,..., z~ ~ k~,-~F(Zl,... , z~), F ( k l , . . . , k~)

(75)

for all n-tuples (Zl,...,Zn) and ( k i , . . . , k~) of natural numbers. This eventually gives w. rk(G) as an upper bound for the computational complexities of all instances G of identity axioms. Pulling together (71), (74) and (75) we obtain w.(rk(A)+l)

(76)

for every non-logical and identity axiom A of N T . The adaption of (39) needs a bit more care. It will, however, clarify the role of the starting function P in the definition of the collapsing function ~ . We show 2.1.4.6. L e m m a . If ~--i(Ul,...,Un) and all free number variables of A ( u l , . . . , un) occur in the list ul,..., un then there is a finite ordinal k such that

IIW'k+zx+'''+zn A(zl, 0

""

. zn) holds for all n-tuples (zl ~

~'''~

zn) of natural numbers.

The proof of Lemma 2.1.4.6 is by induction on m. As prerequisite it needs

I~ A(s) and s N - t N =~ I~ A(t)

(77)

which follows straightforwardly by induction on ~. The only cases in the proof of Lemma 2.1.4.6 which are not immediate from the induction hypotheses are clauses (S) and (V). Let us start with the (3) case. It is of special interest because it needs the starting function P. We have

Set Theory and Second Order Number Theory

Ao(~), G(~, t(~)) ~ o~ Ao(~), (3z)a(~,z)

251

(i)

and obtain by the induction hypothesis a finite ordinal ko such that ,o

Ao (~), G(~, t(zS)).

(ii)

The value t(~) N is computed primitive recursively from Z. By (48) we thus have a natural number kt such that t(_~)N < P~,(z-). Taking k := max{ko + 1, kt} we get w. ko + ~"<< w. k + ~ and t(g) N << w. k + ~'.

(iii)

Hence o

Ao(~), (3x)G(~_, x)

(iv)

from (ii), (77) and (iii) by a clause (3). In the case of an (V) inference

ZXo(~),a(~, v) ~ o~ ZXo(~),(Vx)a(~, ~)

(v)

we have a finite ordinal ko such that

ii~.ko+~+, ,0 Ao(g), G(g,/)

(vi)

for all i C N. Putting k : - ko + 1 we get w . ko + ~' + i <
(vii)

for all i E N and obtain

I1~'*+~ Ao(~) I0

(W)G(g, x)

from (vi)and (vii) by a clause (V).

U

Lemma 2.1.4.6 together with (76) yield 2.1.4.7. Lernma. If N T ~ - F numbers m and n such that II~§

for an s

F then there are natural

F

What is still lacking is a proof that the cut elimination procedure also works for the refined calculus I~ A. We are going to check this step by step. The first step is checking the Reduction Lemma. We claim

I~~ ~ F ,

I~_ F, ~ F and rk(F) _< p =~

II,~ ~

A, F

(78)

The proof is by induction on a ~ fl and follows the pattern of the proof of the Reduction Lemma (Lemma 2.1.2.7). There is, however, a subtle point in it which will clarify the ordinal assignment in the case of (\/)-clauses. We treat only this case. All other cases follow easily from the induction hypotheses and a <
(79)

W. Pohlers

252

[~poA, G(i), (3x)G(x) =~ [~ A, (3x)G(x).

(i)

Then ao << a and i + 1 << a. From (i) and the second hypothesis

r,

(ii)

we obtain

][pnO~ ~ A, F, G(/)

(iii)

by the induction hypothesis. On the other hand we obtain by (60) and (53) from (ii) also

IIP

r,

(iv)

We have

and obtain from i + 1 << a also j3+i+1

<
(vi)

Because of (v), (vi) and rk(G(/)) < rk(F) = p we get

from (iii)and (iv) by cut.

[3

By (78) we obtain a

"2 a't'l

(80)

by induction on a. The proof is that of the Elimination Lemma (Lemma 2.1.2.8). All we need to adapt the proof are a <
(81)

al,a2 << a =~ 2~1+1 ~ 2~2+1 <_<2~+1.

(82)

and

These properties, however, follow easily from the fact that Y(2 ~) _< 2 g(a)

(83)

which in turn is nearly immediate since for a = w . v + n we obtain N(2 ~) N(w V. 2") _< 2". (N(v) + 1) _< 2 N(~)+n ___2 N("). [3 One should observe that (81) and (82) force us to use a fixed point free formulation of the Elimination Lemma (i.e., 2~+t instead of 2 ~). The reason is that in case that a < 2 a, a << /3 and /3 = 2z we do have 2~ < 2z but not necessarily N(2 a) _< P(N(2~)). Now we have all the data for the computation of the II~ of N T . From Lemma 2.1.4.7, (80) and the fact that expn(2,w2 + m) < r we obtain

Set Theory and Second Order Number Theory

N T ~- F =~ I~ F

253 (84)

for all/:N-sentences F and some ordinal a < e0. Together with (66) we have liNT[[no _< eo = [[NTI[ = lINT[In I __ lINT[In o.

(85)

Combining (85) and Theorem 2.1.4.5 we obtain 2.1.4.8. T h e o r e m . specno(NT) = e0 = I[NTII and the provably recursive functions of N T are exactly the functions which are primitive recursive in some function Wa for a < eo. Taking F(x) :ca (3y)[(X)o 6 On V ~i/(x)o((X)I ) -- y] we get NT~-F(_n) for all n E N but N T ~-(Vx)F(x) since cc ((Vx)F(x)) = eo. This sharpens Theorem 2.1.3.4 to 2.1.4.9. T h e o r e m . There is a true ri~ for all n E N but N T ~ (Vx)F(x) . 2.1.5. C o m p u t a t i o n a l c o m p l e x i t y of II~

(Vx)F(x) such that N T [-- F(n)

revisited

Heuristics. We have seen in (65) that IFIno <_ cc ( F ) . Proving also the opposite inequality would be a good argument for the naturalness of the computational complexity. However, if I(Vx)(3y)F(x, y)lno = a we get easily

Iiow,~(i)

(gy)F(i,y)

(86)

for all i E N but there is no ordinal/3 such that W~(i) <
I#o(Vx)(3y)F(x, y).

(87)

for infinite a. Hence w _ IFIno =v cc (F) _< IRis0. The collapsing technique of the previous sections is based on the idea of local predicativity which has been originally developed for the analysis of impredicative axiom systems (cf. Pohlers [1978,1981, 1991] and Buchholz et al. [1981]). It is hardly surprising that the redefinition of <
W. Pohlers

254

the more statical aspect of cut-freeness as e.g. in Beckmann and Pohlers [1997]. The fact that it also needs no alteration of the functions W , indicates the naturalness of Buchholz' approach. In the following section we introduce the concept of operator controlled derivations. We will demonstrate that the computational content of the cut-elimination procedure is measured by the controlling operator. O p e r a t o r c o n t r o l l e d d e r i v a t i o n s . In this section we concentrate on arithmetical sentences and dispense therefore with set variables. We call this language L:~ which we assume to be formulated as Tait-language (cf. Section 1.3). There are two types of arithmetical sentences

9 sentences of A-type, which are true atomic sentences, i.e., sentences in D(N), and sentences of the form (A A B) or (Vx)F(x) and

9 sentences of V-type, which are false atomic sentences and sentences of the form (A V B) or (3x)F(x). For every sentence we associate a characteristic set C (F) of sentences such that

N ~ F We define C(F) :=

r

(VG e C(F))[N ~ G]

if F e A - t y p e

(3G e C(F))[N ~ G]

if F 9 V-type.

{~

{A,B} {G(n)l n 6 w}

(88)

if F is atomic if F is of the shape A o B if F is a sentence (Qx)G(x),

where o 6 {A, V} and Q 6 {V, 3}. To each formula G E C(F) we associate a finite ordinal oF(G) which is 0 if F is of the shape A o S and n if F has the form (Qx)H(x) and G is g ( n ) . In order to obtain a more refined truth complexity for arithmetical sentences we introduce

operator controlled infinitary derivations. For a function F: N

~ N and a finite set M C N we define the function

F[M](n) := F(max(M U {n})). We may interpret F as an operator with values in the collection of finite sets of ordinals by defining a e F(n)

:r

Y(c~) < F(n).

Instead of a E F(0) we write shortly a E F. To simplify notations further we write F[n] for F[{n}] and G e F and FIG] instead of oF(G) e F and FloE(G)], respectively, whenever it is clear from the context to which characteristic set C (F) the sentence G belongs.

Set Theory and Second Order Number Theory

255

2.1.5.1. Definition. Let F: N } N be an increasing function. For a finite set of arithmetical sentences we define the proof relation F ~ A inductively by the following clauses.

(A)

If F E A N A-type, a E F and F[G] ~

(V)

1IF e a n V-type, , ~ , F, a e F, r U a, a ~ e ~ then F ~ A

A, a as well Ola < OLfor all G E C (F)

then F ~ A

< o, Io,-~om~ a e c (F)

We call F the main part in instances of ( A ) and (V)" (cut)

If a E F and F ~v~ A, A as well as F ~po A,-~A for some ao < a and some A such that rk(A) < p then F ~ A

It is then obvious from (88) and the soundness of the cut rule that this infinitary calculus is sound, i.e., we have F~A

=~ N ~ V { F

(89)

I FEA}.

We say that an operator G extends the operator F, written as F c_ G, if F(n) _ G(n) holds for all n. By a straightforward induction on c~ we easily obtain 2.1.5.2. L e m m a .

If a <_ fl, p <_ a, F c_ G g fl, A C_ F and F ~ A then G ~ F.

Another easy observation is the following Detachment Lemma whose proof is again a straightforward induction on c~. 2.1.5.3. D e t a c h m e n t L e m m a .

If F ~ A, A and -~A E D(N) then F ~v A.

One purpose of the operator is to control the witnesses of existential sentences. This is manifested in the following Witnessing Lemma. Lemma. Let (3y)F(y) F ~o (3y)F(y). Then 1N ~ (3y < F(0))F(y).

2.1.5.4. Witnessing

be a E~

such that oto

The proof is by induction on a. The only possible premise is F ~ (3y)F(y), F(n) with c~0 < c~ and n = o(3y)F(y)(F(n_)) e F, i.e. n < F(0). If F(n) e D(N) we are done. Otherwise we have F ~2_ (3y)F(y) by the Detachment Lemma and obtain the claim by the induction hypothesis, n The more important aspect of operator controlled derivations, however, is the control they give on functions as stated in Theorem 2.1.5.6 below. The key property here is the following Inversion Lemma.

W. Pohlers

256 2.1.5.5. Inversion Lemma.

Let F be a sentence of A-type such that F ~ A, F.

Then we get F[G] ~ A, C .for all C E C(F). The proof is by induction on a. In case of an inference according to (V) we get due to F c_ FIG] - the claim immediately from the induction hypothesis. In case of an inference (A) we have the premise F[G] ~2_ A, F, G for some aa < a and get the claim from the induction hypothesis, Lemma 2.1.5.2 and the fact that

-

F[GI[G] = FIG].

rn

Combining the Inversion and Witnessing Lemmas we get the following theorem. 2.1.5.6. Theorem. Let (Vx)(3y)F(x,y) be a n~ such that F ~o (Vx)(3y)F(x, y). Then the associated recursive function J'(x) = #y. F(x, y) is majorized by F, i.e., we have f(n) < F(n) for all n E w. Proof.

Pick n E w and apply the Inversion Lemma to get Fin] ~ (3y)F(n, y).

(i)

Then use the Witnessing Lemma to see that #y. F(n, y) < F[n](0) - F(n).

E]

We have to study the behavior of the controlling operators during the cutelimination process. 2.1.5.7. R e d u c t i o n Lemma.

Let F be a sentence of V-type and p := rk(F).

Let F and G be increasing functions such that 2. m < F(m) < G(m). If F ~ F,-~F and G ~ A , F then (F oG) l,p- ~ I-',A We prove the lemma by induction on/3. First we observe that c~ E F and/3 E G imply a +/3 E F o G since g ( a + fl) <_ g ( a ) + g(fl) _< 2. G(0) < F(G(0)). We also have G c_ F o G. If the last inference is a cut or an inference according to (V), whose main part is different from F, we get the claim immediately by the induction hypothesis and F C_ (F o G). If the last inference is (A)

G[H]p~ - A ' G ' H f ~

E C (G) =~ G ~ A, G

then we get by induction hypothesis (F o G[H])Ipa+~" F,A H

(i)

for all H E C (G) and obtain (F o G ) ~

F A

from (i) by a clause (A) since (F o G[H]) = (F o G)[H]. The interesting case is an inference according to (V)

(ii)

Set Theory and Second Order Number Theory

G~-r,F, G forsomeC e C(F) ~ G~r,F

257 (iii)

whose main part is F. Then we have G E G, flo < fl and obtain (F

oG)lp~+~o F, A

a

(iv)

by induction hypothesis. From the hypothesis

F

(v)

we obtain

FIG] ~ A, ~G

(vi)

by the Inversion Lemma (Lemma 2.1.5.5). But G E G means oF(G) < G(0) which in turn entails F[G](n) - F ( m a x { o f ( G ) , n } ) <_ F(max{G(0),n}) _< F(G(n)), i.e. F[G] C_ (F o G). Hence (F o G) ~~p-

F, A, ~G

(vii)

from (vi) and Lemma 2.1.5.2 and we obtain

(F o G) ~~p- F, A from (iv)and (vii) by cut.

(viii) D

The Reduction Lemma illustrates that avoiding one cut means composing the controlling operator. If we have a derivation F ~ A and we want to reduce the cut rank by 1 we have to iterate the Reduction Lemma a times. This causes no problems as long as a is finite. For transfinite ordinals a, however, we have to decide what we understand by transfinite iterations of operators. There is a big variety of possibilities, e.g., attaching limit ordinals with fundamental sequences and diagonalizing at limit points. However, we already defined a hierarchy of strictly increasing functions by introducing the functions W~ which in turn are based on the collapsing functions CP. First we observe that 2m _ Wa(m). We claim that we also have W a O W fl C Wa ~: ,8+1"

(90)

This turns the members of the hierarchy W~ into good candidates for the controlling operators. To prove (90) we first observe n < w

=~ C P ( n ) = n .

(91)

Then we show cP (a + cP (/3)) _< cP (a ~/3)

(92)

by induction on/3. For a = 0 this is (91). So assume a r 0. For/3 = 0 the claim follows trivially from (91). If/3 r 0 we have CP (/3) = CP (7) + 1 for some 7 <
W. Pohlers

258

~ ( ~ + r (#)) = r + 1+ r _< r (~ + 1 ~ ~) ___r (~ ~ #). Hence W=(W#(n)) = CP (a + CP (fl + n + 1) + 1) <__CP (a ~ ~ + n + 2) = W= ~ #+, (n) and we have (90). [3

I/W# ~

2.1.5.8. E l i m i n a t i o n L e m m a .

A then W ~ + l

.wa+l

~

A.

The proof is by induction on 0/and the crucial case is that of a cut of rank p whose premises are

W # ~p~+ ~ A , A and W # ~p~+ ~A ,~A for some 0/1,0/2 E W# such that 0/i < 0/for i = 1, 2. From the induction hypothesis we obtain [WoK'+1 IWa2+l -~A (i) W ~ 1 + 1 ,p A,A and W~,~2+1 ,a ' which by the Reduction Lemma and (90) entails w a l + l _}.Wa 2 + l

W ~ . ~ + ~ ~ ~ ~-~+~+1 I.

A.

(ii)

Next we show 0/1, 0/2 < 0/ h 0/1, 0/2 E W E

::~

Wwfl~al+l :H:wfl~a2+l-t-1 C Wwfl~:c~.~.1 .

(93)

Since 0/1, 0/2 E W E means N(0/i) < WE(0) = cP (fl + 1) we have W#* ~+t #w #*~2+1 << w # ~ .

2 + cP(fl + 1). 2.

(iii)

This entails by (51) and (92)

< CP (w #~ ~ - 2 + CP (/~ + 1). 2 + n + 2) <_ CP(w E*~ . 2 ~ ~ . 2 + n + 4)

(iv)

_< CP(w # ~ + 1 + n + 1) - ~]wt~ ~+l(n). From (ii) and (93)we finally obtain .wa+l

W.,~ ~ ,~+~~

A.

The remaining cases are immediate from the induction hypothesis and the fact that 0/~ W E implies W~ a + l ~ W~,8~=e~-F1. 1-'1 To reobtain specno(NT) in terms of the indices of functions W~ majorizing the II~ provable in N T by operator controlled derivations, we have to check which operators control the provable sentences of N T . We introduce

par(F(Zl,...,zn)) := {Zl,...,Zn}

(94)

if F ( z t , . . . ,z~) is a sentence with only the shown number parameters z~,... ,z~. Then we prove

Set Theory and Second Order Number Theory

259

10

W2.rk(F) [par(F)] 2.rk(F) A, F, -~F

(95)

by induction on rk(F). First observe that according to (91) we have W0(n) - n + 1. For symmetry reasons we may assume that F has A-type. If F is atomic, then 6 ( F ) = 0 and we obtain W0[par(F)] ~ A, F, ~ F by a clause F - Go A G1 or F - (Vx)G(x). In the second case we get

A" Otherwise it is

A, G ( z ) , - G ( z ) for all z e w W2.rk(G(z_))[par(G(z))] I2.rk(O(~_)) o

(i)

by induction hypothesis. Since z e W2.rk(G(z_))[par(G(_z))] C_ W2.rk(F)[par(F)][z] ~

2. rk(G(z)) + 1

we obtain from (i) W2.rk(F)[par(F)][z]

Io2.,k(C(~_))+1 A, a ( z ) , - ~ F

(ii)

for all natural numbers z by a clause V and from (ii) finally

W2. rk(F) [par(F)]lo2.rk(F) A, F, ~ F by a clause

A" The first case is similar but simpler.

Since we have Wo[Zl,...,zn]~,G(Zl,...,zn) G ( z l , . . . , zn) we obtain

E:] for every true atomic formula

wo ~ (vx,)... (vx,,)a(Xl,... ,x,,)

(96)

for every mathematical axiom (VXl)... (Vxn)G(xl,... ,Xn) by n-fold applications of

(A)

From the method of (95) we obtain also the equality axioms

W2"rk(F(~))-'l-2n-l-2[Z-1I02.,k(F(~))+2n+2 (VZ)(V~7)[Z=g-+ F ( Z ) - +

F(y-')]

(97)

,,,h~re n is the l~ngth oCthe tuple ~ and {~'} := { z , , . . . , z,,,} = p~r((VZ)F(:0). By the same proof as for the Induction Lemma (Lemma 2.1.2.4) we obtain 2.(rk(F(0))-i-n)

W2.(rk(F(O))+n)[par(F(O))][n]I-~

--,F(O), -,(Vx)[F(x) -+ F(S(x))], F(n_.). (98)

But (98) in turn gives

Ww.rk(F(0))[par(F(0))] ~'0

F(0) A (Vx)[F(x) ~ F(x + 1)] -+ (Vx)F(x). (99)

It remains to adapt Lemma 2.1.4.6, i.e., to show that there is a natural number k such that ~- A ( x , , . . . , x , )

=~ W~.~+m[z,,...,z,] 0~ A ( z , , . . . , z , )

(100)

for all tuples Z l , . . . , zn. Since N(w. m) = 2 . m we obtain P(m) << w.m. Hence

P(m)

=

cP (P(m)) < cP (w. m) < Ww.m(0).

(101)

W. Pohlers

260

We prove (100) by induction on m and, as in the proof of Lemma 2.1.4.6, the only critical case is an inference according to (3). We proceed as in that proof and obtain from the induction hypothesis for the premise ~ Ao(g), G(~7,t(~7)) a natural number ko such that W~.ko+mo[z-]~ A(~, G(~', t(~).

(i)

Because t(g) N is computed primitive recursively from ~' we find by (48) a natural number k > ko such that t(g) N < P(k) < Ww.k+m[z-](0). Applying an inference (V) to (i) we therefore get The other cases follow immediately from the induction hypothesis.

O

By (96), (97), (99) and (100) we finally obtain N T ~ F =~ (3k)(3m)(3r) [W~.k~ F] for arithmetical sentences F. This yields a H~ N T ~- (Vx)(3y)F(x, y) for a II~

(102) as follows: If (i)

(Vx)(3y)f(x, y)then

W~.~ ~ (Vx)(3y)F(x, y)

(ii)

by (102). Defining ~n := (A[.exP(W,[+l)) (~) and &~(~,r/) recursively by &o([,r/) := ~ and &~+l(~, r/) "= exp(w,&~([,rl)$Co~(rl)+l) we get from (ii) by r-fold application of the Elimination Lemma (Lemma 2.1.5.8) [~,(m) (Vx)(3y)f(x, y) W~r(~'k,m) ,o

(iii)

Putting a :- &r(w.k, m) < ~o we obtain from (iii) using the Inversion and Witnessing Lemmas (Lemmas 2.1.5.5 and 2.1.5.3) (Vx e N)(3y < W~(x))F(x, y)

(iv)

which shows

I(Vx)(3y)f(x, Y)lno <_a. On the other hand, if a "- [(Vx)(3y)F(x, y)lno then we get W~[i] ~ (3y)F(/, y) for a l l / e N and thus also W~ ~o (Vy)(3y)f(x, y). Hence [F[no = min {a[ (3/3) [W~ ~ F]} for ri~

F.

(103) O

Equation (103) can be taken as evidence for the naturalness of the concept of operator controlled infinitary derivations.

Set Theory and Second Order Number Theory

261

2.2. P e a n o a r i t h m e t i c w i t h additional transfinite i n d u c t i o n 2.2.1. T h e theories NT~t and their H~ Let -~ be a primitive recursive well-ordering. field (-~) = N. By TI (-~ [m) we denote the scheme

Prog(-~, F)

For simplicity assume that

--+ (Vy -~ m)F(y)

F(y) expressing transfinite induction along -< [m. Then

for s

TI(<[) := U Tl(-
denotes the scheme of transfinite induction along all proper initial segments of -~. Let NT.
N:r

~w

satisfying: (N1)

N(0) - 0.

(N2)

N ( a ~ fl) = N ( a ) + N(fl).

(N3)

a C w ~ =~ N(w ~) = N ( a ) + l.

(i4)

For all n E w the set {~ < e I N ( a ) < n} is finite.

Observe that conditions (N1) - N(4) are always satisfiable as soon as we have term notations for the ordinals below c. Using the norm N and the starting function P as defined in Definition 2.1.4.1 we may extend the collapsing function r and the collapsibly-less relation <<~ to all ordinals below ~. Therefore Definition 2.1.4.2 extends to the ordinals below e and we get the same results as in Section 2.1.4. Therefore we need only to know cc ( T I(-< [)) in order to compute specno(NT~r ) . To get this, however, we need to know a little bit more about the relation -<. Call a well-ordering -< a good representation for ~ if its order type is c and there is an order preserving mapping o: field (-<)

satisfying:

> c

W. Pohlers

262 (ol)

(Vm)[o(m)e Lim].

(02)

(Vm < w)[N(o(m)) < P(m) A m < P(N(o(m)))].

Let -~ be a good representation for c. We want to compute cc (TI(-~ r, f ) ) for an /:N-formula F. Let k := rk(F) and put

(k + 1) o(n). Since m -~ n implies am < a~ e Lira and N(am + 4) = g(am) + 4 = 2. (k + 1) + N(o(m)) + 4 < 2. (k + 1) + P(m) + 4 < P(2. (k + 1) + m) < P(N(an) + m) we get m-~n

=~ a m + 4 < < ~ a ~ .

(104)

Because of w _ a , and Y ( n + 1) = n + 1 < P(o(n)) + 1 < P(an) we also have

n + 1 <
(105)

We use (104) and (105) in proving

[~.e_~Prog(~, F), (Vy -~ n)F(y)

(106)

by -~-induction on n. For any m E N we have t

I~ --e--~Prog(-~, F), either by (AxM) with a,~ By (72) we also have '

(Vy -~ m)F(y), ~m ~ n

(i)

0 or by induction hypothesis with a m = am if m ~ n. I

-

[ ~o~ --,F(m), f (m)

(ii)

and obtain [[0~+1 ~Prog(-~, f ) , (Vy -~ m ) f ( y ) A ~ f ( m ) , ~m_ -~ _n, f(m)_

(iii)

from (i) and (ii) by an inference A. By (105) we get

II0~

~Prog(-~, F), ~m _ -~ _n, F(m)

(iv)

from (iii) by an inference 3 and

I1~+4 ~Prog(-~ I0

~

F) ~m -~ n v F(m) ~

- -

__

(v)

from (v) by two inferences (V). Using (104) we finally get

F), (Vy from (v) by an inference V.

(vi) D

Together with the previous section (107) yields that for every sentence F in the theory NT.~r there is an ordinal ~ < c such that

Together with Lemma 2.1.4.6 this means

Set Theory and Second Order Number Theory NT.
263

=~ (3c~ < c)(3m < w)[l~-~ F]

from which we get by cut-elimination and the fact that c is an e-number NT.
=~ (Sc~< ~)[l~ El.

(107)

Hence specno(NT
(108)

Applying Theorem 2.1.4.5 we get the following theorem. 2.2.1.1. T h e o r e m . Let ~ be an e-number and -< a good representation for ~. Then the provably recursive functions of NT.< r are exactly the functions which are elementary in Wa - as defined in (63) - f o r some c~ < c.

2.2.2. Significance of the theories NT.< r A n ordinal analysis of a theory A x yields - among others - the ordinal llAxll < w~ K. W e call an ordinal analysis for a theory A x 2 N T profound if it not only computes llAxll but also provides a primitive recursive well-ordering, say 4, which is a good representation for llAxll such that

Ax~ F

r

N T ~ r ~- F

(109)

holds for all arithmetical formulas. If we have a profound ordinal analysis of A x we know by (108) its II~ and by Theorem 2.2.1.1 also its provably recursive functions. All known ordinal analyses are profound. The general reason for that can be roughly sketched. Recall from Section 1.4 the main steps in an ordinal analysis which are:

9 Designing a semi-formal calculus ~ A which commonly needs a term notation for ordinals. 9 Transforming a formal derivation Ax ~ F into an infinite semi-formal derivation 9 Cut elimination for the semi-formal calculus, yielding ~ F => ~o F. Arithmetizing the term notation gives in general a primitive recursive well-ordering -< which is a good representation for Ilhxll. Unravelling a formal derivation into an infinite one results in a recursive infinite tree. Therefore we may restrict the semi-formal calculus to recursive proof trees. Then there is a recursive predicate, say Proofoo(x, y, z, u), such that Proofoo(e__, ~ , ~f , rA~) expresses that "e is the code of an infinite recursive tree tagged with ordinal notations (i.e., elements in the field of -<,) and codes for finite formula sets which is locally correct with respect to the axioms and rules of the semi-formal calculus witnessing ~ A."

W. Pohlers

264

If we assume ProofAx(e__, rF') then the embedding procedure yields a recursive function g such that

Proofoo(g(e_), n_n_,r, rF~) where n and r are computable from e. This can be done within N T . If we secure that all the manipulations which are done to an infinite proof tree during the cut elimination procedure are locally recursive, we can use the Recursion Lemma to obtain a recursive function, say h, such that

Proofoo(h(g(e_.)), r3~, 0, rFT). Besides N T the Recursion Lemma needs transfinite induction along -~rr37 . Therefore this step can be done within NT.~r. Using the sub-formula property of cut free infinite derivations we obtain

Proofoo(e_, n, 0_, rFT) ~

Truek(rF ~)

by induction on -~ rn where Truek denotes a partial truth predicate for formulas of complexities < k. So this can be done in NT.~r , too. Because of N T ~ - Truek ( rF7) --+ F we get, summing up: hx ~ F

::~ N T ~ ProofAx(e_, F) =~ N T ~- Proofoo (g(e_), n, r, F) =v NT.~r ~ Proo]oo(h(g(e__)), m, O, F) =~ NT.~r ~ Truek(rF ~) =v NT.~ r ~ F

Since we have N T C_ A x and A x ~ Tl(-~ r) we also have the opposite direction NW.~r~-F

=~ A x ~ - F

for arithmetical formulas and the ordinal analysis is profound. Having a profound ordinal analysis for a theory Ax, we can try to sharpen (109) by replacing NT.~r by a more constructive system. If we restrict the scheme of Mathematical Induction in N T to E~ we get the theory E~ Primitive Recursive Analysis - as a second order theory - is a conservative extension of E~ Let -~ be a primitive recursively definable order relation. For m E field(-<) we have the scheme

(PRWO(-
PRWO(-
U m E

field(-<)

PRWO(-
Set Theory and Second Order Number Theory

265

and want to replace NT.~ t by E~ + PRWO(-~ r) in (109). This does not work for arbitrary arithmetical formulas but only for ri~ The proof needs Mints' continuous cut elimination theorem. Let Proofer(e, m, r_, rAT) express that "e is the index of a primitive recursive tree tagged with members of field(-~) (serving as ordinal

notations), numbers (for the cut- rank) and finite formula sets, which is locally correct with respect to the axioms and rules of the semi-formal system (augmented by a replication rule whose premise and conclusion are identical) such that its bottom node is tagged with m (coding the height of the tree), r (coding its cut rank) and A ". By Mints' continuous cut elimination there exists a primitive recursive function, say H, such that E~

~- Proofs

m, r_r_, rA7) --~ Proofs

k, O, rA7)

where k is computable from H(e) in such a way that, provided that -~ is a wellordering, -~rk has order type exp"(2, otyp.~(m)). Giving a sketch of the proof of Mints' theorem would lead us far outside the scope of this contribution. But we want to give a kind of flow chart how to use it in sharpening (109). Thus assume that we have a profound ordinal analysis of A x and let -~ be a good representation for [IAx[[. Let (Vx)(3y)F(x,y) be a ri~ such that A x ~ (Vx)(gy)F(x, y).

(i)

By (109) we thus obtain NT.~r ~- (Vx)(3y)F(x, y).

(ii)

By Theorem 2.1.2.2 and (39) there are formulas F 1 , . . . , Ft which either belong to NT.~r or are identity axioms such that ~0 ~ F 1 , . . . , ~Fz, (3y)F(k, y)

(iii)

for every number k. Looking more carefully at the embedding procedure we observe that the resulting infinitary proof tree is primitive recursive and that an index for that tree can be computed from the formal proof. Since the provably recursive functions of E~ are exactly the primitive recursive ones, this embedding procedure can be formalized within E~ In the next step we observe that all axioms in NT.~r and all identity axioms have primitive recursive proof trees in the semi-formal calculus. The only case in which this is not completely obvious is an instance of Prog(-~, G) --+ (Vy -~ n)G(y). The proof of (106) shows how to construct the tree. Instead of using induction on -~ m E0_IND + PRW0(-~ r) does not know that -~ is w e l l - o r d e r e d - we start with the bottom node and enumerate all possible premises. This gives .., 9 --, Prog(-..<, G ) , --,m -~ n V G(m_.), 9 9 9 -~Prog(~,

G ) , ( V y --< n__)G(y)

Above any of these nodes we decide primitive recursively whether ~m -~ n. If this is

W. Pohlers

266

~Prog(-~, G), -~m -~ n as top node. Otherwise we construct -~Prog(-,:, G), (Vy --,:m)G(y) -~G(m), G(m) -~Prog(-<, G), (Vy 9m)G(y) A -~G(m), G(m)

true then we add

-~Prog(-<, G), G (m__) --,Prog(-.<, C), ~m -,; n v G(m) and repeat the procedure above -~Prog(-~, G), (Vy -~ m)G(y).

Summing up we obtain a primitive recursive function h such that

r~~

~ (W)[P~oof~x(e, ~(3y)F(~, y)') -~ P~oofs

m, ~, ~(3y)F(~, y)')]. (iv)

Together with Mints' Theorem this yields

Ax~- (Vx)(3y)F(x, y) =~ E~ But (3y)F(x, y) is a E~

~ (Vx)[Proofs

m, O, r(3y)F(~, y)7)] (v) A cut free infinitary proof of (3y)F(x, y) cannot

contain instances of a V-rule and is thus finite. Every path in the proof tree is primitive recursive and we may therefore use PRWO(-~ r) to deduce

E~

+

PRWO(-
(vi)

from (v) which in turn entails E~

+ PRWO(-< [) ~-- (Vx)(3y)F(x,

y).

(vii)

By (i) and (vii)we have Ax~ for H~

(Yx)(3y)F(x,y) r

E~

+

PRWO(--~r) i-- (Yx)(3y)F(x,y)

(110)

(Vx)(3y)F(x, y) since the opposite implication holds obviously. O

Call a function f -,:-descendent recursive if it is represented by a function term which is built up from C~, P~n and S by Sub, Rec and the search operator #.~ which is defined by (#.
{ y] ~f (~, Sy) -~ f (~, y) }.

It is not very difficult to show that the provably recursive functions of E~ + PRWO(-~r) are exactly the -<-descendent recursive functions (cf. e.g.,Pohlers [1992] for a proof). Together with Weiermann's result this shows that for a good representation -~ for [[Ax[I a function f is -<-descendent iff it is primitive recursive in Wa for some a < [[Ax[[. A result which can also be proved directly, even under weaker conditions on -~ (cf. Buchholz, Cichon and Weiermann [1994]). A comprehensive study on -~-descendence and proof theory can be found in Friedman and Sheard [1995]. A completely worked out proof of Mints' theorem is in Buchholz [1991]. 3. I m p r e d i c a t i v e

systems

The aim of this chapter is to give upper bounds for the proof-theoretical ordinals of some impredicative axiom systems of Number Theory and Set Theory. We

Set Theory and Second Order Number Theory

267

will restrict ourselves to II~ analyses which are already sufficiently complicated. Moreover, we will also not demonstrate the latest state of the art but restrict ourselves to three axiom systems for Set Theory, K P w , axiomatizing an admissible universe, KP1, axiomatizing a union of admissible universes and K P i axiomatizing an admissible union of admissible universes, and the corresponding axiom systems for Number Theory. Today we know also how to analyse axiom-systems for Mahlouniverses, IIn-reflection a n d - though I have not yet seen the p r o o f s - even for El-separation. T. Arai has announced the analysis of even stronger systems. He uses, however, a different technique which is based on G. Takeuti's methods. 3.1. S o m e r e m a r k s on p r e d i c a t i v i t y a n d

impredicativity

The focus of this contribution is on the ordinal analysis of impredicative systems. In order to distinguish impredicative theories from predicative ones we need a short discussion on predicativity and impredicativity. Limitations of space force us to be rather sketchy. There are two ordinals which characterize a transitive/:(E)-structure ffJl. The ordinal o(gYt) := min {c~ e On I a r if)t} and the least ordinal which cannot be pinned down in 9~t (cf. Barwise [1975,III.7 and VII.3]). We need not to repeat the defnition of "Pinning down ordinals" because we are going to refine it in the following way. Assume that the language s is coded as sets as e.g. in Barwise [1975]. Introduce the notion of an infinitary proof within a semi-formal system for Set Theory as sketched in Section 1.4. Denote by T ~ - F that T is an infinitary proof tree for the formula F. We say that a countable ordinal a is provably pinned down in a transitive s 93~ if there is a well-ordering -~ on w of order type c~ in 9Yr a (possibly infinitary) formula (cf. Pohlers [1989,w for examples of such formulas) Found(-<) in 99~ which expresses the well-foundedness of -~ and an infinitary proof T in 9Yt such that T ~ Found(-<). Define h(D~t) := min { c~ E On] c~ cannot be provably pinned down by 93~}. Of course we always have o(99l) _ h(93~). Now let 99l be the initial part L~ of the constructible hierarchy. Then o(L~) - c~ and we put h(c~) : - h(L~). Then c~ _ h(c0. We call an ordinal c~ autonomously inaccessible if c~ = h(c~). For an autonomously accessible ordinal we have c~ < h(c~) which means that c~ can be provably pinned down by L~. Then we have a formula Found(c~) E L~, expressing the well-foundedness of an well-ordering of order type c~ and a proof T E L~ such that T ~ Found(a). If we denote again , O

by T ~ F that fl is an upper bound for the height of T and the complexity of all formulas occurring in T and p a strict upper bound for the cut formulas occurring in T then there are ordinals fl and p less than c~ such that T ~ F o u n d ( o l ) . If we anticipate that we can construct L~ whenever we have the ordinal e we can interpret autonomously accessible ordinals as ordinals which can be secured by smaller ordinals (cf. Schlfiter [1990] for a fully worked out version of these ideas). The notion of autonomously accessible and inaccessible ordinals is due to Feferman (cf. Feferman [1964]). r

W. Pohlers

268

The Elimination Lemma (Lemma 2.1.2.8) and the Predicative Elimination Lemma (Lemma 2.1.2.9) as well as the Boundedness Theorem (Theorem 1.3.6) carry over. So we get

T~n Found(-<) =~

otyp(-<)<

expn(2,13)

(111)

for n < w and

T ~ Found(-<) =~

otyp(-<)<~Op/3.

(112)

It follows from (111) that w and from (112) that all strongly critical ordinals are autonomously inaccessible. This has first been observed by Feferman [1964] and independently by Schiitte [1965a] who both could also show that these are the only autonomously inaccessible ordinals (cf. Schiitte [1965b]). A proof of this fact which is in the spirit of the above sketch can be found in Pohlers [1989]. In some sense the notion of autonomous accessibility captures the idea of predicativity. First we see that without accepting the ordinal w we stay within the hereditarily finite world. Once we have accepted w as a set we can look for the ordinals c~ which are provably pinned down in L~+I. Then we construct L~, look for ordinals provably pinned down in L~ and so on. This process will stop at the first strongly critical ordinal, i.e., at F0. On the other hand Lro is also exhausted by this procedure. In that sense F0 is known to bound predicativity. We stick to that notation in a very technical manner and call theories whose II~-ordinals are below Fo predicative without further reflection whether there are also possibly stronger principles which can be predicatively justified. But we will see in the following section that there is a completely novel feature in the ordinal analysis of impredicative (i.e., non predicative) systems, collapsing. The simplest theory which needs a collapsing argument in its H~-analysis is the theory of non-iterated inductive definitions which is introduced in the next section. Its ordinal is r (~w1+1), an ordinal which already has been described by H. Bachmann. There are, however, theories whose H~-ordinals are between F0 and r (c~1+1), e.g., the theory ATR introduced by Friedman which axiomatizes autonomous transfinite recursion (which is the axiom (Aut-H ~ introduced on page 276 together with the full scheme of Mathematical Induction). Its ordinal is F, o. Most recently many theories between F0 and r have been analyzed. G. J~iger calls these theories

meta-predicative.

A good summary on predicative theories can be found in the booklet J~iger [1986]. A sample of papers treating meta-predicative theories is J~iger et al. [n.d.], J~iger and Strahm [n.d.], J~iger [1980], Palmgren [n.d.], Strahm [n.d.] and Kahle [1997]. This list has been communicated to me by T. Strahm. 3.2. Axiom systems for number theory In the present section we will introduce some impredicative axiom systems for Number Theory. We will not give an ordinal analysis for these systems directlywhich would be possible in all demonstrated cases - but show that all these systems

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269

can be embedded into axioms systems for Set Theory. The ordinal analysis for the number-theoretic systems will then be obtained via an ordinal analysis of the set-theoretic systems. We start with the most simple example of an impredicative axiom system. 3.2.1. T h e t h e o r y ID1 By a monotone inductive definition on natural numbers we usually understand a monotone operator F: Pow(N)

> Pow(N),

i.e., an operator for which we have S C_ T

=~ F(S) C_ r(T).

A set S c_ N is called F-closed iff F(S) C_ S. We obtain the least fixed-point Ir of F - often called the fixed-point of F - as the intersection of all F-closed subsets of N, i.e., Ir = ~ { S I F(S) c_ S}. A set P C_ 1~ is inductively definable iff it is primitive recursive in the fixed point of some inductive definition. An operator is arithmetically definable iff there is an Z:N formula A(X,x) such that F(S) = {x e N ] N ~ A(S,x)}. If A(X,x) is an X-positive formula, i.e., if its translation into the Tait-language contains no occurrences of t ~g X, then A(X, x) defines a monotonic operator, i.e., an inductive definition. We do not want to go into the theory of inductively defined sets (cf. Moschovakis [1974] and Barwise [1975] for a profound study). All we want to say here is that the fixed-point Ir of an monotone inductive definition comes in stages ICr which are defined by I~r := F(I~ ~) where I~ ~ "= I..Jr ICr 9 By cardinality reasons there is a countable ordinal a such that I~ = I~ ~. One defines

Irl

:-- min {a I I~ - 1~" }

and calls Irl the closure ordinal of F. Thus Ir = II~I. For every element s E I r we may introduce its inductive norm

lair

: - min {~1 s E I~r}.

We then obtain Irl = sup { I~1~ + 11 e I t } . For arithmetically definable operators F we have IFI_ w~K. The theory ID1 axiomatizes the existence of least fixed-points for positively definable arithmetical inductive definitions. Recall the language s of Number Theory. To obtain the language Z:lox we add a set constant IA for every X-positive formula A(X, x) in the language s which contains only the shown free variables. We extend the scheme of Mathematical Induction to all Llox formulas and augment the axioms of N T by the schemes

(IDx) 1 (Vx)[A(IA, X) --+ X elA] and

W. Pohlers

270 (ID1) 2 (Vx)[A(B,x) ~ B(x)]

--+ (Vx)[x 6_ IA -+ B(x)]

where B(x) is an arbitrary s formula. While scheme (ID1)1 expresses that IA is F A(x,x)-closed, scheme (ID1)2 expresses that it is the least F A(x,x)-closed set. The standard semantics for s is obtained interpreting IA by IA. Instead of giving a direct ordinal analysis for ID1 - which is possible e.g. cf. Pohlers [1989] - we will show that it can be easily embedded into axiom systems for Set Theory. 3.2.2. I t e r a t e d i n d u c t i v e definitions The expressive power of first order logic with free set parameters is of course not exhausted by the axiom system ID1. As soon as we have fixed-points of inductive definitions we may use them in the definition of new operators. We are then leaving the realm of 'elementary inductive definitions' on the structure N in the sense of Moschovakis [1974]. To formalize the iteration let -< be a well-ordering of order type v and associate a binary predicate constant JA to every X-positive s formula A(X, Y, x, y) which contains at most the shown free variables. We are going to write a 6_ JbA instead of (a, b) 6_ JA; by a 6_ j~b we abbreviate the formula (=ly -< b)[a 6_ J~]. The language s is obtained by augmenting the language s by a constant for -< and all constants JA. Denote by LO(-<) the formula which is saying that -< is a linear ordering. We obtain the axiom system IDv by taking all the axioms of N T and adding

(TI,,) (ID,,)

LO(-<) A TI(-<,F) (Vy 6_field(-<))[(Vx)[A(JYA, J~,Y,x, y) --+ x 6_ J~]]

and (IDa) 2

(Vy6-field(-<))[(Vx)[A(B,J~Y,x,y)--+ B(x)] --+ (Vx)[x e J~. -+ B(x)]]

where F and B(x) are arbitrary s Since one inductive definition corresponds to one hyperjump the axiom system ID~ may also be interpreted as the system for v-fold iterated hyperjumps. The constructive number classes (.9, for # _< v can be defined in ID~ and their basic properties can also be proved there. We define

ID
UIDe ~
where every ~ < v is represented by a proper initial segment of -<. One may also combine the axioms ID1 and the systems ID~ to obtain the system ID.<. axiomatizing the iteration of inductive definitions along the accessible part of a linear order -<. This is done by taking some arithmetically definable order relation -< and choosing the X-positive formula

A.<(X,x) :r

(Vy)[y -< x --+ y e X].

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271

Its fixed-point ]A~ is called the accessible part of -~ and usually denoted by Acc.~. The axioms of ID.~. are (Acc) 1

(Vx)[A.~(Acc.~,x) ~

x e Acc.~]

(corresponding to (ID1)1) (Ace) 2

(Vx)[A.~(B, x) --+ B(x)] --+ (Vx)[x e Acc..< --+ B(x)]

(corresponding to (ID1)2).

(IDa.)1

(Vy e Acc.~)[(Vx)[A(J~, J~Y, x, y) --+ x e J~]]

modifying (IDv) 1 and (ID.~.)2 (Vy e Acc.~)[(Vz)[A(S, J~Y,x, y)--+ B(x)]

~

(Vx)[x e Acc.~ --+ S(x)]]

modifying (IDv) 2. All these systems of iterated inductive definitions have been introduced by Feferman. There are even stronger iterations of inductive definitions which, however, can be more elegantly formulated within the framework of Second Order Number Theory. 3.2.3. I t e r a t e d i n d u c t i v e definitions in second order In full Second Order Logic we do not have a recursively enumerable notion of provability. Therefore we have to fix a calculus and regard the proof strength of an axiom system relatively to that calculus. This means that we rather use two sorted first order logic than full second order. To fix a calculus we assume that we have a second order Tait language as introduced in Section 1.3. We extend the calculus of Definition 2.1.2.1 by the following second order rules:

(32)

If ~

A, A(X), then ~-- A, (3Y)A(Y) for all m > mo

(V2)

/f ~-~ A, A(X) and Z not free in any of the formulas in A, (VY)A(Y), then

A, (VY)A(Y) for all m > mo. We say that a formula F is provable from an axiom system A x iff there are finitely many instances of identity axioms G 1 , . . . , Gm and finitely many sentences {A1,. . . ,An} C_ A x such that ~-- ~G1,. .., ~Gm, -~A1,. . ., ~ A n , F for some m. The strongest axiom system for Number Theory is NT2 which comprises all the axioms of N T together with the axiom schemes

(CA)

(3X)[(Vx)(x e X ~ F(x))]

(AC)

( V x ) ( S X ) F ( x , X ) -+ (3Y)(Vx)F(x, Yx)

of comprehension and

of choice. We put y e yx :r (y, x / e Y and assume tacitly that F(x) and F(x, Y) must not contain the variable X. If the formulas in the schemes (CA) or (AC) are restricted to a complexity class jc we talk about (~'-CA) and (3v-AC), respectively. By (.T-CA) we denote the

W. Pohlers

272

axiom systems which comprises all the axioms of N T extended to the second order language together with the scheme (Jc-CA). Analogously we denote by (.T'-AC) the axioms of N T together with (Jc-AC). We will also regard axiom systems which are closed under rules. A rule has the form

(R)

F1,...,Fn ~ F

and we say that a theory Ax is closed under the rule (R) if Ax ~- Fi for i = 1 , . . . , n implies F E Ax. For a given rule (R) we define

The least s the rule (R).

(R)

which comprises N T -t-(HI-CA) and is closed under

Observe that the theory (R) is the union of the theories (Rn) where (R0) - N T + (II~-CA) and (1~+1) is obtained by closing (P~) under all applications of the rule

(R).

In NT2 we may replace the scheme of Mathematical Induction by a single axiom (VX)[0eXA (Vy)(yeX--+y+leX)

-+ (Vx)(xeX)].

This is no longer true if we regard axiom systems with restricted comprehension scheme. Therefore we introduce also the axiom systems (bw- CA)o and (.%'- AC)o in which the scheme of Mathematical Induction is replaced by the single axiom. The formula

(VX)[Prog(-<,X) -+ (Vx)(x e X)]

Wf(-~) :r

expresses the well-foundedness of the relation -~. We have Wf(-~) r

(VX)[(3x)(x e X) --+ (3x e X)(Vy-~ x)(y r X)].

(113)

In NT2 the sentence Wf(-~) entails the scheme TI(-~,F) for arbitrary s formulas F(x). This, too, is not longer true for restricted comprehension. Therefore we introduce the scheme

(BI)

Wf(-~)-~ TI(-<,F)

of Bar Induction for definable relations -~ and arbitrary s F(x). A relation -~ is definable iff there is an s G(x, y) such that x -~ y 6+ G(x, y). The formula G(x, y) may contain additional parameters. We will sometimes emphasize this by writing

x "~c,~,2 Y :r

G(x, y, ~, Y,).

Roughly speaking Bar Induction says that a relation which is well-founded with respect to sets is also well-founded for classes. If the defining formulas for the relation -~ in the scheme (BI) are restricted to the complexity class ~" then we talk about (~'-BI). If also the complexity of the allowed classes is restricted to another complexity class Jc2 we notate that as (.T'-BI)r.T'2. If X is a set parameter we may define the binary relation

Set Theory and Second Order Number Theory x.
273

:4=~ (x,y) E X .

Sometimes Bar Induction is formulated as the single axiom

(Bi)

(VX)[Wf(.
which in the presence of

(I]~-CA) has the strength

of

(n~-BI). Weaker than

(BI) is

which is known as Bar Rule. M. Rathjen has shown in Rathjen [1991] that the Bar Rule is of the same strength as parameter free Bar Induction, i.e., the axiom of Bar Induction in which the defining formula for the relation .< must not contain parameters (not even individual parameters). As a basis for nearly all our second order axiom systems we will use (H~-CA), i.e., the scheme for arithmetical comprehension. This system is also known as (ACA). We use both notions interchangeable. Observe that (ACA) proves that a relation .< is well-founded iff it does not contain an infinite -<-descending sequence, i.e.,

Wf(-<) 4=~ -,[(3X)[(3x)(x e X) A tO(-< rX) A (Vx E X)(3y e X)(y -< x)]]. (114) The axiom system (ACA) proves also the equivalence of the schemes (Bi), (IIoLBI) and the following quantifier scheme (cf. Feferman [1970]). (QS)

(VX)A(X) --4 A(F) .for arithmetical A(X) and arbitrary F(x).

Because of this equivalence it has become common to call (QS) also (II~-BI). The iteration of inductive definitions is quite elegantly expressed in a second order language. For an X-positive formula A(X, Y, x, y) we introduce the abbreviation

CIA(X, Y, y) :~::~ (Vx)[A(X, Y, x, y) -+ x 9 X] and define ITA(-<,X) :r162(Vy e field(-<))[CIA(XY, X~Y,y) e

^

e

y))].

Then ITA(-<,X) says ' X is an hierarchy of fixed-points for A(X,Y,x,y) iterated along -<'. 3.2.3.1. Definition. Let .< be a primitive recursive well-ordering of order type v. For an X-positive arithmetical formula A(X, Y, x, y) we introduce the schemes (IT,,)

(3X)ITA(.<,X)

and (B-ITv) (VY)(Vy e field(.<))[/TA(.<, Y) A CIA(B, y-
We define the theories

ID~ := ( A C A ) + TO(.<)+ (ITv) and

W. Pohlers

274

BID~ := (ACA)+ WO(-K)+ (IT.) + (B-IT.)

where WO(-K) stands for LO(-K) AWf(--<) and A varies over all X-positive formulas A(X, Y, x, y) which contain at most the shown variables free. Denote by

(ITA)

(Vx)[WO(-.
the scheme in which F(u, v,x) and A(X, Y, x, y) are supposed to vary over arithmetical formulas without further parameters. We define ID 2. := ( A C A ) +

(ITA).

Finally put

(B-ITA) (Vx)(Vy)(VX)[WO(-KF,=) A ITA(--KF,=,X) A CIA(B,X'
where F(u, v,x) and A(X, Y,x,y, ) are as above and B(x) is an arbitrary s formula. The schemes (B-IT,) and (B-ITA) have the flavor of Bar Induction; therefore the B in their identifiers. Let

BID 2. := (ACA)+ (ITA) + (B-ITA). One should observe that BID 2. allows only iterations along arithmetically definable well-orderings, i.e., along well-orderings of length < w~K. So it may appear weaker than ID.~. which allows iterations along accessible parts of arithmetically definable orderings, i.e., along well-orderings of length < w~K. However, we will sketch in a moment that B I D s* comprises ID.~.. If we drop the restriction to arithmetically definable well-orderings we obtain the schemes

(IT) and (B-IT)

(vx) [wo(-< ) -+

r)]

(Vy)(VY)[WO(-
respectively, where F(u, v), and S(x) may now be arbitrary s and G(X, Y,x,y) is an X-positive arithmetical formula. All formulas may contain additional parameters. We put A u t - I D := ( A C A ) + (IT) and (Aut-BID) := ( A C A ) + (IT) + (B-IT). One easily shows (YI~-CA) ~ WO(-<) A ITA(-<,Y) A ITA(-<,Z) --+ (Yx e field(-<))(Y = = Z=). (115) There is an canonical embedding F ~ F* from the language of IDv into the language of IDZu. We replace every occurrence of s e J~t by (3X)[ITA(-~,X) A s e Xt]. The

Set Theory and Second Order Number Theory

275

theory ID~ shows that there is a set WA such that ITA(-<,WA) and by (115) we obtain that W~ is uniquely defined for t -< v. Hence (s E J~)* iff s E W t for all t E field(-<) which entails J~* = W t and j~t* = W.~t. Since we have CIA(W~, W~ t, t) we get (IDa) 1 * and from B-IT~ also (IDa) 2. So we have shown ID~ C_ BID~.

(116)

An embedding F ~ F* of the language of IDa, into the second order language is obtained similarly. Define

IU(-~,x)

:~

(v=__y)[clA(x',x~',=) ^ (vY)(clA(y,x~',=) ~

(Vz)(z e x 9 -+ z e Y))]

and let X c_ Y stand for (Vx)[x E X ~ x e Y]. Define

Acc(-<,X)

:r

Prog(-<,X) A (VZ)[Prog(-<,Z) -+ X C_Z]

and replace all occurrences of s E Acc~ by (3X)[Acc(-<,X) A s E X] and finally replace all occurrences of s E J~4 by (3X)[IT~(-<,X)A s E Xt]. We indicate that the translations (Acc) 1., (Acc) 2., (lb.<,) 1. and (ID.~,) 2. are all provable in B I D 2.. We argue informally in B I D 2.. Let x _<8 y :r x -< y -< s. Put Ao(X, Y, x, y) :r (Vz)[z -< x -+ z E X]. Then there is a set T such that ITAo(-<~ T) if 0 denotes the least element in -<. We then have Prog(-<,T ~ and define S "- T ~ Moreover we have Prog(-<,X) --+ S c_ X and thus Acc(-<,S). So Acc:<= S and we get (Acc) 1. by Prog(-<,S) and (Acc) 2. from (B-ITA). To prove (lb.<,) 1. we assume s E Acc:<, i.e., s E S. Then we obtain Wf(-<8) because otherwise according to (114) there would be a non empty set Y c_ field(-< 8) containing s and an infinite -<-descending sequence. Since -< is a linear order this entails that if all -<-predecessors of an element x do not belong to V then x cannot belong to V. That means Prog(-<,~V) where ~V denotes the complement of V. But then S c_ ~V which contradicts s E V. Now choose any X-positive formula A(X, Y, x, y). By ITA there exists a set W such that ITA(-<8, W). By the uniqueness property (115) we obtain (Vy)[y --< s --+ (Vx)(x E J~4)* ++ x e WV]. Hence (J~)* = W 8 and (j]8), = W.<8 and we get (lb.<,) 1. from ITA(-
ID.~, C_ BID 2..

(117)

As remarked before the schemes (B-IT~), (B-ITA) and (B-IT) have the flavor of Bar Induction. We are going to substantiate this remark. First we show A x E {BID~, B I D 2., A n t - B I D }

=> A x ~- Bi.

(118)

Assume Wf(-
W. Pohlers

276

ID~ + (II~-BI) I--(B-IT,.,) ID2*+ (n~-Bl) I-- (B-ITA) Aut-ID + (II~-BI) ~ (B-IT).

(119)

Assume Wf(-.<), ITA(-<,X) and CIA(B,X'
= {~1

( V X ) [ C t A ( X ) -+ 9 e

--+ x e X]}

(120)

X]}

Vice versa, every II{-set can be shown to be an inductive set, i.e., a set which is primitive recursive in some fixed-point. So (II{-CA) and inductive definitions are canonically connected. To iterate (II{-CA) we introduce the following notations 3.2.4.1. Definition. Let H(X, x, y) be an s shown parameters. We define

JH(-<,X) :r

which contains only the

(Vx)(Vy)[x e X u <-+H(X'
and call X the jump hierarchy based on H(X, x, y) along -<. For a primitive recursive well-ordering of order type u we introduce the scheme (H{-CA~) (3Z)&(-<, Z) .for H(X, x, y) e II{ of u-fold iterated II{-comprehension. We sometimes express this sloppily by (~Z)JH(I},Z) if we do not want to emphasize the order-relation but its order type. Transfinitely iterated H{-comprehensions are axiomatized by (Aut-II{)

(VX)[Wf(-Kx) --+ (3Z)JH(-Kx, Z)] for H ( X , x , y ) E H{.

For a primitive recursive ordering -< of order type u we define (II{-CA,,) "= ( A C A ) +

WO(-<)+(II{-CA~),

(II{-CA<~) "= U (II{-CA~)

Set Theory and Second Order Number Theory

277

and

(Aut-H~) .= ( A C A ) + (Aut-I1]). Notice that (Ill-CA) and (H]-CAv) have different meanings. Due to the possible presence of set parameters in l-/~-comprehension formulas we have

(IIx~-CA) = (II~X-CA<~) though notationally this looks strange. Recall that (H~- CA)o means (II~-CA) + (ACA)o, i.e., (1-I}-CA) together with the axiom of Mathematical Induction. The theories (II~-CA~)o are defined analogously. We will see that the theories (H~-CA~) and ID~ are equivalent. In a first step we show (ID~)o C_ (II~-CA~)o

and (Aut-ID)o C_ (Aut-II~)o.

(121)

Let -~ be a well-ordering of order type u or assume I/Vf(-~). Let A(X, Y, x, y) be an X-positive arithmetical formula and put

H(X,x,y)

:r

(VZ)[C]A(Z,X,y) --~ x e Z].

(i)

Then g ( X , x , y) e II} and by (II}-CA)v or (Aut-II}) there is a set S such that JH (-~, S). Hence

S y - {x I H(S'~U,x,y)} = {x I (VX)[CIA(X,S'~V,y) --+ x e X]}

(ii)

which in turn implies

(VX)[CIA (X, S "
(iii)

From (iii) we obtain

A(SV, S~V,x,y)--4 C[A(X,S'
Sy C

X

-+ A(X, S "~y,x, y) ---~xEX by the monotonicity of the X-positive formula A(X, Y, x, y). But (iv) means

A(S y, S'~Y,x, y ) +

x

e S y.

(iv)

(v)

Pulling (iii) and (v) together we obtain ITA(-~, S).

D

To prove also the other inclusion in (121) we use the fact that already (ACA)o proves that every II~-formula H(Y, y, 53 is equivalent to the well-foundedness of its associated tree of unsecured sequences, i.e., that there is an arithmetical formula TH (]I, X, y, ~ such that (ACA)o ~ H ( Y , y , ~

++ {x] TH(Y,x,y,~}

is a well-foundedtree.

Defining A(X, Y, x, 5) :r (Vy)[TH(Y, (x)ff" (y), (X)l, z-) -4 ((x)ff" (y), (x)l) e X] we have an X-positive arithmetical formula such that

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278

H(Y, y, 5) <-~ {x[ TH(Y, x, y, z-)} is well-founded ++ <<>,y> e U(~;~ +~ (vz)[c&(z, Y, z-) -+ (<>,y>e z].

(122)

The last equivalence is provable in (ACA)o. To obtain therefrom the opposite inclusion in (121) let H(Y, x, y) be a II}-formula and A(X, Y, x, y) the arithmetical formula such that according to (122)

H(r,x,y)

++ (VZ)[CIA(Z,Y,y)-+ ((>,x> 6 Z].

(i)

Define A'(X,Y,x,y)

:~=>A(X, {z I ((),z) 6 Y},x,y) andlet 9 eitherbyaprimitive recursive well-ordering of order type u or assume WO(-<). Then either ID~ or A u t - I D prove the existence of a set S such that ITA,(-<, S), i.e., CIA(Su, {z I ((>,z> 6 S'~U},y)

(ii)

(vz)[cl,(z, {~1 <<>,z> e s~,},y) ~ s , c z].

(iii)

((),x) 6 S y --+ (VZ)[CIA(Z, {z I ((),z> 6 S' 6 Z] H({z I ((>,z> 6 S'
(iv)

and

Hence

and

H({z I ((>,z> 6 S~Y},x,y) -+ (VZ)[CIA(Z,{z] ((>,z> 6 S~Y},y) -+ ((),x> 6 Z] (v) -+ <<>,~> e s,. So we have ((),x) 6

Sy

~

H({x I ((),x) 6 S'
and putting T := {(x,y>l ((>,x) 6 S ~} we obtain JH(-'<,T). (=~Z)JH(~, Z) is a theorem of ID~ or A u t - I D , respectively.

(vi) Therefore [3

So we have together with (121) and Theorem 3.2.3.2 the following theorem. 3.2.4.2. T h e o r e m . (YI~-CAv)o = (IDa)o, (Aut-YI~)o = (Aut-ID)o, (II~-CA~) = IDa, (YI~-CA~)+(Bi)= ID~ + ( B i ) = BID~, A u t - I D = (Aut-YI~) and A u t - B I D = A u t - I D + (Bi) - (Aut-n~) + (Bi). Regarding (116) we obtain the following chain ID~ C_ BID~ = (YI]-CA~)+ (II01-BI).

(123)

Feferman [1970] has shown that for u = wp with p > 0 the theory ID~ proves the existence of an w-model for (II~-CA
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w-model for all IDa, ~ < v. The scheme (II~-BI) is not needed here since the translation of every/:lD~-formula F is arithmetical in some X y with JH(-4, X). The translation of B-IT~ is thus obtained by (II~-CA). Writing _ for proof theoretical reducibility we get for t, - coP, p E Lira the following chain. ID<~ C_ B I D ~ = (II~-CA<~)+(II~-BI) _ ID<~ _< (YI~-CA<~) _< ID<~. (124) This shows that all these theories are proof theoretically equivalent. To close the section we mention the results of H. Friedman [1970] who showed (A~-CA) = (II~-CA<~o) = (]E21-AC)

(125)

where (A~-CA)

(Vx)[A(x) ++ B(x)] --+ (3X)(Vx)[x e X ~ A(x)]

for A(x) e I11 and B(x) E P~21is the scheme of A21-comprehension. A simpler argument than that in H. Friedman's results is given in Feferman [1970] to characterize the Al-comprehension rule. To define the rule let the class of 'essentially' I1~ formulas be the smallest class of formulas which contains the arithmetical formulas and is closed under the positive boolean operations A and V, first order quantification and second order V-quantification. Dually the class of essentially P~-formulas is the class of formulas whose negation is logically equivalent to an essentially II~-formula. Analogously we obtain the class of essentially II~-formulas when we start with the essentially P~-formulas instead of arithmetical formulas and the class of essentially ~l-formulas as its dual class. The A 1 comprehension rule is defined as follows. (A2LCR)

(Vx)[A(x) ++ B(x)] ~- (3X)(Vx)[x E X ++ A(x)] e A x / o r A(x) essentially II~ and B(x) essentially E~.

Feferman shows (A~-CR) - (YI~-CA
(126)

3.3. A x i o m s y s t e m s for set t h e o r y 3.3.1. T h e a x i o m s y s t e m K P w We introduced the axiom system K P w already in Section 1.2. For ordinal analysis it will be more convenient to restate the axioms of K P w in a more parsimonious way. We keep (Ext) and modify the pairing axiom (Pair) to (Pair')

e z ^ y e z].

It is obvious that (Pair) follows from (Pair') by A0-separation. In a similar way we modify the axiom of union to (Union')

(Vu)(Bw)(Vy e u)(Vz E y)[z E w]

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an axiom which requires only the existence of a superset of the union. Again it is clear that we obtain (Union) form (Union') by A0-separation. Similarly we modify the axiom of infinity to (Inf') (3u)[u r 0 A (Vx e u)(3v e u)(x e v)]. For convenience we introduce an axiom system B S T for Basic Set Theory. It comprises the axioms (Ext), (Pair'), (Union'), (A0-Separation), and the scheme (FOUND) of foundation. Adding (Inf') to B S T we obtain the system B S T w . All systems we will regard here are based on BST. If A x is such a system we denote by A x r the system which is obtained by restricting the foundations scheme (FOUND) to A0-formulas. The system W - A x is the intermediate system between A x ~ and A x in which we have (A0-FOUND) but allow full Mathematical Induction, i.e., the scheme F(0) A (Vx e w)(F(x)--+ F(x + 1)) --~ (Vx e w)F(x) for arbitrary formulas F. Adding the scheme (A0-Collection) to the axioms in B S T we obtain the axiom system K P , adding it to B S T w the system K P w These systems are thoroughly studied in Barwise [1975]. We list some of the most important properties of the system K P w without giving proofs. All proofs can be found in Barwise [1975]. Many of these properties are already provable from axioms which are weaker than K P w ~. However, we do not have enough space to go into more details here. We use class-terms of the form {x I A(x)} freely though they are not regarded as terms of the language. The formula z e (x I A(x) } is an 'abbreviation' for A(z).

3.3.1.1. E-Persistency Lemma.

F aAaCb-+F

b andF a - + F .

3.3.1.2. E-Reflection K P ~ ~ F -~ (3a)F a.

Theorem.

Let F be a P~-formula. Then K P r proves For every E-formula

F

we have

As a consequence of P~-Reflection we obtain that in K P ~ every P~ formula is provably equivalent to a E1 formula. 3.3.1.3. P~-Collection Theorem.

For every E-formula F(x, y) we have

K P r ~- (Vx e a)(3y)F(x, y) --+ (3z)(Vx e a)(3y e z)F(x, y). 3.3.1.4. E-Replacement Theorem.

For every P~-formula A(Z, y) we have

K P r ~ (V~ e u)(3!y)A(~, y) ~ (3f)[Fun(f) A dora(f) = u A (V~ e u)A(~, f(~))]. 3.3.1.5. A - S e p a r a t i o n T h e o r e m .

Then

Let A(x) be a II and B(x) be a E formula.

K P r ~ (Va)[(Vx e a)[A(x) ~ B(x)] --+ (3z)(Vx e a)[x e z ~ x e a A B(x)]]. There are many basic relations which are A0-definable (cf. Barwise [1975] for details). The fact that a is an ordinal, e.g., can be expressed by Tran(a) A (Vx e a)Tran(x).

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Similarly we can express by On(a) A (3x E a)[x E a] A (Vx E a ) ( 3 y E a)[x E y] that a is a limit ordinal. The usual basic notations as Rel(r) (r is a relation), Fun(f) (f is a function ) are A0 definable. See Barwise [1975,pp.14-29] for a more complete list. If F ( X l , . . . ,xn) is a A-formula of K P w then we may introduce a new relation symbol R together with its defining axiom (Vxl)... (Vx,)[R(Xl,... , x , ) ~ F(Xl, . . . , x,)]. Adding defined A relation-symbols to the language/:(E) will not alter the class of E - and A-formulas. If F ( X l , . . . ,x~, y) is a E-formula such that K P ~ (Vx~)... ( V x ~ ) ( 3 ! y ) F ( x ~ , . . . , x n , y ) then we may add an n-cry function symbol F to the language of K P together with its defining axiom (Vxl) .-. (Vz,)(Vy)[F(Xl, . . . ,xy)

=

y ~

F ( X l , . . . ,xn, y)].

Extensions by definitions of E function-symbols will also not alter the class of A and E-formulas of K P w . Details about 'Adding Defined Symbols to K P ' can be found in Chapter 1 5 of Barwise [1975]. One of the most important theorems of K P is the following E-Recursion Theorem. 3.3.1.6. E - R e c u r s i o n T h e o r e m . Let G by an n + 2 - c r y E function-symbol o.f K P . Then there is an n + l - c r y E - f u n c t i o n symbol F of K P such that K P ~- F(~, a) = G(~, a, U F(~, ~)). The above E-Recursion Theorem is a special case of the more general ]E-Recursion Theorem as stated in Barwise [1975]. Because of the axiom of infinity we obtain K P w ~ (3a)Lim(a) and thus a A0-definition of w as a point by Lim(w) A (Vx E w)[-,Lim(x)]. Stronger than E-reflection is the H2-reflection scheme (n2-Ref)

F

--+ (3a)[a r 0 A F a] .for n2 formulas F.

Observe that any model of K P w in the constructible hierarchy already satisfies II2-reflection. To see that let F - ( V y ) ( 3 y ) A ( x , y) be a II2-sentence and assume L~ ~ K P w . For a E L~ there is a least fl0 < a such that a E L/~o and we define fln+l to be the least ordinal such that L~ ~ (Vx E L/~,)(3y E L/~,+I)A(x, y). The ordinal fln+l exists by E-Reflection. This defines a sequence (/~n;n E w) which is E-definable in La. Hence fl := sup {flnl n E w} < a and we have La (Vx E L~)(3y E L~)A(x, y). Due to the ~-Recursion Theorem we can prove the existence of the stages of inductively defined sets in K P w . Let S be an additional unary predicate symbol. We write a E S instead of S(a). We obtain the stages of an inductive definition as stated in the following theorem.

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3.3.1.7. T h e o r e m . Let B(~, y, S) be a A-formula of K P w . Then there is a function-symbol Is such that gPw

t---Is(a,:~') = {yEXll B(~.,y,{zExll (3r

E Is(~,i')]})}.

Putting G(~, y, S) "= {y E Xll B(~, y, S) } we observe that G possesses a ~ definition and get the theorem immediately from the ~-Recursion Theorem. D A A-formula B(~, y, S) and a tuple b of sets induce an operator

Fs,~: Pow(bl)

) Pow(bl)

:= {y e bl I B(b', y, S)} which depends on the parameter list b. Again we say that S occurs positively in a formula F(S) if the formula corresponding to F(S) in the Tait-language (cf. Section 1.3) does not have occurrences of the form t r S. We sometimes denote this by F(S+). For S-positive formulas B(~, y,S) the associated operator Fs, ~ is monotonic, i.e., we have S c T ~ Fs,~(S) C_ Fs,~(T). We say that a set is closed under an operator F if we have F(S) C_ S for all S. For an monotonic operator F: PoT(b) > PoT(b) we obtain its least fixed-point as the intersection of all F closed subsets of b. The fact that a class T "= {x E bll A(x)} E Pow(bl) is Fs, ~ -closed can be expressed by the formula

CIs(b,T) :_= (Vy E bl)[S(b, y, T) -+ y E T]. Defining Is(/~') := {x e bl I (3r 3.3.1.8. T h e o r e m . gPw

~

(127)

e Is(~r b)]} we obtain the following theorem.

Let B(Z, y, S) by an S-positive A formula of K P w . Then

CIB(b, IB(b'))

and K P w t- CIs(b, A) ~

(Vr

E IB(~, b') ~ A(x)].

It follows from Theorem 3.3.1.8 t h a t IB(b') is the least fixed-point of the operator las,b .. To prove the theorem pick a tuple b', c e bl and assume S(b', c, Is(b)). Since S(~, y, S) is S-positive the formula S(b, c, Is(b')) is still a E-formula and by ~ Reflection we obtain a set d such that S(b,c, {x e bll (3~ e d)(x e Is(~, b))}). Now define fl := U { ~ e d I ~ e O n } and a := flU{fl}. Then a is a set by A0Separation, Union and Pair such that {~ E d I ~ E On} C_ a. By E-Persistency it follows S(b,c, {x e bll (3~ e a)(x e Is(~r b'))}) and by Theorem 3.3.1.7 we obtain c e Is(a, b), i.e., c e Is(b'). This proves the first part of the theorem. For the second part we show

s)

c s

(i)

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283

Assuming the hypothesis CIs(b, S) we obtain by induction

U IB(r b) c S. r

(ii)

The monotonicity of the induced operator therefore implies (Vx E bl)[B(b', x, U IB(r b')) =~ B(b', x, S)]. r By Theorem 3.3.1.7 and the hypothesis CIB(b , S) we get (i) from (iii).

(iii) [3

It follows from Barwise [1975] that every primitive recursive function has a A 1-definition in K P w . Therefore we may add function symbols for all primitive recursive functions to the language of K P w . So we may regard the second order language of NT2 as sublanguage of K P w . We mentioned that already in Section 1.3. Interpreting the language /:~ as a sublanguage of s augmented by E function symbols (call this language s turns first order formulas with set parameters into A0 formulas (which in turn are A for the theory K P w ) . It is obvious that the scheme of Mathematical Induction can be easily derived from the Foundation Scheme. All defining axioms for primitive recursive functions are provable in K P w . By Theorem 3.3.1.8 we may also interpret the additional constants lB. So we get the following theorem. 3.3.1.9. T h e o r e m .

The theory ID1 viewed as a theory in the language s

is a subtheory of K P w . 3.3.2. T h e t h e o r y KP1 We are now introducing the theory KP1 which axiomatizes a set universe which is the union of admissible universes. Therefore we augment the language of Set Theory by an additional constant Ad whose intended interpretation is that of an admissibility predicate. The defining axioms for Ad are (Adl)

(Vu)[Ad(u) --+ w e u A Tran(u)]

(Ad2)

(Vx)(Vy)[Ad(x) A Ad(y)--+ x e y V x = y V y e x]

(Ad3)

(Vz)[Ad(x) --+ (Pair') x A (Union') x A (A0-Separation) ~ A (A0-Collection) ~]

3.3.2.1. Definition. The theory KP1 is the system B S T w together with the axioms (Adl) - (Ad3) and the axiom (Lim)

(Vx)(3u)[Ad(u) A x e u].

Let ,~. gt~ enumerate the class Reg U {0} of admissible ordinals (augmented by 0) and their limits. Then the smallest constructible model of KP1 is LN~, i.e., IlgPlll~ = ~ . Since we have g P 1 ~ Ad(u) ~ F ~ for every sentence r e g P w we obtain

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3.3.2.2. L e m m a . K P w r ~- F =~ K P F [-- Ad(u) --+ F ~. Also, K P w ~ F =~ KP1 ~ A d ( u ) ~ F ~. As in K P w the most important theorem in KP1 will be the E-recursion theorem. But we do not have full &0-collection in in KPI. Therefore we will obtain E-recursion only in a relativized version. Let F be a collection of new function symbols. Denote by KPI(F) the theory K P I formulated in the language s together with defining axioms for the function symbols in F. Let A(~,y) be a E-formula such that KPI(F) ~ (V~)(3!y)A(~,y) and KPI(F) ~ (Vu)[Ad(u) --+ (V~ e u)(3y e u)A(~,, y)]. Then we introduce a new function symbol G and its defining axiom

(Da)

(VZ)(Vy)[G(Z) = y ~ A(Z, y)]

and call G a relative E-function symbol of KPI(F). Let KPI(G) be the L(E,FU {G})-theory KPI(F) + DG. By the common techniques we obtain that KPI(G) is an extension by definitions of KPI(F) in the following strong sense. 3.3.2.3. L e m m a . For every formula F(~) in the language f_,(E, G) there is a formula Fo (~) in the language f_,(e, F) such that KPI(G) ~-(VZ)[F(Z) ~ F0(Z)].

If F(Z) is a E-formula of s

G) then Fo(Z) is a E-formula of s

such that

KPI(F) [--(Vu)[Ad(u)--+ (VZ e u)(Fo(Z) ~-~ F0(Z)u)].

If F(Z) is a &o-formula of f-,(e, G) then there is moreover also a H-formula FI(Z) such that KPI(G) ~-(V~')[F(:~) 4+ F~(x)] KPI(F) ~-(Vu)[Ad(u)~ (VZ e u)(Fo(Z) ~ Fo(Z) ~ ~ FI(Z)~)]. Iterating Lemma 3.3.2.3 we see that we can identify the theory KP1 with its closure under extensions by definitions of relative E-function symbols. The E-Recursion Theorem can now be modified in the following way. 3.3.2.4. Relativized E - R e c u r s i o n T h e o r e m . Let G be an n + 2-ary relative E-function symbol of K P F . Then there exists an n + 1-ary relative E-function symbol F such that K P F ~ F(~, a) = G(~, a, U F(~, ~)) To prove the theorem we follow Sarwise [1975] as far as possible. Let C(~, a, v, f) be the A0-formula (a ~ O n A f = O A v = O) V (a E O n A F u n ( f ) A dom(f) -- a A (V~c e a)[f(~r - G(~,r162 f(r A v = G(~,a,(.Jce a f(r We then show g P l " ~ (V:~)(Va)(3Iz)(3 f)C(:~, a, z, f)

(i)

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and KP1 ~ ~ (Vu)[Ad(u) --~ (V3 e u)(Va e u)(3z e u)(3f e u)C(3, a, z, f)]

(ii)

and introduce a new relative E-function symbol F and its defining axiom (V3)(Va)(Vz)[F(3, a) = z ~ (3f)C(3, a, z, f)].

(iii)

To prove (i) and (ii)it suffices to show that g P l r proves

C(3, a , z , I ) A C(3, a , z ' , I ' ) ~ z = z' A I = I'

(iv)

Ad(u) A ~ e u A a e u --+ (3z e u ) ( 3 / e u)C(~,, a, z, I).

(v)

and

We prove (iv) and (v) by induction on a. Since C(~, a, z, f) is A0 we can formalize this in KP1 ~ where we have (A0-FOUND). The proof of (iv) is exactly the same as in Barwise [1975]. For the proof of (v) we also follow Barwise [1975] but use the additional observation that G(3, a, f) E u whenever u is admissible and ~, a, f C u. We do not have full E-replacement in KP1 ~ but we may use its relativized version according to Theorem 3.3.1.4 and Lemma 3.3.2.2 whenever it is used in Barwise [1975]. Once we have the relative function symbol F we may proceed literally as in Barwise [1975]. [3 There is a relativized version of Theorem 3.3.1.7. Let B(3, v,S) be a A0-formula. Then we obtain a relative E-function symbol G such that G(3, v, s) = {y e xl I B(3, y, s)} and we apply the relativized E-recursion Theorem (Theorem 3.3.2.4) to obtain a relative E-function symbol IB such that Is(c~,3) = { Y e X l l B(3, y , { z e x l l is provable in g P l r. Defining 1~(3) := Ur of Theorem 3.3.1.8.

( 3 ~ e ~ ) ( z e IB(~,3))})} Is(~, 3) we obtain the following analogue

3.3.2.5. T h e o r e m . (Inductive Definitions in K P I and KP1 r) Let B(3, y,S) be an S-positive Ao -formula. Then there is a relative E-function symbol Is such that Ad(u) A 3 e u --+ CIB(3, 1~(3))

and Ad(u) A ~, e u A CIB(~, A) --+ I~(~) C {x e ul A(x)}

are provable in KP1 for arbitrary formulas A(x) and in K P I r for A0 -formulas A(x). The proof is essentially that of Theorem 3.3.1.8. We fix an admissible u such that 3 c u and repeat the proof substituting E-reflection by its relativized version. If we only consider A0-formulas A(x) then only (A0-FOUND) is needed. Otherwise we need the full strength of KP1. [3 Observe that Theorem 3.3.2.5 does not immediately follow from Theorem 3.3.1.8 by Lemma 3.3.2.2 since the definitions of the function symbols IB differ slightly.

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Let B ( Z , y , S ) be an S-positive Ao-formula. Then KP1 r proves that.for every admissible set a containing the parameters b and every admissible set u containing a the class I~(b) is a set in u which is the least fixed point of the monotone operator FB, ~ induced by B(b, y, S). 3.3.2.6. Corollary.

Proof. The function symbol ]B is a relative C-function symbol, and we have I~(b) c_ a E c. So by Lemma 3.3.2.3 and A0-Separation relativized to u we obtain I~(b) E u. That I~(b') is the least fixed-point of F B,g follows from Theorem 3.3.2.5. D

3.3.3. The quantifier theorem and axiom/3 The most important tool for embedding subsystems of NT2 into subsystems of Set Theory is the Quantifier Theorem which we are going to present in this section. It is based on a theorem which is commonly known as Spector-Gandy Theorem. First we fix the following notations:

Proga(-<, T)

(Vx E a)[(Vy E a)(y -< x -+ y e T) --+ x E T]

:r

where T may be a set or a class-term,

TP(-<, T) :r

wt"(-<) WOn(-<)

Prog~(-<,T) --+ (Vx

E a)(x e T),

:r

:r

a = field(-.,<)A

LO(-.<) A Wfa(-<)

where LO(-~) says that -< is a linear ordering. For a A0-formula A(~,x, y) we define a relation x - ~ y :r A(Z, x, y) and call it a A0-relation. 3.3.3.1. L e m m a .

Let - ~ be a Ao-relation and define A(b, 3, x, S) as (Vy E b)[y - ~

x -+ y E S]. Then UP1 r ~- Ad(a) A b,Z E a --~ (~/~/fb(~) ~ (Vy E b)(3~ E a)(y E I~(~,b,Z))). To sketch the proof we work informally in K P F . We have Progb(-<~,x) CIA(b,~,x). Let a be an admissible set containing b and all members of ~ as elements. By Corollary 3.3.2.6 we get d := {y E a I (3~ E a)(y E I~A(~,b, ~))} as a set provable in K P F . We have to show

Wfb(-<~) ++ b C_ d.

(i)

For the direction from left to right observe that Wfb(-~) implies Progb(-<~,d) -~ b C_ d. But by Theorem 3.3.2.5 we have CIA(b,Z,d) which is equivalent to Progb(-<~, d). For the opposite direction we use the second part of Theorem 3.3.2.5 to get CIA(b,Z,x) -+ d c_ x for any x. Together with b C_ d this implies Progb(-<~,x) ~ b C_ x which is Wfb(-~). [3 The following notion is motivated by the H{-completeness of well-foundedness.

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3.3.3.2. Definition. Let Ax be a theory in the language of Set Theory. We say that a formula A(Z) is II ~(Ax) iff there is a A0-relation -<~C w x w such that Ax ~-(Vs163

Wf(-<~)].

Observe that the folklore fact that every II~-formula is equivalent to the wellfoundedness of its associated tree of unsecured sequences can be proved in B S T r. Therefore every II~-formula is a II I ( B S T r)-formula such that in the defining formula for its corresponding A0-relation all quantifier are restricted to w. 3.3.3.3. T h e o r e m . such that K P F

C(s

We indicate

For every IIl(KPF)-formula A(~.) there is a E-formula ~ Ad(a) A ~ E a --+ (A(Z) ~ C(s

proof. Let -<~C_ w x w the Ao-relation such that Choose an admissible a such that s E a and define :4=~ (VyEw)[y-<~x --+ y E S]. Then A(s ~ Wf~(-<~) ++ ( V y e w ) ( 3 ~ E a)(y e Is(~,s by Lemma 3.3.3.1. Since IB is a relative E-function symbol, Ad(a) and ~ E a we get A(s ++ C(s a for

A(s B(s

~

the

Wf(-<~).

(vye

e

n

3.3.3.4. Q u a n t i f i e r T h e o r e m . For every II~ -formula A(f,, s in the language s there is a E-formula C ( s s so that Ad(a)-~ (V)~ E a)(Vs E w)[A(.P,,s ~]

, o,aU gPr. :o mula A(f,,2) a r,-fo m la C( 7, 2) such that A(X,s ~ C ( 2 , ~ ) is provable in K P F . Dually for every YI~-formula A()~, 2) there is a II-formula C(X, ~) such that A(fi,, ~) o C(X, ~) is provable in KPF. Recall that we regard s as a sublanguage of s by restricting all first order quantifiers to w and all second order quantifiers to subsets of w. As already remarked every II~-formula is a II 1( B S T ~) and hence also a II 1( g P F ) - f o r m u l a . Assume Ad(a) and )~ E a. By Theorem 3.3.3.3 we obtain a E-formula C(.Y,x) such that A()(,~) ++ C()~,s a. For the second claim assume A()~,s r (3Y)[Y c_ w A B()~, Y, s for a II~-formula B()~, Y, s By the first claim there is a E-formula C'()~, Y,Z) such that for Ad(a) and )~,Y e a we get B()~, Y,Z) e+ C'()~, y,s provable in K P F . By axiom (Lim) we always find such an a and thus obtain the claim bydefining C(.X,s :r (3Z)[Ad(Z) A )~ E Z A (3Y E Z)(Y C_ w A C'(fi,,Y,s The last claim follows from the second by taking negations. El As a corollary of Theorem 3.3.3.4 we get YI1-comprehension 1 in K P F . 3.3.3.5. I I ~ - C o m p r e h e n s i o n T h e o r e m . Let H(.~, s x) be a II~-formula. Then K P F proves (V.~)(V~ e w)(3y)[y = {z e w I H()(, ~, z)}]. Proof. By Theorem 3.3.3.4 there is a Z-formula C(.,~,Z,x) such that KP1 r ~ Ad(a) A )~ E a A ~, x E w --+ (H()~, ~, x) +-~ C()~, s x)a). Using axiom

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288

(Lim) we find such an admissible a and another admissible u such that a E u. Now we apply A0-Separation relativized to u to obtain d := {x E w I C()~, Z, x) a} as a set. Obviously d is a witness for y in the claim, r7 The Quantifier Theorem is a good example for reducing the complexity of 'analytical' formulas by translating them into the language of Set Theory. (The reason for this reduction is the presence of the axiom (Found) in Set Theory). Another example is Axiom/3 which turns the H1 notion of well-foundedness into a/k-notion. We will show that Axiom fl is provable in K P F . This needs some notations. Define

Found(a, r) :r

(Vx)[x C_ a A x ~ 0 --+ (3z E x)(Vy E x)((y, z) r r)]

(128)

expressing that r is a well-founded relation on the set a. To increase readability we use the infix notation for relations, i.e., x r y instead of (x, y) E r. Recall that Found(a, r) entails (Vx E a)(x //x).

(Axfl)

(Vx)(Vr)[ Found(x, r) --+ (3f)(3a)(Fun(f) A dom(f) = x A rug(f) = A (Vu E x)(Vv E x)(u r v --4 f(u) < f(v)))].

3.3.3.6. T h e o r e m .

Axiom fl is a theorem of K P F .

P r o o f . We start with the obvious observation K P I " ~- Found(b,r)-+ Wfb(r).

(i)

Then we obtain from Lemma 3.3.3.1 KP1 r ~- Ad(a) A b, r E a --4 [Found(b, r) +4 (Vy E b)(3~ E a)(y E I~(~, b, r))] (ii) for S(x, S) :r (Vy)[y r x --4 y E S]. Let b and r be given. By (Lim) we choose an admissible a such that b, r E a. For x E b there is by (ii) a ~ E a such that x E IB(~, b, r) and we define f(x) := min {~ e a[ x E I~(~, b, r)}. This defines a function f with dora(f) = b. Defining a := sup {f(x) + 1[ x E b} we see that f is an (r, E)-homomorphisms from b onto a. Q Observe that the function f whose existence is required by (Axl~) is uniquely determined and has for transitive well-founded r the property that f(x) = {f(y)[ y r x}. We often denote this function by otypr and define otyp(r) := rng(otypr). As a corollary of the proof of Theorem 3.3.3.6 we obtain KPF

~- Ad(a) A Found(b, r) A b, r E a --4 otyp~ E a.

(129)

As soon as we have (Axfl) we see that well-foundedness for sets is in KP1 extensible to well-foundedness for classes.

Let - ~ be a Ao-relation. Then KP1 ~ Wf~(-<~) --~ TI~(-.<~, T) .for any class-term T. For Ao-classes T this is already provable in KPI r . 3.3.3.7. T h e o r e m .

P r o o f . Assume Wfa(-~). We have to show that from the hypothesis Proga(-~, T) we o b t a i n a C _ T . Definer := {(x,y) l x E a A y E a A x - < ~ y } . Then r is a set

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by A0-separation and we obtain Found(a, r) from td/fa(-~). By (Axle) there is an onto ordinal c~ and an order preserving mapping f: a ) ~. Put B(~) "r ~ < ~ -+ (Vy E a)[f(y) - ~ -+ y E T]. Since we have the scheme (FOUND) we have induction on ordinals, i.e., especially (V{)[(Vr < {)B(r

--+ B({)]

+

(V{)B({).

(i)

But from (V{)B({) we immediately obtain (Vy E a)[y E T ] , i.e., a C_ T. So assume (Vr < { ) B ( r for { < ce. We have to show B({). Let x E a such that = f(x). For all y E a such that y r x we get f(y) < { and thus B(f(y)). Hence (Vy E a)(y r x --+ y E T) which by Proga(-~, T) implies x E T. Hence B({) and we are done. If T is a A0-class then (A0-FOUND) suffices to have (i) which allows to formalize the proof in K P F . [:] 3.3.3.8. Corollary.

The schemes

(/]~-BI)

as well as (Bi) are theorems of KP1.

P r o o f . Let -~)?,~ be arithmetically definable in Z:~. Then its defining formula is A0 in the sense o f / : ( E , . . . ) . The formula Wf(-~Z,~) as/::(E,...)-formula becomes Wf~(-~,~). Hence g P l ~ Wf~ -~ TI~(-~)?,~,T) for every class term T by Theorem 3.3.3.7 which implies (II~ As a special case we obtain the provability of k~/fw(-
Definition.

The theory K P i is the union of the axioms in K P w and

KP1. The axioms in K P i describe a universe which is admissible and simultaneously the union of admissible universes. So we obtain IIKPiI]~ - I where I denotes the first recursively inaccessible ordinal, i.e., the least ordinal which is admissible and the limit of admissible ordinals. Most of the properties of the theory K P i follow from the previous sections. 3.3.4.2.

Theorem.

1 (A2-comprehension in K P i r) Suppose that A(X, ~, y) is a

H~- and B(~',Z',y) a ~l-formula. Then ( V y E w ) ( A ( X , ~ , y ) ~ B ( f , , ~ , y ) ) (3z)(z = (y E w I A()(, ~, y)}) is provable in g P f f .

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290

Proof. From the hypothesis (Vy E w)(A(f,,Y~,y) ++ B(.~,Y.,y)) and the Quantifier Theorem 3.3.3 is follows that A()~, ~, y) is a A-formula of K P F . Therefore z "= {y e w[ A()~, ~, y)} is a set by A-comprehension. [:] As an immediate consequence of Theorem 3.3.4.2 we obtain the following theorem. 3.3.4.3. T h e o r e m . The theory ( A ~ - C A ) o is a subtheory of K P F , the theory (A~-CA) is a subtheory of W-KPi and the theories (A~-CA)+(Bi) and ( A ~ - C A ) + (Bi) are subtheories of KPi. We have seen in the previous section that KP1 proves (Ax~). We will now show that K P w cannot prove (Axfl). It will moreover become clear that augmenting K P w by (Axfl) will give a theory of the strength of KPi. 3.3.4.4. Definition.

Let KP/3 be the theory K P w + (Axfl).

Since we have already shown that K P F ~ (Axfl) we get immediately 3.3.4.5. T h e o r e m . The theory KP/3 r is a subtheory of K P i r. W-KP/3 is a subtheory of W-KPi and KP/3 is a subtheory o / K P i .

The theory

We will show that conversely KP/3 is of the same proof theoretical strength as KPi. That, however, does not mean that both theories coincide. Though ]lKP/311oo = I = []gPilloo there are ordinals a such that L~ ~ KP/3 but L~ ~ g P i (e.g., a = R~ + cf. Platek [1966] 5.11). We will first check that KP/3 allows the embedding of the same s as KPi. Therefore we need an equivalent for the Quantifier Theorems of KP1. 3.3.4.6. L e m m a . For every IIl(KPi3r)-formula A(Z) there is a E-formula B(2) such that KP/3 r ~ (V2)(A(2) ~ B(2)). For every E~-formula A(.~,Y~) there is a E-formula B(fi,,Y.) and dually for every II~-formula, a H-formula such that KP/3 r ~ (V)~)(V2 e w)[)~ C_w --+ (A()~, ~) ~ B()~, ~))]. Proof. Let "~2,~ C_ w x w be a A0-relation such that KP/3 r ~- Wf(-~:~,~) ++ A()(,~).

Then r := {(x,y)l x e w A y e w A x - ~ 2 , ~ y }

is a set and we have

Wf(-~2,~) ~ Found(w,r). By (Axfl)we obtain

Found(w, r) ~ (:]f)(:la)[f: field(r)

onto

a order preserving].

The right hand side, however, is Et. For the second claim assume A()~,s ~:~ (3Y)[Y C_ w A Ao()~, ]I, s where Ao()~, Y, s is II~. The 'unsecured sequences' argument is of course formalizable in KP/3 r (it needs only arithmetical comprehension which is covered by Ao-separation) and we therefore get Ao()~,Y,s as a Hl(KP/3~)-formula. By the first claim we therefore have a E-formula, say Co()~, Y, s which is in KP/3 ~ equivalent to Ao()~, Y, ~). Defining C()~, ~) :~=~ (3Y)[Y c_ w A Co()~, Y, s we get a E-formula

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291

which is in KPf~ r equivalent to A()~, Z). The dual claim follows by taking negations. [3 Now having also a Quantifier Theorem in K P ~ ~ we get with the same proof the equivalent of Theorem 3.3.4.2.

3.3.4.7. Theorem. (A~-Comprehension in KPf~ ~) Let A(f,,Z,y) be a I-I~- and B(:~, ~, y) a 2 ~ - f o ~ l a . Th~

(Vyew)(A(f,,i,y) ~ B()~,i,y)) -+ (3z)(z= {year I A()~,i,y)}) is provable in KPf~r. As in Theorem 3.3.3.7 we get from (Ax;3) the well-foundedness for classes for every A0-relation which is well-founded for sets. I.e., we have with the same proof 3.3.4.8. T h e o r e m . Let - ~ be a Ao-relation. Then KP/3 ~ t'Vfa(-~) --+ 7-1(-~, T) for any class-term T. For Ao-classes T this is already provable in KPf~r. Summing up we obtain 3.3.4.9. T h e o r e m . The theory (A~_ CA)o is a subtheory of KPf} r, the theory (A~-CA) is a subtheory of W-KP/3 and the theories (A~-CA) + (Bi) and (A~-CA) + (Bi) are subtheories o f g P ~ . It follows from Theorem 3.3.4.9 that for every ordinal c~ for which we have La KPf~ we also have L~ N PoT(w) ~ ( A ~ - C A ) + (Bi). But again the opposite claim is not true. S. Simpson in a private communication to G. J~iger constructed the following counterexample. Let c~ be the least admissible ordinal such that L,~ ~ (3~)[~c is uncountable]. Then c~ = (R~) +. While L,~ ~ NT2 and therefore also L~ ~ (A~-CA) + (Bi) we have L~ ~= (Axle). 3.3.5. Theories of iterated admissibility

There is a tremendous gap between KP1-even K P i r - and KPi. Within this gap there is a whole zoo of theories. M. Rathjen in his thesis [1988] studied these theories exhaustively. Unfortunately his thesis never appeared in English. But there is not enough space to introduce all these theories. As examples, however, we will introduce some theories for iterated admissibility, i.e., theories which axiomatize universes containing a certain number of admissibles. As in Barwise [1975] we denote by T~ the ~th admissible, i.e., )~. T~ enumerates the class of admissible ordinals including w. So we have To = w, T1 = W~K = f21, T~ = ~ + 1 etc. 3.3.5.1. Definition.

Let

ll:Ad(~,f) :4=> Fun(f)

A oL E O n A

dom(f)

= c~ A (V~c < c~)[Ad(f(~c))

A ('v'(~< ~c)(f((~)E f(~)) A ('v'xE f(~c))(Ad(x) ~ (::I(~< ~)(x = f((~)))]

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292

express that there are at least a many admissibles. We define KPIv := K P I + (V~ < v)(3f)ltAd(~, f). If -< is a well-ordering we put KPI.~ := K P I + WO(-<) + (V~ E otyp(-<))(3f)ltAd(~, f). The theory A u t - K P I := KP1 + (Va)(3f)ltAd(a, f) axiomatizes a universe which contains as many admissibles as ordinals. Observe that we can define x = Lnl :r Ad(x) A (Vy E x)-~Ad(y). The formula x E Ln~ is a A-formula since x E Ln~ r (Vy)[Ad(y) --+ x E y] r (3y)[y = Lnx A x E y]. We define the theory KPI* := KP1 + (V~ E Lnx)(3f)ltAd(~, f). It follows by (A0-FOUND) that the enumerating function of the admissibles is uniquely determined, i.e., we have KP1 r ~ ItAd(a, f) A ItAd(a, g) --+ f = g.

(130)

As a consequence of (130) we obtain KPI* ~ (Va E Lnl)(3!f)ltAd(a, f). The formula

RItAd(a, f, a) :r

a E O n A Ad(a) A Fun(f) A dom(f) = a A f(O) = a A (V~ < a)[Ad(f(~)) A (V( < ~)(f(() E f(~) A (Vx E f(~))(Ad(x) A a E x -+ (3( < ~)(x = f(())))]

expresses a-fold iteration of admissibility relative to a start-admissible a. It is easy to show that A u t - K P 1 r ~- Ad(a) -+ (Va)(3!f)RItAd(a, f, a).

(131)

The uniqueness follows again by (A0-FOUND). To prove the existence of the function f we choose by axiom (Lim) two admissible sets b and u such that a, a E b E u. By A0-separation relativized to u we obtain ~ := b fq O n e u. Then ~ @ b and thus r a. There is a function g such that ItAd(13 + ~, g). By (A0-FOUND) we obtain (V~ < ~ + ~)(~ e g(~ + 1)). Because of ~ E g(~ + 1), ~ r a and axiom (Ad2) we get a E g(~ + 1). Therefore there is a p < ~ such that a = g(p). We may now define f (~) := g(p + ~) for ~ < a < Z and easily check RItAd(a, f, a). We will now show that the theories for iterated inductive definitions can be embedded into the theories of iterated admissibility. In the following lemma we use the same notational conventions as in Section 3.2.3. So x E X y stands for (x, y) E X and for a r e l a t i o n r we write x E X ry for (3z)(z r y A x E XZ). The capital letters serve only to improve readability and to emphasize the close connection to Section 3.2.3. Their meaning is that of ordinary variables for sets.

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293

3.3.5.2. Lemma. Let A(S, T, x, y, ~ be an S-positive A0-formula, r C w x w and define C]~(S, T, y, ~ :r (Yx 9 w)[A(S, T, x, y, ~ ~ x 9 S] and IT~(r,X,z-')

:4=~ r C w x w A (Vyew)CI~(XY, XrY, y,z-*) ^ (vy)(y c ~ -~ a% (y, z ~, y, ~ -~ z~ c_ y).

Then we obtain KPI.~ ~-(3Z)lT~(-~,X,z-')

(132)

KPI* ~- WOW(r) A r,Z 9 La 1 -+ (3Z)lT~(r,X,z~

(133)

and Aut-KP1

~ WOW(r) -~ (3X)IT~(r,X,Z).

(134)

The proof of all three claims is essentially the same. We sketch the most complicated case of (134). Assume WOW(r) and chose by (Lim) an admissible a such that r , ~ 9 a. Then field(r) N w 9 a and we get by (129) otyp~ 9 a. Hence also c~ "- otyp(r) 9 a. By (131) there is function f such that R/tAd(a, f, a). By (Lim) we obtain an admissible set u such that a, f, a 9 u. For ~ < a let u~ "= f(~), so u0 - a. By E-Recursion relativized to u we obtain functions g and hu for y 9 w satisfying

hy(rl)

-

{~e~,l A(U{hy(r

r

( = e ~ l A(U{h~(r

r < v),U{g(r

< n),O,x,y, zD} and

dom(hy)=O n n a if y r field (r)

r < otyp~(y)},x,y,~}

(i)

and dom(hy) : O n N Uotypr(u) and g(otyp~(y)) : Urng(hy) if y 9 field (r). Put

s :- ((~, u) l u e field(r) A U{ (x,

Y) I Y

9w A y ~

x

9

g(otyp~(y)) }

field(r)

A x

9 Urng(hy)}.

By construction we have S, hy, g, { (hy, Y)I Y 9 w} 9 u. We show

c/x (s~, s~, y, z-)

(ii)

(vx)[x g ~ -~ a%(x, s ~, y, ~ -~ s~ g x].

(iii)

and

First assume y ~ field(r). Then Sry = 0. The function hy is E-definable in the admissible set a. We have S ~ - {x I (3~ E a)(x E hy((~))}. Thus S y corresponds to the class IA(~ of Section 3.3.1 and we show (ii) and (ii) as in the proof of Theorem 3.3.1.8. Now assume y E field(r). Let ~ : - otypr(y). We show

g r~ e u~

(iv)

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294

by induction on ~. From the induction hypothesis we obtain g r( E ur E u~ for all ( < ~. The function ~ > ( ~ g r( is thus E-definable in u~ and because of e a E u~ we get { ((, g I()l ( < ~} e u~ by E-replacement relativized to u~. This proves (iv) in case that ~ E kim. If ~ = ( + 1 then ( - otypr(z) for some z E field(r) and dom(hz) -- ur as well as gI( are elements of u~. By E-recursion relativized to u~ we obtain hz e u~. Therefore gI~ = gI( U {((,Urng(hz))} e u~. This proves (iv). By (iv) and the definition of S we obtain S ~ e u~ and S y - [.J rng(hy). For r] e u~ we have hy(~?) = {x e w I A((J~<,hy((),SrY, x , y , ~ } which shows that hy is definable in u~ by E-Recursion. Therefore the formula A(SY, S ~y, x, y, z-') is still E and we obtain (3r/e u~)A([.Jr hy((), S ~u, x, y, ~ by E-reflection relativized to u~. Hence x ~ S ~. This proves (ii). To prove also (iii) we show

Cl~(X,S~Y,y,z-') -+ hy(~) C_ X

(v)

by induction on r/ E u~. From the induction hypothesis we have [.Jr hy(() C_ X which implies by Z-positivity A([.Jr hy((), Sry, x, y, ~ ~ A(X, Sry, x, y, ~ . Together with the hypothesis C~(X, S ~y, y, ~ this yields hy(~) C_ X. El The following theorem is a consequence of Lemma 3.3.5.2. 3.3.5.3. Theorem.

(i)

(rt~-CA~)o =

5i)

(H~-CA~) = ID~ c_ W-KPI~

(ID~)o c_ KPlr,

ID~ C_ ( H 1 - C A ~ ) + (Bi)= BID~ C_ g P l ~ (Aut-H~)o = (Aut-ID)o C_ A u t - K P F (Aut-YI1~) = A u t - I D C_ W - A u t - K P I (Aut-H~) + (Bi) = A u t - B I D C_ A u t - g P 1 ID.~. C_ BID* C_ KPl* Proof. Claims (i), (ii), (iv), (v) and (vii) follow immediately from Lemma 3.3.5.2 and Theorem 3.2.4.2. Claims (iii) and (vi) follow from Lemma 3.3.5.2, Theorems 3.2.4.2 and 3.2.3.2 and Corollary 3.3.3.8. D The opposite inclusions in claims (iv), (v) and (vi) are also true. In (i), (ii) and (iii) this can of course only be true for limit ordinals u. And this is the case. 3.4. O r d i n a l analysis for set-theoretic axioms systems 3.4.1. Ramified set theory

We introduce the language l:as and of Ramified Set Theory. The basic symbols are E, ~, Ad, ~Ad and a constant [~ for every ordinal a.

295

Set Theory and Second Order Number Theory

3.4.1.1. Definition. (Inductive definition of the set terms of Z:as) Every constant In is an atomic set term of stage c~. If a l , . . . , an are set terms of stages < c~ and F(x, x l , . . . ,xn) is an s formula without further free variables then

{xeL,

}

is a composed set term of stage ~. We denote the stage of a set term s by stg (s). There are only sentences in Z:as. An s is obtained from an Z:(E, Ad)formula (in 'Tait'-style) by replacing all free variables by Z:as-terms and restricting all unbounded quantifiers to Z:as-terms. To have a uniform notation we refer to Z:as-terms and-sentences as Z:as-expressions. Let T~ := {t I stg(t) < c~}.

(135)

We do not count the equality symbol among the basic symbols of s

but define

We transfer the Levy hierarchy to the language Z:as. 3.4.1.2. Definition. Let jc be a complexity class in the Levy hierarchy. We call an Z:as-sentence F an ~'~-sentence if there is a ~'-formula G(Z) in the language Z: which has only the shown free variables and a tuple ~ o f / : a s - t e r m s of stages less than c~ such that F - G(g) L,. If F L~ is a ~'~-sentence, we denote by F z the sentence which is obtained by replacing all quantifier restrictions E / ~ by E z. The standard interpretation of s

is given by

LaL = La

e c,, I F(x, al,..., an)L" } r,,

__

{x e Lo, I Lo, ~ F(x, a~,..., a n ) ). L

It is obvious that for every set term s of stage a we have s ~" E L~+t. We have L~sEL, L~Ad(t)

r r

L~s-t

for some set term t with stg (t)
L~t=k~

for s o m e a E R e g s u c h t h a t a _ < s t g ( t )

(136) (137)

L ~ t = s A F(t)

(138)

and

L ~ s e {x e L~] F(x) } r

for some set term t with stg(t) < ~. This motivates the following definition. 3.4.1.3. Definition. We say that/:as-sentences of the shape s E r, Ad(t), A V B and (3x E r)G(x) have V - t y p e . Dually sentences of the shape s ~ t, -~Ad(t), A A B and (Vx E r)G(x) are said to have A-type. We put

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296

C(s e r) :=

{t = ~1 stg (t) < a} {t = s n F(t) l stg (t) < a}

if r = La if r = {x E kol F(x)},

6 (Ad(t)) := { k,~ = t l a e R e g A K; _< stg (t) },

C(A v B):= {A,B} and

C((=txEr)G(x)) :=

{a(t) l

stg (t) < a} { F ( t ) A a(t)l stg(t) < a}

if r = L~ ifr= {xek~lF(x)}.

This defines the characteristic sub-sentences-set 6 (F) for all sentences F of V-type. Dually we define

c (F) :=

{-~al a e c ( - F ) }

for sentences F of A-type. If F is not a conjunction or disjunction then every G E C(F) is of the form H(t) for some characteristic set term t. We define rE(G) := t and oF(G) :-- stg(t). For F = A0 o A1, o E {V,A} and G E C(F) we put tF(G):= ki and oF(G):= / i f G = Ai for i E {0,1}. The following lemma is an immediate consequence of Definition 3.4.1.3 and its preceding remarks. 3.4.1.4. L e m m a .

L~F

~

For every sentence F of V-type we have L~

V

a

G E C(F)

and dually L~F

r162L ~

A G G E C(F)

for sentences of A-type. Lemma 3.4.1.4 is the basis for the following infinitary calculus. 3.4.1.5. Definition. We define the relation ~ A for finite sets of s A inductively by the following two clauses:

(V)

,s~ ~ os V-,,,,~ o,,,~ ~ " , ~ some

o~,,,~,,,,~

~,o ~ , , a,,d o,~(~) ~ , , ,,o,dSo,-

G E 6(F) then ~ A , F

and

(A) isF ~ os A , - ~ o~ ~ ", a ~A,F.

o~ ~ , ,

o~ ~

< ,,

~o~so~ o. a~ c(~),,,o,,

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From Definition 3.4.1.5 we get for L:Rs-sentences F

L~F

r

(3a)~F

(139)

and for a El-sentence F and an ordinal -y we have

~ FL~ =~ L~ ~ F.

(140)

If we put to(F) := { minoo{aI ~ F } otherwiseifL ~ F

(141)

for/:as-sentences F we obtain from (140)

IFI~,

< tc(F L~)

(142)

for all El-sentences F and all ordinals 7. We are going to define a rank function for L:Rs-expressions in such a way that all sentences in the characteristic sub-sentences set of a sentence F get lower rank. 3.4.1.6. Definition. clauses:

For an L:Rs-expression E we define rk(E) by the following

rk(L~) : - w. o~ rk({x E L~I F(x)} ) := max{rk(L~)+ 1, rk(F(L0))+ 2}

rk(Ad(t)) := rk(-~Ad(t)):= rk(t)+ 5 rk(s e t ) : = rk(s ~ t ) : = max{rk(s)+ 6, rk(t) + I} rk(A V B ) : = rk(A A B ) : = max{rk(A), rk(B)} + 1 rk((3x E s ) F ( x ) ) : = rk((Vx E s ) Y ( x ) ) : = max{rk(s), rk(F(L0)) + 2} The crucial property of the rank function is stated in the following theorem. 3.4.1.7. T h e o r e m .

For G E C(F) we have rk(G) < rk(F).

The proof of the theorem is a bit lengthy and we will not give all details. Let a, b and c be/:Rs-expressions. First we show

stg(a) < stg(b(Lo)) =~ rk(b(a)) < rk(b(Lo))

(i)

by an easy induction on rk(b(Lo)). The next step is to show stg(c) < a =~ rk(b(c)) < max{w .a, rk(b(Lo))+ 1}

(ii)

by induction on rk(b(Lo)). From (ii) we get easily stg(c) < a =~ rk(F(c)) + 1 < rk(s E {x E La I F(x)}). Now we compute

(iii)

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298

rk(a = b) = max{rk(a), rk(b)} + 4.

(iv)

Finally we show G E C (F)

=~ rk(G) < rk(F)

(v)

by distinguishing cases on the shape of F. We only consider sentences of V - t y p e . The case of A - t y p e is dual. If F - Ad(t) then G - (t = L~) for some ~ E R e g N s t g ( t ) + l . Hence rk(G) = max{rk(t), ~} + 4 < rk(t) + 5 = rk(F). If F = (s E La) then G - ( t = s ) for some t such that stg(t) < c~. But then rk(G) = max{rk(t), rk(s)} + 4 < max{w 9c~ + 1, rk(s) + 6} = rk(F). If F =_ (s e { x e k ~ l H ( x ) } ) then G = (s = t A H ( t ) ) for some t such that stg(t) < a. Hence rk(G) = max{rk(s = t),rk(H(t))} + 1 = max{rk(s) + 5, r k ( t ) + 5, r k ( H ( t ) ) + 1}. But rk(s) + 5 < rk(s) + 6 _ rk(f), rk(t) + 5 < w . a _ rk(F) and by (iii) also r k ( g ( t ) ) + 1 < rk(F). The claim is obvious for F - (A Y B). If F = (3z e L~)H(x) then G = H(t) for some t with stg(t) < a. Then by (ii)

rk(G) < max{w .oL, rk(H(Lo))+ i} _< max{w .o~,rk(H(Lo))+ 2} = rk(r).

If F =_ ( 3 x e {yE k~[H(y)})K(x) then G = H ( t ) A K(t) for some t such that stg(t) < a. Then by (ii) rk(G) = max{rk(g(t)),rk(g(t))} + 1 < max{w.

a, rk(H(Lo))+ 2, rk(K(Lo))+ 2} = max{rk({x EEL~,IH(x)}), rk(K(Lo))+ 2} = rk(F). Because of

L~F

rk(C) < ~

rk(F) for all a e C (F) we get by induction on rk(F)

Irk(F) F .

(143)

Hence tc(F) < rk(F) for all s

F.

(144)

3.4.2. A s e m i - f o r m a l calculus for ramified set t h e o r y It follows by (139) that the rules (cut)

~ A , A and ~ A , ~ A

=~ ~ A

and (Ref)

~ A, F L~ =~ ~ A, (3z e L~)[z ~: 0 A f z] for ~ e Reg, F L~ a II~-sentence

are admissible for some ordinal 6. However, we do not yet know how to compute 6 from c~ or c~ and/~ respectively. Therefore we design a semi-formal system having these rules as basic inferences. 3.4.2.1. Definition. Let A be a finite set of s We define the relation ~ A by the following clauses:

and c~ and p ordinals.

Set Theory and Second Order Number Theory (A)

If F E A n A-type, ~

(V)

I: F ~ ~ n

299

A, G and ac < a for all G E C (F) then ~ A.

V-tup~. ~ A. c a.~

o~(C)..~ < ~ Io~ ~om~ C ~ C (F) t ~ .

(Rely) IS~ e l e g , F L~ e n;. (~z e L~)[z # 0 ^ F "] e ~ , ~ A . F L~ a - d ~ . ~ o + l < a then ~ A. We call F the main part in instances of ( A ) and ( V ) and (3z E/~)[z --~ 0 A F z] the main part in an instance of (Rely). (cut)

If ~po A, A, ~pOA,-~A for some ao < a and some A such that rk(A) < p then

We say that A is semi-formal derivable in s if there are ordinals a and p such that ~ A. As an immediate consequence of the soundness of all rules we get the soundness of the semi-formal calculus, i.e., A => L ~ V

A.

(145)

Since ~ A is a correct calculus which derives sentences we get cut elimination nearly for free. We prove:

If ~

A, r and L ~ F

for all F

E r then ~ A

(146)

by induction on a. The claim is immediate from the induction hypothesis if the last inference is according to ( A ) or ( V ) and its main part does not belong to F. If the main part is in F we have in the case of an inference according to ( V ) an F E F M V - t y p e and the premise ~2_ A, F, G for some G E C (F). Then L ~ G and A follows by induction hypothesis. In the case of an inference according to ( A ) there is an F E F M A - t y p e and therefore a G E C (F) such that L ~ G. But then there is a premise ~ A, F, G with c~c < c~ and we obtain ~ A by the induction hypothesis. If the last inference is a cut with cut-sentence F then either L ~ F or L ~ ~ F . We pick the corresponding premise and obtain the claim by induction hypothesis. If the last inference is according to (Ref~) we distinguish the cases that for its main part we have L ~ (3z E L~)F z or not. In the second case we also have L ~ F [~ and obtain the claim by the induction hypothesis. In the first case we have (3z E L~)[z ~: 0 A F z] E A and get ~ A from rk((3z E L~)F z) = a < a and (143). [:] Summing up and taking F as the empty set in (146) we get the following theorem.

The semi-formal calculus is sound and allows cut elimination. Especially we get ~ A from ~ A.

3.4.2.2. T h e o r e m .

300

W. Pohlers

3.4.3. O p e r a t o r - c o n t r o l l e d derivations

It follows from Theorem 3.4.2.2 that cut-elimination alone cannot be crucial for the ordinal analysis of theories Ax for which we have A x e - Wf(-~) r

A x ~ (3~ e L~,xc~)H.~(~),

i.e. for theories in which the well-foundedness of a A0-definable ordering can be wOK expressed by an E11 -sentence. This is true for all theories comprising K P w . The main problem there is to collapse the ordinals which arise canonically in the embedwOK ding procedure of lC1x -sentences into ordinals below co~K. Collapsing is therefore the Leitmotiv of Impredicative Proof-Theory (but we will see that cut-elimination will be needed for collapsing). We will use the technique of local p r e d i c a t i v i t y , first introduced in Pohlers [1982a,1982b], Buchholz et al. [1981], but we are going to use an essential simplification of the original technique which has been introduced by Buchholz [1992]. We already used collapsing techniques in the YI~ of Sections 2.1.4 and 2.1.5. We will, however, not give II~ for impredicative theories. Already for II~-analyses the matter is sufficiently complicated. We just beg the reader to believe that the refinement to II~ can be done by modifying the techniques of Section 2.1.5 (cf. Blankertz and Weiermann [1996], Blankertz [1997] for more details). Another way to extend the following analyses to ri~ is to apply Section 2.2.2. Our presentation will follow quite closely that of Buchholz in Buchholz [1992]. (Those who have tried know that it is hardly possible to improve Buchholz's presentations.) 3.4.3.1. Definition.

7/" Pow(On)

An

(ordinal-) operator

is a map

~ Pow(On).

We introduce the abbreviations MC_7r

:r

MC_7-/(0)

7/C_M

:~

7/(0) C_M

(147)

n c n' :,~ (vx)[n(x) c n'(x)] and call an operator 7/closed under a function f" On n (VX e Pow(On))(Vr162 In case that 7t is closed under

e 7/(X) r

> On if {r162

f ( a l , . . . , a n ) := w ~1 # "'" # w '~"

C_ 7/(X)].

we call it

Cantorian-

closed.

A set M C_ On and an operator 7t induce a new operator ~

7t[M](X) := 7t(M u X). For an s

E let

by

(148)

301

Set Theory and Second Order Number Theory

par(E) : - - {~1 L~ o ~ u ~ i~ E } If {3 is a set of s 3.4.3.2. Definition. conditions:

(149)

and 7i an operator we define ~

:= "//[par(O)].

An operator ~ is acceptable if it satisfies the following

o ~ n(O) 7/is Cantorian-closed (VX E Pow(On))[X C_ ~

(150)

(vx e pow(On))(VY e Pow(On))[X c_ ~ / ( y ) ~ ~/(X) c_ n(y)]. 3.4.3.3. Definition. Let 7r Pow(On) ---4 Pow(On)be an operator. For a finite set A of s we define 7/ ~ A iff par(A)t0 {a} C_ 7t and one of the following conditions is satisfied: (A)

There is a sentence F E AM A - t y p e such that 7-l[tF(G)] ~ - A, G and ar < a for all G E d(F).

(V)

There is a sentence F E AM V-type such that 7-I ~ - A, G and oF(G), av < a for some G E d(F).

(Ref~) There is a H~-sentence F [~ such that (3zEl~)[z 7-l ~po A, F [~ and a, ao + 1 < a. (cut)

r

0 A F ~] E A,

There is a sentence A such that rk(A) < p, 7-I ~po A, A and 7-I ~po A,-~A for some ao < ~.

We say that A is operator controlled derivable if there are an acceptable operator 7t and ordinals a and p such that 7t ~ A. From now on we will only regard acceptable operators without mentioning it explicitly. The following properties of operator-controlled derivations are immediate consequences of Definition 3.4.3.3. /f 7t ~ A, 7/C_ 7~', A C_ F, a _< 13 E 7/', p _< a and par(F) C_ 7~'

(151)

then 7t' ~ F. /f 7t ~ A, (31 E L6)F(x) and 5 <_ ~ E 7-I then 7t ~ A, (31 E L~)F(x). (152) If 7/ ~ A, (Vx E L~)F(x) and ~ E a M 7-I then 7/ ~ A, (Vx E L~)F(x). (153) If 7i ~ A, F V G then 74 ~ A, F, G.

(154)

We refer to (151) as Structural Rule, to (152) as Upward Persistency, to (153) as Downward Persistency and to (154) as V-exportation. All these properties are easily proved by induction on a. The predicative cut-elimination procedure works also for operator controlled derivations. First we prove

W. Pohlers

302 3.4.3.4. I n v e r s i o n

Lemma.

If

7-l ~ A , F

7t[tF(a)] ~ A, G for all G e C (F).

and F

E

A -type

then

The proof is a straightforward induction on c~.

If F E V-type, rk(F) - p r R e g and 7/ ~ A, ~ F

3.4.3.5. R e d u c t i o n L e m m a .

as well as 7/ ~ F F then 7/

A F

,p

~

9

The proof is that of the Reduction Lemma (Lemma 2.1.5.7) of Section 2.1.5. Since we restrict ourselves to II~-ordinal analysis we consider finite ordinals as trivial. Therefore we don't have to compose the controlling operators. This makes the proof simpler. The hypothesis p ~ R e g is needed to exclude the case that F is the main part of an inference according to (Ref). As in Section 2.1.2 we obtain the Predicative Elimination Lemma as a straightforward consequence of the Reduction Lemma. 3.4.3.6. P r e d i c a t i v e E l i m i n a t i o n L e m m a . Let 7-l be an operator which is closed under the function ~, t3 ~ qo~t3 , 7-l I~+~p A, [13,t3 + w p) fq R e g -- 0 and p E 74.

Then 7-l ~i~

A

The proof is that of the Predicative Elimination Lemma (Lemma 2.1.2.9) with some extra care on the controlling operator. As an example we treat the case of a cut. There we have the premises ~o

o~o

7t I~+wp A, F and 7i I~+~p A, ~ F .

(i)

Using the induction hypothesis we obtain 7/ [Z~2_ A F 't~

'

and

7/ [Z~2ZA ~F. '8

'

(ii)

If rk(F)
n~A by a cut since qOpC~0< ~Op~E ~ . If/3 _ rk(F) =NF /~ + w px + ' ' " + wP" < fl + wP we first obtain I~~176 A

qt~ i rkCF)

(iii)

by the Reduction Lemma (Lemma 3.4.3.5). Since par(F) C_ 7t we also get rk(F) E 7/ and therefore { p l , . . . , p,} c_ 7/. From (iii) we first get 7/ ~irk(f) A and finally

n~A

(iv)

Set Theory and Second Order Number Theory

303

by n-fold application of the main induction hypothesis since ~om~Opa - ~Opa for i- 1,...n. []

Let 7"L ~p A , F L~ for a ~-sentence F L~. Then 7-l ~ A, FL~ for all ~3 E [a, ~) N 7-l.

3.4.3.7. B o u n d e d n e s s T h e o r e m .

The proof is by induction on a. The claim follows immediately from the induction hypothesis if the main part of the last inference is different from F L~. So assume that F L~ is the main part of the last inference. If F L~ E A - t y p e then every member of C (F L~) is a Z;~-sentence of the form G L~ and we have the premise

7-l[tfL~ (GL~)] U A, F L~, G L~.

(i)

7-l[tFL~(GL~)] ~ A, FL~, GL~ for all G L~ E C (F L~)

(ii)

Hence

by induction hypothesis. But since F L~ is a 2~-sentence we have tfL~ (G L~) --tEL~ (G L~) and get the claim from (ii) by an inference according to (A)" If F L~ E V - t y p e and F L~ ~ (3x E L~)G(x) then every member of C ( F k~) is again a 2n-sentence of the form G L~ and we have the premise 7t ~2_ A, F L~, G L~

(iii)

for some a0 < a. Applying the induction hypothesis to (iii) we get the claim by an inference (V). if F L~ - (3x E k~)G(x) L~ then we are either in the case of an inference (V) whose premise is

74 ~po A, FL,,G(t)L.

(iv)

with a0 < a and stg(t) < a or in the case of an inference (aef~) "t/ ~ A, F k', (Vx E L~)(3y E L~)G(x, y) =~ 7/ ~ A, (3z E L,)(Vx E z)(3y E x)G(x, y)

(v)

for a0 + 1 < a. In the first case we get

7-I ~o~ A, FL,,G(t) L~

(vi)

by induction hypothesis. We have G(t) L~ E C ((3x E [.~)G(x) L~) because of stg(t) < a _ / 3 and we get the claim from (iv) by an inference (V)" In the second case we get by the Inversion Lemma and the induction hypothesis 7~[t] ~ A, f L~, (3y E k~o)G(t, y) for all/:as-terms t such that stg(t) < a. Since a0 < a < fl < a we obtain

(vii)

W. Pohlers

304 "Jr/ ~I ~

A ' FEe ' (Vx e k~o)(3y e L,o)G(x, y)

(viii)

from (vii) by an inference (A)" But (Vx e L~o)(3y e L~ola(z, y) e C ((3z e La)(Vx e z)(3y e z)G(x, y)) and we obtain the claim from (viii) by an inference ( V ) . 3.4.4. C o l l a p s i n g f u n c t i o n s In order to define operators which allow the collapsing of derivations we need to know more about ordinals. This theory turns out to be very complicated. To simplify things we are going to use an abstraction. Instead of using admissible, i.e., recursively regular ordinals, we will develop the theory on the basis of just regular ordinals. This has the advantage not to have to bother about the complexity of the ordinal functions which we are going to define. Complexity arguments will be replaced by simple cardinality arguments. The disadvantage, however, is that the replacement of regular ordinals by recursively regular ones is not at all easy. See Rathjen [1995] and Schliiter [1993,n.d.] for details. It is outside the scope of this contribution even to give a hint how this can be performed. All we can indicate is that the segment below wl, the first regular ordinal, is recursive and therefore already a wlcK segment below w~K. This yields at least a correct computation of the E 1 -ordinals of the analyzed axiom systems. In this section we denote by R e g the class of regular ordinals above w. By R e g we denote its topological closure, i.e., the class of uncountable cardinals. Let ~ . f2~ denote the enumerating function of R e g U {0}. Then f~0 = 0, ftl = R1 = wl, f ~ = R~ etc. We reserve the letters n, 7r, h i , . . . , 7rl,... to denote members of R e g exclusively. We put a +:=min{~EReg

I a<~}.

For the following we assume that there exist a weakly inaccessible ordinal. Let I be the least such ordinal, i.e. I is a regular ordinal for which we have f~I - I and a < f~,, for all a E R e g A I. Then we have ~ = f~,,+i for all ~ E R e g A I. 3.4.4.1. Definition.

r

We define sets Cl(a, fl) and functions

:= min {/51 ~ E Cl(a,/3) A Cl(a, ~) A ~, C_/~}

by recursion on a. The set Cl(a,/~) is the least set which contains/3 tJ {0, I} and is closed under ordinal addition +, the Veblen-function A(. A~. ~o~r/, the enumerating function A~. f~ of the class R e g and the function A~ < a. ATr. r We call Cl(a, fl) the a-iterated closure of ft. There are some immediate consequences of Definition 3.4.4.1.

Set Theory and Second Order Number Theory

305

(155)

C~o < c~ and flo <_ fl =:> Cl(c~o, ~o) c Cl(c~, ~)

(156) Icl( ,

cl( ,

(157)

<

n

=

(158)

Since a < ft~+l = a C_ Cl(a, ,~) for ~ < I we get

~; E Cl(a, ~) for all a.

(159)

A little more effort is needed to show

r

< ,~ and r

r Cl(c~, r

(160)

By (156) and (159) we have r/0 "= min {~1 ~ E Cl(a,~)} < ~. Putting r/~+l := min {~l Cl(a, rln) M ,~ C_ ~} we obtain r/~ < ~ from (157) by induction on n. Hence r/ := supne~ r/n < ~ and

Cl(a, rl)M~ - U~e~oCl(a, rln)M~ C_ sup~e~ r/~+l = 77. This shows r _< r / < ~ and, since Cl(a, r M~ = r also r r Cl(a, r Putting I r "= min {7 E SC I I < 7} it follows from (160) that Cl(a,~) c_ I r for

/ 3 c _ I r.

C~o < a and ao E Cl(a, r follows since r We obtain r

E Cl(a, r

< r

(161)

M ~.

r (ft~ I a < ft~}

since the assumption r contradicts (160). We have r

=> r

= ft,, > a E Cl(a, r

(162) leads to r

E Cl(a, r

E SC

which

(163)

since ~, r / < r C_ Cl(a, r entails ~o~rl E Cl(a, r M~ = r By definition we have f ~ E Cl(a, ~) for a E Cl(a, ~). This extends to

f ~ E Cl(a, ~) r

a E Cl(a, ~)

(164)

which is obvious for a = ft~. If we assume ft~ > a ~ Cl(a,~) then we get ft~ r flU{0, I} as well as ft~ # r all ,~ and r/. If ft~ = ~+r/or ft~ = qo~r/for some and r/then gt~ E {~, 77}. This shows that the set Cl(a, Z) \ {ft~} contains ~U {0, I} and has the same closure properties as Cl(a, fl). Hence Cl(a, fl) = Cl(a, fl) \ {f~}, i.e., ft~ r Cl(a, ~5). We define the set of strongly critical components SC(a) of an ordinal a by

W. Pohlers

306

sc(~) .=

0

ifa =0

{~}

~f ~ e sc if a =NF ~('r/ ~f ~ = ~ ~ + . . . + ~..

SC(r U SC(r/)

sc(~l)

sc(~.)

u...

(165)

Then obviously SC(a) c SC and SC(a) c_ a 4- 1. Now we get

e cl(~, ~) ,~ sc(~) c cl(~, ~).

(~66)

The claim is trivial for 7/ E SC. So assume 77 ~ SC. The direction "r is clear by definition. We show the opposite direction by induction on 7. Assume SC(r/) Cl(a,~). Then r/ r /~U {0, I}. If 77 =NF q0~1~2 or 77 = ~1 4- ~2 for ~i < r/ we get SC(r/) C_ SC(~Cl)U SC(~2) and SC(~,) g Cl(a, fl) =~ ~ ~ Cl(a, fl) by induction hypothesis. Therefore we can't have r / = qor or 77 = ~1 4- ~2 such that ~i E Cl(a, ~) for i - 1, 2. Thus Cl(a,/~) \ {r/} contains/3 U {0, I} and satisfies the same closure conditions as el(a, 1~), i.e., r/~ Cl(a, 1~). If ~ = f~+l then a E Cl(a, r by the definition of r and (164). Hence ~'~a E Cl(a, ~)~a) N ~ = r < ~'~a4-1 and we have

Let ~ < a. By (167) we obtain ~ E Cl(~, r = r we get r162 <__r and therefore

r < a =~ r162 < r

A

Cl(~, r

Since Cl(~, r c el(a, r

M ~ c_ el(a, r

M (168)

Together with (161) this gives

a <~

A a E

Cl(7, r

for some 7 <--/~ =~ r

It follows from (167) that for n < I the ordinal r situation is different. There we have nr

< r

(169)

is not in Reg. For n = I the

= r

(170)

showing that r is a limit of regular ordinals. To prove (170) choose a such that ~'~a _~ ~/3Ia < flaW1" Then ~"~a-t-1 < I which entails ~'~a+l r Cl(a, r Hence a ~ Cl(a, r But then e t a <_ a <_ f]~ <_ r Since f~ E SC and f~ ~- r for 7r r I by (167) we also get

f ~ = a E Cl(a, ~) => a = I or a = CPl for some ~7. So I and the ordinals of the form r fixed points of A~. f~r

[ ~ , ~+~)

n

c~(~, ~) # 0 ~

(171)

are the only ordinals in Cl(a, ~) which are

~ e c~(~, ~).

(17~)

We prove (172) by contraposition. Assume that f~ r Cl(a,Z). We are going to show that then M :- Cl(a,/~) \ [f~, f~+l) satisfies the same closure condition as Cl(a, 1~). Then M - el(a, ~) which shows el(a, Z) n [f~, f~+l) = O. First we have /3 U {0, I} C_ M. If ~, r/ r [f~, ~'~a+l) we get ~ + 77 r [f~, ~'~a4-1) as well as

Set Theory and Second Order Number Theory

307

~or162 [f~,f~+l). If r E [flo, flo+l) for {~,~} c_ Cl(a, fl) but tr r [flo, f~o+l) then we have ~ - f~+l by (167). Hence f~o E Cl(a, ~) by (164). A contradiction. That M is closed under ,k~. f~r is obvious. Next we observe cl(

,

= cl(

,

(173)

To prove (173) put ~ : : min {77[ ~/r Cl(a, f~)}. Then ~ c_ Cl(a, fl,)M ~'~a+l which entails Cl(a, ~o) = Cl(a, ~). Therefore we have ~ r Cl(a, ~) and ~ < fl,+l which implies r a _< ~r because fla+l E Cl(a,~o) C_ Cl(a,~). Hence Cl(a,f~a) c_ Cl(a,r C_ Cl(a,~) = Cl(a, fl~) by(167). From (173) we obtain especially

Cl(I r, 0) M ~'~1

----- r

(ir).

(174)

It follows from (174) that all ordinals below r can be represented by terms which are built up from O, I, by the unary function ,~(. f~r and the binary functions ~ . It/. ~ + 77, I ( . Ar]. ~oCr] and 1~r. )~. r solely. We already defined the notions a =Ne al + ' ' ' + a~ and a =Ne ~r This gives unique term notations for terms not in SC. If we define a--NF

~"~a

:r

a--

~'~a A G < a

and =,F

:,,

=

A

Cl( ,

we get a : N F ~"~ /k a : N F ~'~r/ ~

~ = 77

(175)

as well as

a=NFr162

=~ ( a = f l

r

~=77).

(176)

While (175) holds obviously we get (176) because ~ < r/and ~ E Cl(~, r implies r _< r (168) and therefore ~ E Cl(~, r M 77. Hence a < fl by (161). This entails also the opposite direction. As a consequence of (176), (167) and (170) we obtain that a --NF r determines r and r/uniquely. In order to decide whether r is in normal form, we have to decide a E Cl(a, r By (173), however, it suffices to decide a E el(a, Ir where lena] denotes the cardinality of r For # E R e g we get more generally fl E Cl(a, #) if and only if one of the following conditions is satisfied:

#e u{0,I} Z r SC and SC(fl) C_ el(a, #)

# <_ fl = r

77 < a and {Tr,77} C_ Cl(a, #)

# <_ fl = flo and a E Cl(a,#). This gives raise for the following definition.

(177)

W. Pohlers

308 3.4.4.2. D e f i n i t i o n .

K,,(,~)

For ordinals a, # E Cl(I r, 0) we define finite sets K,(a) by

U {K.(Z) I Z e sc(~)} :=

0

{~} u K,,(r

K~,(a)

K,,(~)

ifa ifa if # if #

it SC E # U {I, 0} _ a = r _< a = ft~.

It does not matter that the above definition is not deterministic in the case that a is an ordinal below # which is not strongly critical. We get K,(a) - 0 regardless of the clause we apply. From (177) and Definition 3.4.4.2 we get ~ E e l ( a , # ) r (V~ E K~(/?))[~ < a]. Putting Kt,(a ) < ~ :r (V~ E K~(a))[~ < ~] we get O/--NF

~)~r~

r

a -

~)lr?~

and KIr

(178)

r/.

Observe that for ordinals a E Cl(I r, O) the cardinality is determined by a

I~1=

max{l~l,..., I~.1} Ivl

if a = ~t~ or a = r if a = a 1 -[-- 9 9 9 -4- a n if a = ~r if a = r I > ~r =

(179) ~a+l.

All that opens the possibility to define simultaneously a term-system T together with an evaluation function I iv: 7" > O n and a "less than" relation < on the ordinal-terms such that a < b r ]alo < Ibio and the "less-than" relation on the ordinal terms becomes primitive recursive. We will not do this in all details but only indicate the essential steps. There are the following sets of ordinal-terms

9 the set T comprising all ordinal terms 9 the set P of principal terms denoting additively indecomposable ordinals 9 the set SC denoting strongly critical ordinals in SC 9 the set K of cardinal terms denoting ordinals in R e g 9 the set F of fixed-point terms denoting ordinals which are fixed-points of the enumerating function of R e g 9 the set R of regular-terms denoting ordinals in R e g which are defined by:

9 RcKcSCcPc

T

9 FcK

9 O_E T,

IOlo:-0

9 / E R n F, IZlo := z 9

al,...,a,

E P and al >_ . " >_ a,

I~xlo + ' "

+ I~.lo

=~ al + . . . + a, E T, la~ + ' "

+ a.lo :=

Set Theory and Second Order Number Theory

309

9 a, b E T =~ - ~ a b E P , ]-~ab[o:-~l~lolb[o

where ~ is the fixed-point free version of the function ~, i.e. 9= ~ qp~(fl + 1) L~ f l

if fl = "), + n for some n < w and -), such that ~o~q, - ,), otherwise,

9 I.fp E R, a ~ I, a E T and KilCpalI (a) < a then Cpa e SC and ICpalo "- r

lalo

9 If a E T and KIICL~II(a) < a then CLa E F and ICLalv "- r 9 If a E T \ F then f ~ E K and f~a+l E R and If~lo ' - f~l~lo where 1 " - 9~o0. The definition of the sets Kp(a) for p E K and a E T should be obvious from Definition 3.4.4.2. Similarly obvious is the definition of the "cardinality" licit = ilalo] of an ordinal term a from (179). Finally we have (omitting some obvious cases) a = 0 A b -7(=0, or Ilall < libl], or licit - ]]bi] and one of the following conditions is satisfied

a
4=~

a = r A b-end A c < d a =-~cd A b E 5 C A c,d < b a E S C A b - - ~ c d A (a < c V a <_ d) {c
This shows that the set T together with the sets Kp(a), the cardinality Ilal] and the relation a < b can be simultaneously defined by course-of-values recursion. It follows that T and the < -re l a t i on on T are primitive recursive and it is pretty easy to realize that

{ lalo I a E 7-} = C l ( I r, 0). Since Cl(I r, 0) Nf~l is a segment of the ordinals we obtain Cl(I r, O)Nf~ C_ w~ K which entails r < w~ K by (174). This shows that we at least may replace f~l by w~ K. It needs, however, considerably more effort to show that we can replace all regular ordinals by recursively regular ones without changing the segment C l ( I r, O) N w~ K. Nevertheless, we will pretend that we have done that and interpret the ordinals in R e g as recursively regular, i.e. admissible ordinals. 3.4.5. T h e C o l l a p s i n g T h e o r e m We are now prepared to define the controlling operator which allows to collapse the controlled derivation.

W. Pohlers

310 3.4.5.1. Definition.

Put

n . ( x ) .= N { c l ( . , Z) l x c cl(., Z) ^

< .}

We certainly have 0 E 7/7 for all 7 and obtain from (166) that all 7/7 are Cantorian closed and closed under ~. By definition we get X C_ 7/7(X) for all X C_ On. If we assume X C_ 7/~(Y), ~ E 7/7(X) and Y C_ Cl(a, fl) for some 7 < c~ then we obtain also X C_ Cl((~, ~) and therefore also ~ E e l ( a , fl). Hence ~ E 7/7(Y) and we have 7/~(X) c_ ~7(Y). Pulling this together we have the following lemma.

The operators 7-l7 are all acceptable and closed under the Veblen-function ~ and the function ~ . f~r .

3.4.5.2. L e m m a .

But we also get the closure of the operators 7/~ under the functions r following sense.

<7^

n

in the

(180)

(x)

From (172) and (164) we get

[~"~a,~"~o'+1]N n 7 ( X ) # 0 =:~ {~"~o-,~"~o-+1}C_ n T ( X ) .

(181)

The operators are also cumulative. We have 7_<5

~

(182)

7/~c_7/6.

The aim of this section is to prove the Collapsing Theorem, i.e., to show that every derivation of a set of E~-sentences can be collapsed into a derivation whose derivation length is less than n. Obviously such an derivation must not contain cuts whose cut-sentence has a complexity above a. Therefore collapsing below a has to come together with the elimination of all cuts of complexities above n. Cut elimination as stated in Theorem 3.4.2.2 is of no help since it doesn't say anything about the controlling operators. However, we know already from the Predicative Elimination Lemma 3.4.3.6 that we can eliminate cuts whose cut-ranks are in (F/h, ~,k-t-1] for E I_im or in (Q~ + 1,~+1] without losing information about the controlling operator. Therefore we introduce the class of left initial points of these intervals, i.e., we put Reg:={I+l}U{f~l

aEICIkim}U{~,,+l[

aEI\Lim}

The crucial situation which is not yet covered by the Predicative Elimination Lemma are derivations of the form

In the case that A is a set of E~-sentences we want to collapse this derivation below a. We are going to show the following theorem.

Set Theory and Second Order Number Theory

311 A

3.4.5.3. Collapsing T h e o r e m . Let A be a set of E~-sentences, # E Reg, {n,.y,#} C_ 7/7 and assume 7-l~ ~ A. Then this derivation is collapsed to 0~(~+~+~) A To make the induction work we assume that e is a set of s more general claim

and prove the

A C_ 2 ~, # e R e g

c N{cl(

+ 1,

+ 1))1

->

[el

A (i)

by main induction on # with side induction on c~. To simplify notations we abbreviate the first three lines in the assumptions of claim (i) by Asmp(A; e; #; a; 7). So (i) becomes Asmp(A; O"' #; to; ~/) A 7/~[O] ~ A =~ 7/~+~,+~ [O] Irr

A"

(ii)

To prepare the induction we first observe that by Lemma 3.4.5.2 we have Asmp(A; O; #; ~;; ')') A c~ E 7-/.y[O] =~ 7 + w~'+'~ e ~

].

(iii)

From (iii), (182), "f' _< ~' + w~+~ and (180)we then obtain Asmp(A; O; #; a; ")') A ~ E 7/~[O] =~ 0~(~/+ w"+~) E 7/~+~+. [O]. Most important is the following collapsing property of the function r

(iv)

+ w~+~).

Asmp(A; O; #; a; "7) A c~ E 7-/~[O] A ~'+w ~+a < fl =~ 0,(~, + w"+a) < r

(v)

To obtain (v) it suffices by (169) to find some 5 _ fl such that 7 + w"+~ E Cl(5, 0,5). But we have 7 + 1 _ 7 + w"+~ and 7 + w"+~ e 7/~[O] c_ Cl(7 + 1, 0 , ( 7 + 1)) by (iii) and the assumption par(O) C_ Cl(9/+ 1, 0~(7 + 1)). So we may choose 5 "= ")f + 1. To prove claim (i) we run through the cases. If the last inference was by ( A ) then there is a sentence F E A M A - t y p e and we have the premises 7{~[O U tF(C)] ~

A, G and ac < a for all G E C (F).

(vi)

Since F E E~MA-type there is aS E par(F)M~ C 7/~[O]M~ such that par(tF(G)) C 5 for all G E C(F). For T _> ~ we get 7/~[O] MT C_ Cl(~/+l,r V = Or(')' + 1). Hence par(tf(G)) C_ 5 < r 1) C_ el(7 + 1, 0r(7 + 1)) which shows par(O U tf(G)) C_ Nr>, Cl(~/+ 1, r + 1)) for all G E C(F). So we have Asmp(A, G; O U tf(G); #; a; Z/) for all G e C(F) and get

7-l.~+~,+,o[OUtf(G)] [0"(~+~"+"~ A G

(vii)

for all G E C(F) by the side induction hypothesis. By (v) we have r + w'+"G) < r + w'+~) and r + w'+~) E 7/~+~,+~[O] by (iv). Therefore we obtain

312

W. Pohlers

"H.,.,,+,.,.+,:,,[0] I '/''' ,r C~§

+'' ))

from (vii) by an inference (A). In the case of an inference ( V ) there is a sentence F E A N V-type, a sentence G E C (F) and some So < e such that 3/~[0] ~ - A , G

and oF(G)< oL.

(viii)

By side induction hypothesis we obtain

n~+~.+oo [o] Irr176176 ~ c ,+~o) '

(ix)

From "),+w u+~~ < " y + w u+a and So e 7/7[0 ] we obtain r r u+") by (v). Since oF(G) e par(A,G)Cl ~ C_ 7/~[0] Cl n C_ r r + wu+a) we obtain

~/~+~.+~ [0] ,rIr

u+a~ < 1) _

A

from (ix) by an inference (V). In the case of an inference (Ref~) there is a II~ sentence (Vx E L~)(3y E L~)Y(x, y) such that (3z E L~)[z ~ 0 A (Vx e z)(3y e z)F(x, y)] e A, an ordinal ao such that n, C~o+ 1 < a and 7/~[O1 ~2_ A, (Vx e L~)(3y e L~)F(x, y).

(x)

By inversion we obtain from (x) 3/~[0, t] ~2_ A, (3y e k~)f(t, y)

(xi)

for all t e T~. Let r/:= r + wu+~~ and 7n := 7 + wu+~~ By #, 7, ao E 7/7[0 ] we get ~n e 3/~[0] C_ 3/~.[0, t] for all n E w. For t E T~ there is an n e w such that stg(t) < r We have par(O) C_ Cl(7 + 1, r + 1)) c_ Cl(Tn + 1, Cr(Tn + 1)) and obtain par(t) C_ stg(t) < r E 7"IT,[O]NT C_C/(Tn+l, r + 1))Fl~- = Cr(Tn + 1) for all T > n. Hence par(t) C_ Cl(% + 1, r + 1)) for all T _ n. So we have Asmp(A, (3y E / ~ ) r ( t , y); (9, t; #; n; 7n)

(xii)

for all t E T~. Observing that 7, + wU+"~ = %+1 the induction hypothesis applied to (xi) yields

n..,,,,+, [e, t] I.,rr

A, (3y e L.~)F(t, y)

(xiii)

for all t E T~. Using the Boundedness Theorem 3.4.3.7 we obtain

n~,,,+,[e, t] Irr

~ ~, (3y e t.)F(t, y)

for all t E T~. Since ~/. < ? + wu+~~

"H..,,+,.,.,.+..,,o+,[e] r

o ,) A,

< "y + wu+~ we get

(xiv)

313

Set Theory and Second Order Number Theory

by an inference (A). Since no ~0 (3x E k,)(x E In) for some 5 < r (191) below) we obtain r

+ w~+(') (cf.

A, (3z e L~)[z # O A (Vx e z)(3y e z)F(x, y)]

from (xv) by inferences ( A ) a n d (V). In the case of an inference (Ref~) with ~T < ~ we obtain the claim directly from the side induction hypotheses. The real crucial case is a cut. There we have a sentence A with rk(A) _< # and an ordinal a0 such that ?-/~[e] ~2_ A , A and ?t~[e] ~2_ A,-~A.

(xvi)

The simple case is rk(A) < a. Since n~[O] is Cantorian closed and par(A) C_ 7-/7[0 ] we get rk(A) E HT[O] n a c_ r + 1) _< r + w~+") 9This together with the side induction hypotheses applied to (xvi) yields the claim by a cut. Now assume a __ rk(A) _< #. First we consider the sub-case that rk(A) Reg. Then g _< rk(A) < rk(A) + ='~T _< #. As before we have rk(A) E ~7[O] and thus also r E 7/7[0 ] by (199) and trivially A U {A} U {-~A} C_ E ' . Hence Asmp(A, (-~)A; O; #; ~T;7) and the induction hypothesis applied to (xvi) yields 7-/7+,.,,.+,,,0[0] Jr176 'r

'

A and nT+.,,`+o,o[O] 1rr176

-~A. (xvii)

In this situation we want to make a cut, apply the Predicative Elimination Lemma (Lemma 3.4.3.6) and then use the main induction hypothesis. Since the same situation will return we are going to state this in a more general form. Let "7 <_ ~ < "7 + w "+~, ~l E ?-/,[O], rk(A) < 7r _< #, 7-/,7[0] ~ A, A and

n,[o]

Z<

9

Ir

I r I,f',,( 7 "i- ~Jl,~,` "~"0' )

A

(xviii)

Applying (xviii) with rl = ff + w'+~~ and/? = r + w"+~~ to (xvii) will then finish the case. To prove (xviii) put 5 := max{rk(A),/~} + 1 < r and choose p e Reg such that p < 5 < p+. Defining t~ := p i f p r Reg and jh:= p + l otherwise we get ItS,~ + w6) n Reg = 0. From the hypotheses we obtain ?-/,110] ',~+J ~ A by a cut. Since par(A) C_ 7-/,7[0] and ~ e 7-/,110] we have 5 e n,[O]. Predicative Elimination Lemma (Lemma 3.4.3.6) we therefore obtain

n,,[o]

V35(~+1) A .

(xix) Using the

(XX)

By (199) we get ~ E 7/,[O]. From ff < r] we also obtain par(O) _C ~r>~_ C/(-y+l, r + 1)) c Nr>~ C/(rl+l, Or(r] + 1)). So we have Asmp(A; O; t~; ~; r/) and f~ < #. The main hypothes]-s applied to (xx) thus yields

314

W. Pohlers

r

~6(

))

"

Since/5+ ~o6(fl + 1) < r < # we have w~+~6(~+1) < co". From r/< 7 + w "+~ we either get 77 <_ 7 and thus also r/+w ~+~(~+1) < 7+w" _< 7+w "+~ or 77 = 7+~ with r < co"+~ which entails 77+ w~+~(~+1) < 7 + r + cow < 7 + w"+~. Hence r w~+~6(~+1)) _< r + w"+~) and we obtain the claim from (xxi) by a structural rule. Now we consider the sub-case that ~ <_ rk(A) =: r E Reg. Then either A or ~A has the shape ( 3 x e L~)G(x). Since ~ _ ~r we easily check Asmp(A, (3x e L~)G(x); O; #; r; 7) and obtain thus by the induction hypothesis ?-/.y+.,.+oo [el I'rr176 Defining r := r

A, (Sx e k~)V(x).

(xxii)

+ co"Y+~~ we have

e 7/~+~,+~o[O] n lr

(xxiii)

by (iv) and get from (xxii) by the Boundedness Theorem (Theorem 3.4.3.7)

n~+~.+oo[O] Ir176176 r

~ ' (3= e Lr

(xxiv)

Respecting (xxiii) we may apply Downward Persistency (153) to the second premise in (xvi) and obtain 7/~+~,+.o[e] ~- A, (Vx e L()~C(x).

(xxv)

Since 7 < ")'+ w"+a~ e 7/~[(~] we obtain Asmp(A, (Vx e Lr O; #; r; "y+w "+e~ and may therefore apply the side induction hypothesis to (xxv). This yields

,a, (v= e Lr "/L~+,,.,,,+,,o+,,,,,+,,o[~1] I'~'r176176176176 r Putting 77 "- 7 + c~176 + w"+~~ and fl "= r (xxiv) and (xxvi)

(xxvi)

+ co"+~~ + w"+~~ we obtain from

~[e] ~ z~, (~ e ~)c(~) a~ ~[e] ~ ~, (w e ~)~a(~).

(xxvii)

We realize that rk((~x ~ L~)G(x)) < ~r, fl < ~r, ~/ _ 77 < -~ + w"+~ and obtain by (xviii) the claim. ~] One should notice that the collapsing procedure not only collapses the derivations but also removes applications of the rules (Ref~). 3.4.6. C o n t r o l l i n g o p e r a t o r s for a x i o m s y s t e m s of set t h e o r y The aim of the following section is to determine controlling operators for different axiom systems for Set Theory. We will see that this is a fairly straightforward procedure which parallels the last part of Section 2.1.5. Due to extensionality of sets, however, it will turn out to be more painstaking. We start with pure logic. C o n t r o l l i n g o p e r a t o r s for p u r e logic. First we show an analogue of (95)

Set Theory and Second Order Number Theory

315

Let A be a finite set of f~Rs-sentences and F be a sentence such 2.rk(F) A holds.for every acceptable operator 74. that {F,-~F} C_ A. Then 74[par(A)] 10

3.4.6.1. L e m m a .

The proof is by induction on rk(F). -type. Then we have

Without loss of generality we assume F E

Io2"rk(G) A , G,~G

(i)

V

7-/[par(A, G)]

for all G E C (F) by the induction hypothesis and obtain first

n[par(A,

Io

G)] ~.rk(a)+l A F,-~G

(ii)

for all G E C(F) by an inference ( V ) and, since 2. r k ( G ) + 1 < 2. rk(F) and par(A, G) c par(A u tF(a)), from (ii) finally 12"rk(F) A F, ~ F ,0

,

by an inference ( A ) .

[-1

For a finite set A of LRs-sentences we define rk(A) := max {rk(F)] F E A}.

(183)

Now recall the cut-free Tait-calculus introduced in Section 2.1.2. We obtain the following lemma. 3.4.6.2. L e m m a . Let A ( x l , . . . , X n ) be a finite set of f_.(E, Ad)-formulas whose free variables occur all in the list {Xl,... ,Xn} such that ~-- A ( x l , . . . ,Xn). For any ordinal )~, any n-tuple ( a l , . . . ,an) of LRs-terms of stages less than )~ and any acceptable operator 74 we obtain 7-/[par(A(a,,..., an)Lx)] Io2.(~+m) A(al , .. , an) Lx

for a :-- r k ( A ( a l , . . . , an)Lx). The proof is by induction on m. We abbreviate A ( a l , . . . , a n ) L~ by A'. In the case of an axiom (AxL) we obtain the claim from Lemma 3.4.6.1. In the case of an inference (V) there is a sentence A 0 ( a l , . . . , a n ) Lx V A l ( a l , . . . , a n ) L~ E A'. Then par(Ai(al,...,an) L~) C_ par(A') and rk(Ai(al,...,an) L~) < rk(A') and we

12'(a+mo)A', A i ( a l , . . . , an) L~ for some get from the induction hypothesis 7/[par(A')] ,0 i E {0, 1}. By an inference ( V ) we finally obtain 7/[par(A')] [02.(~+m) A'. The case of an inference (A) is treated analogously. In the case of an inference (3) there are two subcases. First assume that we are in the situation of an unrestricted quantifier. Then there is a sentence (3x E L~)F(x,~) L~ E A'. For a E T~ we have F(a,~) L~ E C((3x E L~)F(x,~)L~), O(3xeL~)f(x,~)L~ (F(a, d) L~) = stg(a) < A _< a and rk(F(a, d)) < a. Moreover, we have

316

W. Pohlers

the premise ~

A(Z),F(y,Z-). If y e {xl,...,xn} then par(F(a,,~) L~) C_ par(A'). 2.(a+mo )

Otherwise we replace y by L0. In both cases we get 7/[par(A')] 10

A', F(a, g)L~

by the induction hypothesis and obtain the claim by an inference (V). In the case of an restricted quantifier there is a sentence (3z E ai)F(x, d) L~ in A'. Assume that a, = { x e L~I G(x,~)}. From the premise ~ A,y e ~i A F(y,~) 2.(~+mo) A', a E ai A F(a, ~) we obtain by the induction hypothesis 7/[par(A')] 10 for a

E T~ as in the previous subcase.

By A-inversion this implies 2.(aTmo)

"]-/[par(A')] 12.(~+m~ A' ~a e ai and "]-/[par(A')] 10 0

A', F(a, d) We easily prove

( 84) 2.(.+mo)+~ A' G(a, ~) /X F(a, ~) by induction on a. Using (184)we obtain "]-/[par(A')] 10

and obtain the claim by an inference (V). In the case of an inference (V) we have the premise ~ A(~), F(y, :~) with y r 2.a+mo The induction hypothesis implies 7/[par(A'),b] Io A',F(b,d) t~

{xl,...,xn}.

2.a+m for all b E T~. Using an inference (A) we obtain 7/[par(A')] 10

/V.

['7

Regarding identity axioms as part of Pure Logic the next step is to deal with these axioms. We already mentioned that, because of the extensionality of sets, this is by far not simple. The tedious point is the bookkeeping of derivation lengths. To obtain precise bounds we are forced to derive all axioms step by step which is a bore. However, we do not need absolutely exact bounds. The collapsing procedure will equalize too precise bounds anyway. We already observed that the rank of a sentence is always an upper bound for its truth complexity. So we will introduce a more liberal derivation calculus ~ A and show afterwards that ~ A entails 7/[par(A)] ~ A where a is computable from rk(A). For a set (9 C_ On we define O to be the closure of (9 tJ {w} under successor, i.e., ~ ~ + 1 and regular successor, i.e., ~ ~ ~+. We define the relation ~- A by the rules

(A')

[-A, GlorallG~C(F)

~

~A,F

and (V')

~- A, F, F c_ C (F) and par(A, F) C_ par(A, F)

=~ ~- A, F.

Here we want A, ..., to denote multi-sets, i.e., sequences which are independent from the order of their elements but count their multiplicity. To avoid distinctions by cases we introduce for a E Tstg(b) the relation

F(a) A G(s) if b = {x e L~I F(x)} if b - L,.

Dually we put

(185)

Set Theory and Second Order Number Theory

a f b Y G(s) :r

{

-~F(a) V G(s)

G(s)

if b = {x E k, I F(x)} if b = k~.

317

(lS6)

For multi-sets we define analogously

S {.'.,-~F(a),G(s),...}

( . . . , a f b, G ( s ) , . . .} : - ~ {

,G(s),...}

ifb = {z E k~[ F(x)} ifb-k~.

This has the notational advantage that C (a E b) = {t e b A t = a I stg(t) < stg(b)} and C((3x E b)F(x)) - {t e b A F(t)l stg(t) < stg(b)} independent of the shape of b. There is a number of inference rules which are derivable or admissible within the calculus ~ . We list the most important ones (Str)

~A

andA C_F =~ ~-F

(Taut) ~ A,-~A (Sent) ~ A , A

=~ ~ A , ~ B , A A B

(E)

~-A, t e b A t - a f o r s o m e t E T s t g ( b )

(r

~- A, t f b, t # a for all t E %tg(b) :=~ ~- A, a r b

(V~)

~- A,F(t) for all t E T~ =~ ~ A , ( V x E L ~ ) F ( x )

(3 ~)

t E T~, par(t) C_ par(A,F(Lo)) and ~- A , F ( t )

(Vb)

~- A, t/[ b, F(t) for all t E Tstg(b) =~ ~- A, (Vx E b)F(x)

(~b)

t E Tstg(b), par(t) C_ par(A, F(L0)) and ~ A, t e b A F(t)

=~ ~--A, a E b

::~ ~- A, (3x E La)F(x)

We refer to (Str) as Structural Rule, to (Taut) as Tautology Rule, to (Sent) as Sentential Rule, to (E), (r as E-rule or ~-rule, etc. The proofs are all obvious. For a multi-set A of/:as-sentences we define ~:A "~----~ Wrk(F) FEA and observe that in rules according to (A') and (V') we always have #Ap < #Ac if Ap denotes a premise and Ac the conclusion of the rule. This will be the main argument in showing

Let 74 be an acceptable operator which is closed under ~ ~+ ~+. Then ~- A implies 74[par(A)] 0 ~ A.

3.4.6.3. L e m m a .

The proof is by induction on the definition of ~ A. In the case of an inference (A') we get the claim immediately from the induction hypothesis, the previous remark and the fact that for G E C(F) we have par(G) c par(F U tf(G)). In the case of an inference ( V ' ) we have 7/[par(A, F)] I#(A'r) ,0 A,F , F C - C(F) and

318

W. Pohlers

par(A, F) C_ par(A, F). Since ? / i s Cantorian closed and closed under ~c ~_~ ~c+ the _ ?/[par(A,F)]. latter implies ?/ [par(A, F)] C

So we get ?/[par(A, F)] I,0#(~'r) A , F

which entails ?/[par(A,F)] l0#(~,r)+lrl A F where IFI denotes the cardinality of the finite set F. Since rk(A) < rk(F) for all A E C (f) we get ~ w rk(A) + iFI < wrk(f), hence #(A, F) + IFI < #(A, F) and this yields the claim.

AEF

D

We are now going to derive a series of sentences which will be needed in the computation of the truth complexities of identity and non-logical axioms of Set Theory.

a r a for all s

terms a.

(187)

The proof is by induction on rk(a). We obtain ~ b ~ b for all b E Tstg(,) by the induction hypothesis. Hence ~-b f a, b e a A b ~( b by (Sent) which in turn gives ~-b X a, (3x E a)[x ~( b] by (3b). This implies ~-b f a,b ~ a for all b E Tstg(,). Hence ~-a ~( a by (r D

a C_ a hence also ~- a = a .for all s

a

(188)

The proof is by in induction on rk(a). We get ~-b C_ b for all a E Tstg(,) by the induction hypothesis. For symmetry reasons this implies ~ b = b. So ~ b f a, b e cabb by (Sent) and ~-b t a , b E a for all b E T~tg(,) by (E) which implies ~- (Vx E a)[x E a] by (V'). D As a corollary of the proof of (188) we obtain

F- b ,~ a, b e a for all b e Tstg(~).

(189)

By (Sent) we have ~- (3x E a)[x r b], (3x E b)[x ~ a], (Vx E b)[x E a] A (Vx E a)[z E b] which entails b a :fi b, b = a.

(190)

Since we have ~- b = b for all b E T~ we get

b b E L~ for all b E T~.

(191)

Now we show

[--- Tran(L,~).

(192)

For a E T~ and b E Tstg(a) we get stg(b) < e. Hence by (191) ~- b f a, b E L~ which in turn gives ~- (Vy E a)[y E L~] for all a E T~. Thus ~- (Vx E L~)(Vy E x)[y E L~]. D We prove the next item stg(b)<~

=~ b a # b ,

L6#a

(193)

by induction on &. Put fl := stg(b). For t E 7~ and s E ~tg(a) we obtain: ~- s :~ t, L~ # s

by induction hypothesis

(i)

Set Theory and Second Order Number Theory

319

from (i) by (Str)

(ii)

~- t 1[ b, t 7s s, 1/3 ~ s for all t e T~

~-s~b, LzCs ~-seaAsCb,

from(ii) b y ( r s;ga, ks ~ s

(iii) from(iii) by(Sent)

(iv)

~- (3x e a)[x r b], s f a, LZ ~ s from (iv) by (3 ~)

(v)

Sincea=b-aCbAbCa-(Vxea)[xEb]

A (Vx e b) [x e a] we may continue

a ~ b, s f a, Le ~ s for all s e Tstg(a) from (v) by ( V ' )

(vi)

a#b,L~r

(vii)

from(vi) b y ( r

~- a ~ b, (3x E L~)[x r a] from (vii) by (3 ~) ~- a -J= b, [_~ :/= a

(viii)

from (vii) by (V').

r-1

The most painstaking part is the proof of the following theorem 3.4.6.4. Equality Theorem. a and b we have

F(k0) and all f-.Rs-terms

For every s

a ~ b, ~F(a), F(b).

To prepare the proof we show a more general lemma which entails the Equality Theorem. Let a -J:' b denote the multi-set --a C_ b,--b C_ a. 3.4.6.5. Lemma.

Assume that A ( X l , . . . , x n ) is a Ao-formula in /:(E, Ad) in which every variable X l , . . . , Xn occurs at most once and let a l , . . . , an and b l , . . . , bn be/:as-terms. Then

al ~k' bl,..., an yk' bn, ~ A ( a l , . . . , an), A(b~,..., bn). We prove the lemma by induction on r k ( A ( a l , . . . , a n ) ) ~ rk(A(bl,...,bn)) and distinguish the following cases: A(Xl,...,xn) = xl E x2. For stg(a) < stg(a2) and stg(b) < stg(b2) we have rk(a = al) < rk(al e a2) and rk(b = bl) < rk(bl e b2). So we get for a e %tg(a2) and b E Tstg(b2) al 7k' bl, a ~ al, a ~s b, b - bl

by induction hypothesis

el :/:' bl, a :/: al, b f b2, a ~-' b, b e b2 A b - bl

(i)

from (i) by (Sent)

(ii)

al -J:' bl, a -J: al, b f b2, a r b, bl e b2 from (ii) by (e) and (V')

(iii)

al ~' bl, a -J: al, a r b2, bl e b2 from (iii) by (r

(iv)

~- al ~:' bl, a t a2, a r al, a e a2 A a r b2, bl e b2 from (iv) by (Sent) al ~' bl, a f' a2, a -J: al,-,(a2 C_ b2), b~ e b2 from (v) by (3 a'-)

(v) (vi)

W. Pohlers

320

~-al r bl, a2 -J:' b2, al ~ a2, bl E b2 from (vi) by (~).

A(Xl,...,Xn) -- Ad(xl). For ~ < min{stg(al),stg(bl)} ='1~ we have rk(L~ = al) < rk(Ad(al)) and rk(L~ = bl) < rk(Ad(bl)). Therefore we obtain: ~- al ~' bl, L~ r al, L~ - 51 for all ~ <_~ al ~-' bl, L~ r al, Ad(bl)

by induction hypothesis

from (vii) by (V')

al -J='bl, L~ ~ al, Ad(bl) ]or all ~ < stg(al)

(vii) (viii)

from (viii) and (193)

(ix)

from (ix) by (A'). For A(xl,...,xn) -- A0(xl,...,Xn) A Al(Xl,...,Xn) we obtain the claim easily from the induction hypothesis. A = (3y E Xl)B(x2,...,xn, y). Putting ~ "= a~,...,an and defining b~ analogously we obtain for a E Tstg(al) and b E Tstg(bl)" ~- a~ -~' b~, a g:' b, ~B(a~, a), S(b~, b)

by induction hypothesis

(x)

~- ~ ~' ~, b ~ bl, a r b,-,B(c~, a), be bl A B(~, b) from (x) by (Sent) (xi) a~ ~:' b~, b t bl, a -~ b,--B(a~, a), (3y E bl)B(b~, y) from (xi) by (3 hi)and (V')

~- c~ r ~,a r bl,-,S(c~,a), (3y E bl)S(~,y)

from (xii) by (r

(xii) (xiii)

~- 4 #' g, a e al A a r bl, a t al,-,S(4, a), (3y E bl)B(~, y) from (xiii) by (Sent)

(xiv)

~- a~ =/='b~,--(al c_ bl), a ,/al,--B(a~, a), (3y E bl)B(b~, y) from (xiv) by (3 a' )

(xv)

a~ ~:' b~, al ~-' bl, (Vy E al)~S(4, y), (3y E bl)S(~, y) from (xv) by (V~') and (Str). Since the claim is symmetrical in -,A(al,..., an) and A(bl,..., bn) the distinction by cases is complete. 0 To infer the Equality Theorem from Lemma 3.4.6.5 let G(Xl,..., xn) be an A0formula in which every variable xi occurs at most once such that F(a) =_G(a,..., a). Then apply the lemma to G(xl,... ,xn) and infer the theorem by an inference (V')" Observe that by (188), (190) and the Equality Theorem we have

~- A L~ for all identity axioms A.

(194)

If F(xl,...,xn) is an s Ad) formula and a l , . . . , a n are all/:Rs-terms of stages < ~ we obtain that rk(F(al,..., an)) < w. ~ + n for some n < w. So putting together Lemma 3.4.6.2 and (194) we obtain the following theorem.

321

Set Theory and Second Order Number Theory

3.4.6.6. T h e o r e m . Let F ( x l , . . . , x n ) be an f_.(e, Ad)-formula which is valid in Pure Logic with identity and 7{ an acceptable operator. Then there is an m < w such that

n

p r(a,) u

F(al,...,a,)

Controlling o p e r a t o r s for s e t - t h e o r e t i c axioms. Having established controlling operators and derivation lengths for logically valid sentences we will now determine controlling operators and derivation length for axioms of Set Theory. This is fairly easy for the axiom of extensionality. Since we defined a = b to stand for a C b A b C a we obtain (Vx e L~)(Vy e L~)[x = y 4-+ (Vz e

x)(z

e y) A (Vz e

y)(z e

x)].

(195)

We derive the remaining axioms in their modified forms as introduced in Section 3.3. We prove (Pair') L~ for A E Lira.

(196)

Let a, b e T~ and/3 := max{stg(a), stg(b)} + 1. Then ~-a e L/~ A b S L/~ from(191).

(i)

Since/3 9 par(a) U par(b) and/3 < A we get from (i) by (V') (3z e L~)[a e z A b e z].

(ii)

Hence by twofold (Vx) (Vx e L~)(Vy S L~)(3z 9 L~)[x 9 z A y 9 z].

[]

Next we show (Union') L~ for A e Lira.

(197)

Let a e T~ and c~ := stg(a). For t e T~ and s e Tstg(t) we obtain: s f t, s e L~ from (190) by (Str)

(i)

~-(Vx e t)[x e L~] from (i) by (~)

(ii)

~- t ~/a, (Vx 9 t)[x e L~] from (ii) by (Str)

(iii)

}-- (Vy 9a)(Vx e y)[x e L~] from (iii) by (Va)

(iv)

F (3w e L~)(Vy e a)(Vx e y)[x e w] from (iv) by (qLx)

(v)

~- (Vu e L~)(3w e L~)(Vy e u)(Vx e y)[x e w] from (v) by (vL~).

D

We prove the set existence axiom-schemes of Separation and Collection in the form (A0-Sep)

(Vg)(Va)(3z)[(Vx e z)(x e a A F(x, if)) A (Vx e a)(F(x, if) -+ x e z)]

322

W. Pohlers

(Ao-Col)

(vg)(w)[(w 9 ~)(3y)F(~, y, v-3 ~ (3z)(W e ~)(3y e z)F(~, y, g)]

for Ao-formulas F(x, g) and F(x, y, ~), respectively. We first prove (198)

!--(Ao-Sep) L~ for A E Lim.

Let {a, a l , . . . , a n } C_ k~ and a := max{stg(a), stg(az),..., stg(an)} + 1. Define b : - {x E k~l x E a A

F(x, a i , . . . ,

an)}.

(i)

by (Taut)and (Sent)

(ii)

Then we obtain for t E T~: F-t t b, t E a A F(t, az,... ,an)

(Vx E b)[x E a A F(x, a i , . . . , a,)]

from (ii) by (Vb)

F- t f a, ~F(t, ~), t E a A F(t, g) A t = t for t E Tstg(,)

(iii) (iv)

from (188)and (189) by (Sent) t f a,-~F(t, ~), t e b A t = t }--t f a,-~F(t,~) V t E b

reformulation of (iv)

(v)

from (v) by (E) and (V')

(vi)

~- (Vx E a)[F(x, ~) -4 x E b] from (vi) by (Va)

b (3z E L~)[(Vx E z)(x E a

A

F(x, ~)) A (Vx E a)(F(x, ~) --+ x E z)]

from (iii)and (vii) by (A') and (3~). From (viii) we finally obtain (Ao-Sep) L~ by inferences (Y~).

(vii) (viii)

D

We show

I- (Inf') Lx for w < ~ E Lim.

(199)

Let a E T~ and c~ := s t g ( a ) + 1 . Then ~ a E L~ and ~-a E l~ by (191). This entails ~(3yEL~)[y E L~] and ~-(3zEk~)[a E z] for all t E T~. Hence k~ ~= 0 A (Vx E k~)(3z E L~)[x E z]. Since w < A we get the claim by an inference

(V')We summarize Lemma 3.4.6.3, (194), (196), (197), (198) and (199).

3.4.6.7. Lemma. Let ~ be a limit ordinal and A be one of the axioms (Ext), (Pair'), (Union'), (Ao-Sep) or (Inf')and 7-l an acceptable operator. Then there is an n < w such that 7/[{A}] Io(x+") ALx

Next we prove that for acceptable 7/and n E Reg there is an n < w such that H I { 4 ] ~'0

(~o-CoD ~.

(200)

Let n E Reg and {a, al,... ,an} C_ T~. By Lemma 3.4.6.1 we obtain 7/[par(~, a)U{n}] ~'0

~(Vx E a)(3y E L~)F(x y, ~),(Vx E a)(3y E L~)F(x, y, ~). (i)

Set Theory and Second Order Number Theory

323

By (Ref)~ this implies 7-/[par(E, a)

U {n}]

~I 0

-~(Vx E a)(3y E L,)F(x, y ~ g) (~ e L,)(W e a)(3y e z)F(~, y, ~).

By two clauses (V) and some clauses 74[{n}] ~'0

(ii)

(A) this entails

(Vg E L,)(Vu E L,)(Vx E u)(3y E L,)F(x y, 0") --+ (3z e L,)(Vx E u)(3y E z)F(x, y, g).

To deal with the foundation scheme (and axiom) we prove a lemma. 3.4.6.8. Foundation L e m m a .

operator 7i. Then

Let (A} U par(F(Lo) L~) C_ 7-/ for an acceptable

Io

7-/[par(a)] 2.rk(F(a)'~)+3.(str

_~F(a)L~ ' (3x E Lx)[F(x) L~ A (Vy E x)-,F(y) L~]

for all a E T~. We prove the lemma by induction on stg(a) and get 2.rk(f(b)'~)+S.(,tg(b)+l) _~F(b)Lx' (3x e n[p~r(b)] Io

L~)[F(x) Lx A (Vy E x)-,F(y) Lx] (i)

for all b E Tstg(a). By the structural rule (151) this entails

12"rk(F(b)L~)+3"(stg(b)+l)b f a,-~F(b) L~ 7-l[par(a, b)] ,o ' (3z e L~)[F(~) ~ ^ (Vy e ~)~F(y)L~].

(ii)

Using (A) this implies (Vz E a)~F(z) L~ ' (3x e L~)[F(x) L~ A (Vy e x)-~F(y)L~].

7-/[par(a)] Io2.rk(F(b)4,)+3.(,tr

(iii)

By Lemma 3.4.6.1 we have

12"rk(fCa)L~) F(a) L~, -~F(a) L~, (3x e

7-/[par(a)] ,0

L~)[F(x) L~ A (Vy e x)-~F(y)L~]. (iv)

Putting a := 2. rk(F(a) L~) we obtain "//[par(a)]

la+3'(stg(a))+2 ,; F(a) L~ A (Vz E a)-~F(z) L~, ~F(a),

(v)

(3x E L~)[F(x) L~ A (Vy E x)-~F(y) L~] from (iii)and (iv). Hence ~+3.(st,(,))+3 ~F(a)L~, (3x E L~)[F(x) L~ A (Vy E x)-~F(y) L~] 7-/[par(a)] 1o by (V)" We get the following theorem as a corollary of the Foundation Lemma.

(vi) []

324

W. Pohlers

3.4.6.9. Foundation Theorem. Let F(x,Z) be an L:(E, Ad) formula without further free variables, A E Lira and 7-l an acceptable operator with A E 7-l. Then there is an n < co such that ?t I~'~+~+n (Vs e L~)[(3x e L~)F(x,s

E L~)[F(x,s

'0

E x)~F(y,s

To prove the theorem we introduce the abbreviation Found({x e L~I F(z, s e

e

}) :r

^ (Vy e

Observe that for {a, a l , . . . , a , } C_ 7"~ we have rk(F(a, a l , . . . , a n ) [~) < co. A + k for some k < co. From the Foundation Lemma we obtain Found({x e k~[ F(x, ~)Lx}) ?/[par(al,...,a~)] [w.~+~ ,0 by an inference (A)" Applying n inferences (A) we finally obtain the claim.

D

So far we are already prepared to compute an upper bound for [[KPwl[~cK. If K P w ~ F then we have finitely many axioms A 1 , . . . , A n of KPw, such that A1 -+ - " -~ An -+ F is valid in first order logic with identity. Hence I *'~ AlL, --+ "'" --+ AnL, --->FLfl for f~ : - co~K by Theorem 3.4.6.6. 7/0 '~+no By Lemma 3.4.6.7, (200) and the Foundation Theorem (Theorem 3.4.6.9) we therefore obtain 7to [n.2+~ FL . by cuts. Applying the Predicative Elimination Lemma ft+n (n- 1)(fl.2+w)

(Lemma 3.4.3.6) we then obtain 7t0 [~o fl+l

F L" Assuming that F is a ~1-

formula and putting a : = ~n-1)(~.~. 2 -[-co) < Cf~_l_1 we get by the Collapsing Theorem (Theorem 3.4.5.3) 7/~,+~ ]rr ]r

F L" . Finally applying Theorem 3.4.2.2 we obtain

F L,. Therefore we have shown

3.4.6.10. Theorem.

For f~ := co~K we have IIKP~,IIn _< r

To obtain also an ordinal analysis for the theories KPI and K P i we need controlling operators for axioms (Adl), (Ad2), (Ad3) and (Lim). Let 7t be an acceptable operator which is closed under regular successors and A an ordinal satisfying (V~
(i)

for some a < A depending on a. By an inference (A) we obtain from (i) n[{A}] ~ (Vx E L~)[Ad(x) --+ co E x A Tran(x)], i.e., 7/[{A}] ~ (Adl) L~. (201)

325

Set Theory and Second Order Number Theory

Similarly we obtain ~-L~ E LuVL~ = LuV Lu E L~ by (191)and (188)and a~: Lu, b r L~,--(L~ E LuV L~ = LuV Lu E L~),aE b V a = b V b E aforall #, n E A MReg. As in the previous case this implies "k/[{A}] ~ (Ad2) L~.

(202)

Now let A be one of the axioms (Pair'), (Union'), (A0-Separation) or (A0Collection). For any n E AMReg we get by Lemma 3.4.6.7 and (200) n[{~}] ~ A L" for some a < ~+. As in the previous cases this implies "k/[{A}] ~ (Ad3) [~.

(203)

For b E T~ we have n := stg(b) + < A. By (188) and (191)it follows ~- Ad(L~) A be L~. By (V') it follows ~- (By E L~)[Ad(y) A b E y] and by (A') we finally obtain ~ (Vx E L~)(3y E In)lAd(y) A x E y]. Hence . co.X+n

N[{A}] ~ - - ( L i m ) L~.

(204)

Since ~co satisfies (VxEf~co)(3~E~co)[~ E Reg A x E ~] we obtain from Lemma 3.4.6.2, Lemma 3.4.6.7, Lemma 3.4.6.3, the Foundation Theorem 3.4.6.9 and (201) through (204) that for every sentence F there is an m < co such that KPI

~ F

=~ "kt0 [n~.2+coFL.~

(205)

n~+m

r CK

If we assume that F in (205) is a ~11 -sentence we can apply the Predicative Elimination Lemma 3.4.3.6, the Collapsing Theorem 3.4.5.3 and Theorem 3.4.2.2 to obtain KPI ~ F and F

COCK

E Y]I 1

==~

~ F for some a < Cn(~'ftw_}_l).

So we have shown the following theorem 3.4.6.11. Theorem.

IIKPlll a < Cn(cn~+l).

The ordinal I not only satisfies (Vx E f~co)(3~ E f~co)[~ E Reg A x E n] but also I E Reg. So we obtain by (200), Lemma 3.4.6.2, Lemma 3.4.6.7, Lemma 3.4.6.3, the Foundation Theorem (Theorem 3.4.6.9)and (201)through (204) KPi ~F

=~ 7/0 Ir2+coF L' I+n

for some n < w. As before this implies that there is an a < Cn(r for ~l~-sentences which are provable in KPi. So we have 3.4.6.12. Theorem.

such that ~ F

IIKPill n _< Cn(r

To obtain also an analysis of the theory KP1 r we have to do a bit more. First we show ~+.+2k

~--,Lim, A(e) =~ (3n
W. Pohlers

326

for a limit ordinal A and a finite set A(Z) of E-formulas which contain only the shown free variables and an acceptable operator 7/which is closed under ~ ~-~ ~+. The proof is by induction on k and runs essentially as that of Lemma 3.4.6.2. But there is the additional (and critical) case that the main formula of the last inference is -~Lim. Then we have the premise ~-~-~Lim, (Vy)[-~Ad(y) V xi q~ y], A(~) which by inversion yields ~-~Lim,-~Ad(y) V x, r y, A(Z) for a free variable y not occurring in A(Z). By induction hypothesis there is an no < w such that

n"+"~176 , n[,~, b] I~,,+,.o+~+

---,Ad(b) V a, ~ b, A(g)L""o +~+1

for all g E 7"~, and all b E Tn,+~. So we obtain 7/[d] ,n,+.o+,+ll n"+"~176

(Vz e Ln,+,)[~Ad(z) V a/it z], A(g)L%+"o+~ .

By (188) and (191) we have 7/[{a/}] ~'02 _ (3z e Ln,+l)[Ad(z) A ai e z] and obtain

~u+no+l+2k

n[~] I.,.§

/,,(~)'~247247

by cut. Next we observe that the axioms (Pair'), (Union') and (A0-Separation) are dispensable. They can be derived from (Ad 3) together with (Lim). The same is true for (Inf'). The scheme of Ao-Foundation can be formulated as (Ao-Found)

(Vu)[Tran(u) A (Vx e u)[(Vy e x)F(y) -+ F(x)] -+ (Vx e u)F(x)]

where F(x) is a A0-formula. So if K P F ~ A(Z) for a finite set A(Z) of E-formulas then there are finitely many instances of axioms A1,..., Ak such that ~- -~Lim, ~ A , , . . . , ~Ak, A(Z). With the exception of axiom (Lim) every of these formulas -~A/is a El-formula. For a limit ordinal A and a tuple ~ E Ls~ we get by (206) #
n[~]

]::-I-k

.Lo.

.Lo.

+, -,A 1 ,...,-~A,

,a(g)'~

]n,+k A(~)L", " So we have shown Applying some cuts this implies ~[g] 'a.+l

~"+~ ~x(~)'~ KPI" ~-A(~7) =~ (V~ e Lim)(V~ e fix)(3# < A)(3k < w)[n0[d] In.+1

(:07)

For a E-sentence F this entails KP1 r ~ F

=~ (3mEw) [7/0 lu~+~n~+lF

Ln~

(208)

which for a E~-sentence F by the Collapsing Theorem and Theorem 3.4.2.2 implies ]r

F . Since limm~ Cn(gt~. w~) = Ca(~o,) we have

Set Theory and Second Order Number Theory

3.4.6.13. Theorem.

327

IIgPl~[In_
Though there is a tremendous gap between KP1 and K P i there is no difference between KPff and KPff. If we try the same analysis as we did for KPff the axiom of Ao-Collection will spoil the argument. This, however, can be remedied by augmenting the logical calculus by the following non-logical rule. (A0-Collection Rule)

~- A, (Vx e a)(3y)F(x, y) zx, (3z)(W e a)(3y e

y).

By [Ao] ~ A we denote that A is provable in the extended calculus. We obtain for a finite set A(zT) of E- formulas and an ordinal A ~,+2,~ 1 A(g)L.~,+2m] (209) [A0] ~- ~Lim, A(g) =v (V# e A)(Vde Tn,)[n[d] la,+~m+ The proof parallels that of (206). The additional case is an application of the A0-Collection Rule. By the induction hypothesis we then have (Vx e ai)(3y e La,+2mo)f(x, y).

n[~]

By an inference (Refa,+2~o+l) and Upward Persistency this implies 7/[g] ,a,+2mo+~+ll a€176 A(d)a~+2~o+2, (3z e La,+~o+2)(Vx e a,)(3y e z)F(x, y) Now replacing (206) by (209) we obtain with the same strategy as in the case of KPF KPff ~ F =v (3nEw)[7io In"a.+l F] for E~ sentences F. So we have 3.4.6.14. Theorem.

[IKPi~lln_ Cn(f~).

We will roughly sketch how an analysis for W-KP1 and W - K P i can be obtained without going too much into details. The stumbling block here is the scheme of Mathematical Induction. The remedy is to augment the calculi ~and [A0] ~ - , respectively, by an w- and a cut-rule. Call these extended calculi [w] ~ and [A0,w] ~ , respectively. Then if W-KP1 ~ - F or w - g P i ~ F we obtain [(Ao)] ~----,A1,...,~Ak,~MI1,...,~MIt, F for finitely many instances MIi of Mathematical Induction. Since [(A0) ' w] ~' 0 MI, we obtain by cut [(A0),w] ~-~---~A1,...,-~Ak, F which by the usual cut-elimination for w-logic (cf. Lemma 2.1.2.9) entails [(A0),w] ~ -~A1,..., ~Ak, F for some a < Co. Now we are in a situation similar to that in the analyses of KP1 r and KPff. While (209) modifies directly to

(2 o)

328

W. Pohlers

we cannot directly adapt (206) to the calculus [w] ~ derivations. We have to refine the argument and prove

because we have infinite

[w] ~-~Lim, A(~) =~ (VA e Lim)(V~, b e T~)[A(~) ~ c_ E~ =~

I'~+1 for a finite set of E-formulas A(~) by induction on a. The crucial case is again that the main formula of the last inference is -~Lim. Let ~,b such that A(~) b C_ E ~. From the induction hypothesis we obtain 7/~,~+3~o [~, b, c] [a~+a~o -~Ad(c) V ai r C,/k(~) b for all c E T~+ which by an inference

'~++1

(A) implies ~/o,n~+3~o[~,b] ]flX+3c~o+l~(3Z E L~+)[Ad(z) A ai E z], A(a) b By cut we K+-I-I

obtain n~,x+3oo[~,b] 1a~+3~o+2 A(~)b. Since wn~+3~~ + w~++n~+3~o+2 < w~+3~ we ',r

obtain 7/~,~+3~[~, b] I~+1 "~+3~ A(~) ~ by the Collapsing Theorem. By the now familiar technique we obtain from (210) W-KPi~-F

=~ ( 3 a e c ~

7/~ In~n~+lFL"~]

(212)

for ~l-sentences F. By the Collapsing Theorem and Theorem 3.4.2.2 this implies

3.4.6.15. Theorem.

IIW-KPilln _< Cn(~o).

From (211) we obtain as in the proof of (208) W-KP1 ~ f

=~ (3a e Co) no,,~+~ [~+~+t EL"]

for ~l-sentences F. Since Cn(w ~+n~+~) -- r w~) < r get from (213) by the Collapsing Theorem and Theorem 3.4.2.2 3.4.6.16. Theorem.

(213) Co) for a < Co we

[IW-KPlll ~ _< Cn(fl~" c0).

The analysis of theories for iterated admissibles needs serious extra work which has first been done by M. Rathjen. We have to show that there are operator controlled derivations for the axioms ItAd(a, f). This is prepared by a Lemma which is provable in K P F . 3.4.6.17. Lemma. (KPI r) Let a and u be admissible sets such that a E u and let a e a. Then (V~ < a)(3f e a)ltAd(~, f) implies (3g e u)ltAd(a, g). Proof. Since ItAd(~, f) is a Ao-formula we get from the hypothesis by Ao-Collection relative to a a set b e a such that (V~ < a)(3f e b)ltAd(~, f). By (130) for every < a there is exactly one such f~ E b. By A0-Separation relative to a we get h := U~<~ f~ E a. Clearly h is a function. Let c be an E-least element of { x e u I Ad(x) A x ~ r n g ( h ) } . Since Ad(a) A a e u the set c exists by A0Foundation and we define

Set Theory and Second Order Number Theory

h h U { ( 7 , c)}

g:=

329

i f a E Lim U {0} ifa=9'+l.

Then g is a function with domain a and we easily check

ItAd(a, 9).

0

Recall the function A~. T~ which enumerates the admissible ordinals. We have w fl,~

Ta =

Qa+l

if a = 0 for 0 < a < w for w < a < I

I

fora=I

From Lemma 3.4.6.17 and (207) we get for a limit ordinal u and a < u a k < w such that

I

~Ad(L~) -~Ad(L~-o+,) a ~ L~,---,(V~ E a)(3f E L~)[ItAd(~ f)] L~ r L,~+,, (39 E L~+,)[ItAd(a, 9)1.

Ad(L o+,) ) For a < I we have 7~0[{a}] ~ IT~+I Ad(L,~) ) ~/o[{a}] ,~.+~ L~ E L~+ 1. So we get by some cuts

i,o+,,+,," ~. +,+,, -,(V~ e a)(3.f

E

e

L,~)[ItAd(~, .f)], (3g E L,~+l)[ItAd(a, g)]. (214)

Now we show by induction on a < I "~/o[{a}] ,n,~+.,

(3f E

L~+,)ltAd(a, :).

(215)

Ig16+,, n'+~+4"(~+1) a r On, (3f E L,,+,)/tAd(~, f) , Fta+~+4a+3 for all ~c < a. This implies 7/0[{a}] In~+~ (V~ < a ) ( 3 f E Lr~)ltAd(~, f) by two By the induction hypothesis 7/0[{a,~r

inferences ( V ) , an inference ( A ) and Upward Persistency. From this and (214) we obtain the claim by a cut. [3 Since ~ < v implies ~ + w _ v for v E Lira we obtain from (215) n o [ { 4 ] In~ ''" +" (V~ <

u)(3f E Lfl~)ltAd(~, f)

(216)

for u E Lira. The strategy we applied for the ordinal analysis of KP1 together with (216) now yields KP1, ~-F(g)

n~+m F(g)L"~ =~ 7/o[~1 In''2+''

(217)

for a limit ordinal u and all ~ E T ~ . By the meanwhile familiar argument this entails 3.4.6.18. T h e o r e m .

IIKPI.IIa

--<~CD(Ef~u+l)

An ordinal analysis of the theory KPI~ for limit ordinals u is obtained in in a similar way as that of KP1 r. We modify (206) to

~- -,Lim,-,(V~ < u)(3 f)ltAd(~, f), A(~) =~

(w <

e "-r,o)

(218)

{o,}1

s

A(~)Lfla+ w]

W. Pohlers

330

for a finite set A(g) of ~-formulas and an acceptable operator which is closed under A~. f ~ . The proof is analogous to that of (206) with the additional case that the main formula of the last inference is ~(V~ < u)(3f)ltAd(~, f). Then we have the premise -~Lim,-~(V~ < u)(g f)ItAd(~, f), ui e O n A ui < u A (Vf)-~ltAd(ui, f). Applying inversion we obtain for c~, ai < u, ~ e T~ and f e T~.+I by induction hyn~+~+4(~+,+ko) -~ltAd(ai, f), A(~)L"~+~ . By an inference ( A ) pothesis 7-/[~, {c~}, f] In.+ ~ in~+~+4(~+ko+l)+l this yields "k/[~, {c~}] 'a.+~ -,(3f e L~.+l)ltAd(c~, f), A(g)L"-+ ~ which together with (215) entails the claim by a cut. If KPlr, ~- F for a 2-sentence F then there are 111 sentences A1,..., An - the axioms different from Lim and the iteration axiom - such that by (218) we obtain f~a+~+ft,, n0[{~}] I,~ ~A~-Lna+~ ,..., ALnn~+o, , FLn~+o,

for some a < u. Applying some cuts we get

no+~.2FL~ =~ (3a < u) [n0[u] Ifl~+~

KPI~ ~ F

]

(219)

for E-sentences F. If we assume that w < u and u is additively indecomposable then a < u implies c~ + w < u. So we obtain from (219) 3.4.6.19. T h e o r e m .

[]KPI~IIn _< Cn(ft~)for additively indecomposable ordinals

V.

We can also say something for limit ordinals which are additively decomposable. We generalize (211) to [w] ~ -~Lim, ~(V~ < u)(3f)ltAd(~, f)A(~)

=~

(v~, b e T~)[~X(~) ~ c r ~ ~ n~~176 --

(220)

b] I"~+'" ~x(~) ~] ~;-I-I

"

Therefore we obtain from KPI~ ~ F for a ~-sentence F a k < co such that Ln

7~,..+~ In~+k F n-l-1

"

IIKPl',,lln <

Thisshows

Cn(~ .~)

for limit ordinals which are additively decomposable. I have, however, never checked whether this ordinal is the precise one or in other way meaningful. In all relevant applications the ordinal u is additively indecomposable. We also obtain an analysis of W - K P I ~ . If W - K P I ~ ~ F for a ~-sentence F we obtain similarly as in the case of W - K P I an c~ < c0 such that by (220) 7-/~o,~+.[{oL}] If~+~ E L" Applying Collapsing and Theorem 3.4.2.2 we obtain n+l 3.4.6.20. T h e o r e m .

For limit ordinals u we have IIW-KPluI[~ < Cn(f~v" Co).

Set Theory and Second Order Number Theory

331

Recall from (169) and (170) that Cx0 is the least fixed point of the function A~c. R e . By (215) we get therefore 7/o ~ 'r

(Vc~e Lr

e Lr

f).

(221)

By the now familiar procedures we obtain from (221) [_r

A u t - K P 1 ~ F =~ 7t0 ir

n - FL+,o

(222)

I W - A u t - K P 1 ~ F =~ (3a < e0)(Vb E Lr for E-sentences F. Therefore we obtain

[Hr

(223) ~

'f~+l

FL,] (224)

3.4.6.21. Theorem.

(i) IIAut-KPF Ila _< Ca(r (ii) [IW-Aut-KPF II~ _< r162

9~'o)

(i) IIAut-KPlll a < r162 From (215) we get also 7i0 ~ our previous work shows 3.4.6.22. Theorem.

(V~ e l a , ) ( 3 f e la,)ltAd((, f) which together with

[IKPl,[I a <~ ~)~t(E~n.C1) "

These examples should suffice to demonstrate the technique of ordinal analysis. We already mentioned that it is not the most recent state of the art. On the one hand the analyses of much stronger axiom systems such as Ha reflection and even El-Separation - which corresponds to IIl-comprehension _ are known today. The collapsing functions needed there are based on a H~ ordinal which - when we stick to the simplification of using regular cardinals instead of recursively regular ones - corresponds to the first weakly compact cardinal. The cardinals which are in this way connected to collapsing functions for El-separation are considerably larger. On the other hand we also omitted a whole zoo of theories which are between (A~-CA) and ( A ~ - C A ) + (Bi), i.e., between KPleo and KPi. Examples for such omitted systems are W - K P i + (E-FOUND) which on the side of subsystems of Number Theory corresponds to autonomously iterated A~ comprehension (Aut-A~) (and whose ordinal is r162 We arranged the theories in Table 1 in such a way that in every row the theories are embeddable from left to right. We have shown that the ordinal in the column IIAxlIwcK is an upper bound for the "strongest" theory, i.e., the Set Theoretic theory. To show that the bounds are precise it suffices to arithmetize the notation system (whose construction is indicated on page 308) and to give a well-ordering proof for the segment below IIAxll~cK within the theory in the leftmost column. All the shown

332

W. Pohlers

Restricted Comprehension Set Theoretic and Choice in NT2 Theories

Theories for Inductive Definitions lSt-order

2nd-order

ID1

ID~

ID
(ID~,)o

W-ID~t

ID,.,

(Ill-CA) -

KPw

Cn(~n+l)

(II I- CA)o ( A1- CA)o

KPff KPi r KPfiF

Cn(n~)

ID~

(II1-CA)

W-KPI

BID~

(I-II-CA) + (Bi)

KPI

BID~w~

ID
(III-CA<~)

ID~

(A~-CR)

BID~6 o

(II1-CA<~o)

ID
iD2eo

ID<~

BID~. I D l y + (ai)

(A~-CA) ( ~ - A C ) '

KPI~

Cn(~)

KPI~o W-KPi W-KP/3

Cn(~,o) !

(III-CA<~) (III-CA<~) + (Bi)

i KPlr' ,

JI

Cn(nv)*

(ID~)o

(Illl-CAv)o

KPIr,

Cn(nv)**

W-IDv t

ID~

(Illl-CAv)

W-KPlv

Cn(nv" eo)tt

IDv

BID2v ID 2 + (Bi)

(III'CAv) + (Bi)

KPlv

Cn(en~+l)tt

ID.<,

I BID2, ID 2. + (Bi)

KPI* = KPI~

Cn(enn+l)

(Aut-ID)o

(Aut-Illl)o

Aut-KPff

r162

Aut-ID

(Aut-II 1)

W-Aut-KP1

r162

Aut-BID i

IIAxll~o~

i

(Aut-Hll)+(Bi) i

Aut-KPI i

(A~-CA) + (Bi) ' (E~-AC) + (Bi)

co)

r162 i

KPi KPf}

Cn(el+l)

Fig. 1: H~-ordinals of some impredicative theories. Notice that ~ : - w~K tWe did not define this theory. For details cf. Buchholz et al. [1981] ~For v = wP, p E Lira ttFor v E Lira ~For additively indecomposable v

333

Set Theory and Second Order Number Theory

upper bounds are precise ones. Unfortunately there is not enough space to indicate the well-ordering proofs. Details for the well-ordering proof for IDv are in Buchholz and Pohlers [1978] and Buchholz et al. [1981]. More well-orderings proof are carried through in Rathjen [1988]. Notice that the remark in the beginning of Section 2.2.2 applies to all the ordinal analyses given in this article. So these analyses are all profound and can therefore be extended to II~ References P. H. G. ACZEL, H. SIMMONS, AND S. S. WAINER [1992] eds., Proof Theory, Cambridge, July, Cambridge University Press. J. BARWISE [1975] Admissible Sets and Structures, Perspectives in Mathematical Logic, Springer-Verlag, Berlin. A. BECKMANN AND W. POHLERS [1997] Application of cut-free infinitary derivations to generalized recursion theory. To appear in Annals of Pure and Applied Logic. B. BLANKERTZ [1997] Beweistheoretische Techniken zur Bestimmung yon H~ Westf~ilische Wilhelms-Universit~it, Mfinster.

Funktionen, Dissertation,

B. BLANKERTZ AND A. WEIERMANN [1996] How to characterize provably total functions by the Buchholz' operator method, in: GSdel '96: Logical Foundations of Mathematics, Computer Science and Physics - Kurt GSdel's Legacy, P. Hhjek, ed., Lecture Notes in Logic #6, Springer-Verlag, Berlin, pp. 205-213. W. BUCHHOLZ [1991] Notation systems for infinitary derivations, Archive for Mathematical Logic, 30, pp. 277296. [1992] A simplified version of local predicativity, in: Aczel, Simmons and Wainer [1992], pp. 115-147. W. BUCHHOLZ, E. A. CICHON, AND A. WEIERMANN [1994] A uniform approach to fundamental sequences and hierarchies, Mathematical Logic Quarterly, 40, pp. 273-286. W. BUCHHOLZ, S. FEFERMAN, W. POHLERS, AND W. SIEG [1981] eds., Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof- Theoretical Studies, Lecture Notes in Mathematics #897, Springer-Verlag, Berlin. W. BUCHHOLZ AND W. POHLERS [1978] Provable well orderings of formal theories for transfinitely iterated inductive definitions, Journal of Symbolic Logic, 43, pp. 118-125. S. FEFERMAN [1964] Systems of predicative analysis, Journal of Symbolic Logic, 29, pp. 1-30. [1970] Formal theories for transfinite iteration of generalized inductive definitions and some subsystems of analysis, in: Kino, Myhill and Vesley [1970], pp. 303-326. H. M. FRIEDMAN [1970] Iterated inductive definitions and ~ - A C , in: Kino, Myhill and Vesley [1970], pp. 435442.

334

W. Pohlers

H. M. FRIEDMAN AND M. SHEARD [1995] Elementary descent recursion and proof theory, Annals of Pure and Applied Logic, 71, pp. 1-45. J.-Y. GIRARD [1987] Proof Theory and Logical Complexity, vol. 1, Bibliopolis, Naples.

G. J~.GER [1980] Theories.foriteratedjumps. Handwritten notes. [1986] Theories for Admissible Sets. A Unifying Approach to Proof Theory, Studies in Proof Theory, Lecture Notes, #2, Bibliopolis,Naples. G. JAGER, R. KAHLE, A. SETZER, AND T. STRAHM [n.d.] The proof-theoretic analysis of transfinitely iterated fixed point theories. To appear in Journal of Symbolic Logic. G. J)i.GER AND W. POHLERS [1983] Eine beweistheoretische Untersuchung von (A~-CA) + (BI) und verwandter Systeme, Bayerische Akademie der Wissenschaften, Sitzungsberichte 1982, pp. 1-28. G. J XGER AND T. STRAHM [n.d.] Fixed point theories and dependent choice. Submitted for publication. R. KAHLE [1997] Applicative Theories and Frege Structures, Dissertation, Institut fiir Informatik und angewandte Mathematik, Universitiit Bern, Bern. A. KINO, J. MYHILL, AND R.. E. VESLEY [1970] eds., Intuitionism and Proof Theory, Studies in Logic and the Foundations of Mathematics, Amsterdam, North-Holland. Y. N. MOSCHOVAKIS [1974] Elementary Induction on Abstract Structures, Studies in Logic and the Foundations of Mathematics #77, North-Holland, Amsterdam. E. PALMGREN [n.d.] On universes in type theory. To appear. R. PLATEK [1966] Foundations of Recursion Theory II, dissertation, Stanford University. W. POHLERS [1978] Ordinals connected with formal theories for transfinitely iterated inductive definitions, Journal of Symbolic Logic, 43, pp. 161-182. [1981] Proof-theoretical analysis of ID~ by the method of local predicativity, in: Buchholz et al. [1981], pp. 261-357. [1982a] Admissibility in proof theory; a survey, in: Logic, Methodology and Philosophy of Science VI, L. J. Cohen, J. Los, H. Pfeiffer, and K.-P. Podewski, eds., Studies in Logic and the Foundations of Mathematics #104, North-Holland, Amsterdam, Aug., pp. 123-139. [1982b] Cut elimination for impredicative infinitary systems II. Ordinal analysis for iterated inductive definitions, Archiv flit Mathematische Logik und Grundlagenforschung, 22, pp. 69-87. [1989] Proof Theory. An Introduction, Lecture Notes in Mathematics #1407, Springer-Verlag, Berlin. [1991] Proof theory and ordinal analysis, Archive .for Mathematical Logic, 30, pp. 311-376. [1992] A short course in ordinal analysis, in: Aczel, Simmons and Wainer [1992], pp. 27-78.

Set Theory and Second Order Number Theory

335

M. RATHJEN [1988] Untersuchungen zu Teilsystemen der Zahlentheorie zweiter Stufe und der Mengenlehre mit einer zwischen A~-CA und A~-CA +BI liegenden Beweisstiirke, Dissertation, Westfdlische Wilhelms-Universit~it, Miinster. [1991] The role of parameters in bar rule and bar induction, Journal of Symbolic Logic, 56, pp. 715-730. [1995] Recent advances in ordinal analysis: II1-CA and related systems, Bulletin of Symbolic Logic, 1, pp. 468-485. A. SCHLUTER [n.d.] What is provable using first order reflection. To appear in Annal of Pure and Applied Logic. A. SCHLOTER [1990] Autonom erreichbare Mengen, Miinster.

Diplomarbeit,

Westf'eilische Wilhelms-Universit~it,

A. SCHLUTER [1993] Zur Mengenexistenz in formalen Theorien der Mengenlehre, Dissertation, Westf'~lische Wilhelms-Universit~it, Miinster. K. SCHtITTE [1960] Beweistheorie, Springer-Verlag, Berlin. Translated into English as Schiitte [1977]. [1965a] Eine Grenze fiir die Beweisbarkeit der transfiniten Induktion in der verzweigten Typenlogik, Archiv fiir Mathematische Logik und Grundlagenforschung, 7, pp. 45-60. [1965b] Predicative well-orderings, in: Formal Systems and Recursive Functions, J. N. Crossley and M. A. E. Dummett, eds., Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, July, pp. 280-303. [1977] Proof Theory, Springer-Verlag, Berlin. W. STRAHM [n.d.] First steps into metapredicativity in explicit mathematics. In preparation. G. TAKEUTI [1975] Proof Theory, North-Holland, Amsterdam. A. WEIERMANN [1996] How to characterize provably total functions by local predicativity, Journal of Symbolic Logic, 61, pp. 52-69.