Reflection Principles and Indescrib Ability

SETS AND CLASSES; On the work by Paul Bernays

@ North-Holland Publishing Company (1976) pp. 277-323.

REFLECTION PRINCIPLES AND INDESCRIBABILITY Klaus GLOEDE Heidelberg, West Germany 0. Introduction

The aim of this paper is to present some results supplementary to P. Bernays' investigation of a second-order reflection principle (BERNAYS [ 19611). This schema is classified within the hierarchy of reflection principles for various classes of formulas, beginning with the first-order language of Zermelo-Fraenkel set theory, which leads to the principles first introduced and studied by LCvy, and ending with the language of finite (or even transfinite) type theory, which is related to the notion of indescribability in the sense of Hanf and Scott. Taking this view, one is led in a natural way to take into account two aspects in the investigation of various set theories: (i) the distinction between class terms (in the sense of ZermeloFraenkel set theory) and class variables (in the sense of Bernays and von Neumann-Bernays set theory) and (ii) the classification of set-theoretical formulas according to their logical complexity. Montague-Vaught's theorem is the most illuminating and striking illustration of (i), and we shall encounter a similar phenomenon at a higher level of type. (ii) Leads us to consider some subtheories of the BernaysLCvy theory BL, in particular the theory EL, which corresponds to II,'-indescribability. A comprehensive treatment of the subject as indicated would require a whole monograph. Therefore, we restrict ourselves primarily to those results which have not yet been published in detail (although they seem to be well known to a large extent). 1. Reflection principles in set theory without class variables 1.1. Reflection principles in the language of ZF set theory 1.1.1. The language of ZF set theory, denoted by YZF,is the usual first-order language with (set) variables vo, v , , ...,atomic formulas v, = v,

277

278

KLAUS GLOEDE

and v,,E v,, and the logical symbols 7 , A , v, +,t*,V and 3.We also use lowercase letters a, b, c ,... for free, and u, v , x ,... for bound variables (possibly with indices). Formulas of this language are denoted by cp, $, . . . . Some of the notation is self-explanatory. 1.1.2. Relativization. If cp is a formula of LEzF, the relativization of cp to a, denoted by Re1 (a,cp)or simply cpa, is the formula obtained from cp by replacing any quantifier Vx or 3 x occurring in cp by Vx E a and 3x E a respectively. Here, Vx E a $ and 3 x E a t,b are regarded as abbreviations of the formulas Vx (x E a + $) and 3 x (x E a A $) resp. Though the operation of relativizing is a purely syntactical notion (to be defined by recursion on the subformula relation), it has an obvious model-theoretic interpretation: Let T be a sufficiently strong set theory, e.g. Zermelo-Fraenkel set theory ZF or the Kripke-Platek theory KP of admissible sets (which we usually assume to include the axiom of infinity). One can define in T a set Fml,, which intuitively represents the set of all formulas of the language of ZF. Every formula cp of ~ Z corresponds to a suitably chosen term ’cp7 of T (the “Godel number of cp” which may be a natural number or, more generally, a hereditarily finite set) such that T rcpl E FmlzF. Moreover, in the usual recursive manner one can define in T a formalized notion of satisfaction in a structure ( a ,E ) by a formula $o(vo,v1,v2) of 2 z F such that

(a) in T +ho(a,e,b)intuitively expresses the statement

“ e is the Godel number of a formula cp of LEzF with free variables among v,, . .., v,,for some natural number n, b is a sequence of n + 1 elements of a and cp is true in the structure (a,E ) if b ( i ) is assigned to vifor i = 0,. . . , n”, (b) one can prove in T the following schema:

for any formula cp with Godel number e and any assignment b as in (a). (For more details see, e.g. JENSEN-KARP[1971],p. 147, or LEVY119651, 05, for formulas of rank sj, it is rather obvious how to remove this restriction.) In order to remind the reader of the intuitive interpretation of the formula we write for $,,(a,e,b),

F

REFLECTION P R I N C P L E S A N D INDESCRIBABILITY

or-somewhat

,279

imprecise-

( a , E ) F e [b(O),...,b(n)I (if n is suitably chosen). Thus, we may read cp a (an,. ..,a,) as “cp holds in a ” (under the obvious assignment and with the obvious interpretation of the membership relation). However, it should be kept in mind that cpa(an,..., a,) is a formula of TZF (in contradistinction to the metamathematical statement “cp holds in a”) and hence corresponds to a formalized notion of satisfaction. 1.1.3. Reflection principles (introduced into set theory by LBVY[ 19581). These are schemata of the following form: PR, (9)

cp(ao,..., an)+3u [$(U)Aao,..., a, E u Acp’(ao ,..., a,)] (partial reflection)

or

C R, ( 2 )

3~ [ a E u A t , h ( u ) v V x o * * -Eu(cp(Xn, x. ...,X n ) (py(xn,...,xn)>l (complete reflection )

where in both cases cp is any formula of 9 with free variables among v,, ..., v, and $ is some fixed formula (containing possibly free variables in addition to the indicated variable). If 9 contains formulas which are not in the basic language ZzF,the notion of relativization has to be extended in a suitable way to apply to all formulas of 2;examples of such extensions are discussed later. In view of the model-theoretic interpretation of the notion of relativization (1.1.2) one can interpret these principles roughly as follows:

any property (not necessarily in the one-place sense) which is expressible by a formula of 2 and which is true in the universe (under some given assignment) is true (under the same assignment) in some set satisfying $ (PR,(9)); or in the case of complete reflection (CR$(9)):

for any property expressible by a formula cp of 6p and any set a there is a set b which contains a as an element, satisfies $ and is a complete “reflection” ofthe universe with respect to the property expressed by cp. It is shown in the sequel that reflection principles can conveniently be applied to yield the existence of sets satisfying certain closure properties.

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KLAUS GLOEDE

In standard axiomatizations of set theory, the existence of such sets is proved by means of the axiom schema of replacement and certain types of the axiom of infinity. Conversely, the latter axioms, together with some basic axioms of set theory, imply certain types of reflection principles. (This fact should be rather obvious to anyone familiar with the Lowenheim-Skolem theorem.) Axiomatizing set theory by means of reflection principles has several advantages. First, from the model-theoretic interpretation of these principles, one can easily deduce metamathematical properties of the corresponding systems (e.g. in the case of CRTrans(ZzF): Z F is not finitely axiomatizable).’ Second, the principles of partial and complete reflection €or the language of ZF set theory admit straightforward generalization to richer languages (higher order, infinitary languages, etc.), thus yielding stronger axiom systems, or else can be weakened by restricting Z to suitable classes of formulas. 1.1.4. For the sake of completeness and later reference, we indicate main steps for a proof of the well-known fact that the axioms of ZF can be derived from a principle of complete reflection together with some basic axioms. The axioms of ZF are the following:’ Ext Pair

Sum Pow Inf RePIS Fund

-

V x ( x E a ++ x E b ) + a = b (Extensionality) 3 x V y ( y E x t,y = a v y = b ) (Pairing) 3x Vy(y E x (3z E a ) y E z ) (Sum set) 3 x V y ( y E X t,y a ) (Power set)

3 x @ Y ( Y E x ) A V Y E x ({y)Vx)l (Infinity) VX,Y,Z (cp(X,Y,...)A(X,Z,...)--, Y = z ) + (Axiom schema 3 z V y ( y E t t,3 x E a ~ ( xy , . . .) of replacement) (Axiom of foundation, 3 y ( y E a ) - + 3 y E a Vz E yz$Z a Fundieru ng )

Here and in the following we use the common set-theoretical notations: ( a } = {a,a}, ( a & ) = {(a},(a,b}} (ordered pair), C (inclusion), P ( a ) (power set of a ) . A Ao-formulacp is a formula of TEZF in which all quantifiers are restricted to sets, i.e. of the form Vv, E v,,, or 3v, E urn (LEVY [1965]).For example Rel(rp,a) is a Ao-formula in which all quantifiers are restricted to the same set a ; in

0 (empty set), {a,b) (unordered pair),

I A proof of this statement (due to R. Montague) can be found e.g. in FELGNER [1971al, p. 20 and KRIVINE(1971), p. 54. For a generalization, cf. LEVY[1%0]. ’The symbol “=” is regarded as a logical symbol.

28 1

REFLECTION PRINCIPLES AND INDESCRIBABILITY

general, quantifiers may be restricted in A,-formulas to different set variables. Thus, the definition of a transitive (or complete) set may begiven by the following A,-formula: Trans ( a ) : f* Vx E a Vy E x (y E a). We also need the following definition: Strans ( a ) :-Trans (a) A Vx E a V y (y C x + y E a ) ( “ a is supertransitive (supercomplete)”; cf. BERNAYS [1961] and SHEPHERDSON [19521).

We frequently make use of the fact that Ao-formulas are “absolute” with respect to transitive sets, i.e. if cp(ao,...,a,) is a A,-formula with free variables as indicated then

-

Trans (a ) -+ Vxo. x. E a

(cp (x,,

. ..,x, )

-

Re1 (a,cp (x,, .. . ,x,)).

The Aussonderungsschema (schema of subsets)

AusS

3XVy(yEXt*yEU

A

cp(y, ...))

is provable in ZF. For our purposes, it is often sufficient to consider the weaker

AO-AUSS

3X

v y (y

E

E X f*y

where cp is any Ao-formula. We now turn to the principles PR,

U A

cp(y, ...))

= PR,(LF’zF)

and CR, = CR,(LF’ZF):

1.1. PROPOSITION

(a) (b)

Ext + A,-AusS Ext + A,-AusS

+ PR, Pair (for any formula + PRTrans Pair + S u m + Inf.

+),

PROOF. Applying PR, to the formula Uo

= UoA

Ul

= Ul

we obtain the formula 3X

(UoEX

A

UlEX)

and hence Pair by means of A,-AusS. This proves (a). The proof of (b) is similar and is given in BERNAYS[1961], pp. 127-128. REMARK1.1. Let PRJ be the schema cp(ao,..., & ) + 3 u

($(u) A ( P U(ao,...,an))

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KLAUS GLOEDE

(cp any formula of LZZF),as considered in BERNAYS [1961], p. 126. Though this is a weakening of PR,, both schemata are in fact equivalent. (To deduce an instance of PR, from PRI, apply P R I to the formula cp(ao,..., an)A 3 x (x = ao)A . . - A3 x (x = a n ) . )

+ PRs,,

PROPOSITION 1.2. Ext + A,,-AusS

Pow.

PROOF(cf. BERNAYS[1961], p. 139). By PRs,,,,, there is, for any set a a set b such that a Eb

A

Strans ( b ) ,

hence a Eb

A

Vx E b Vy (y C x + y E b),

p(a)Cb.

Hence 9 ( a ) is a set by A,-AusS. Using the axiom of pairing, if $(a)+Trans(a), CR, implies PR,. However, whereas the derivation of the axiom of pairing from PRJ, is obvious, we first need the axiom schema of replacement to derive Pair from CR* Without using Pair one can prove as in Propositions 1.1 and 1.2: PROPOSITION 1.3. (a) (b)

+ + AO-AUSS+ CRStransk POW.

Ext + Ao-AusS CRTrans Vx 3 y (y = {x}) + Sum + Inf, EXt

Before proving the axiom schema of replacement we need a lemma in which the iterated application of CRTranscan be replaced by a single one: LEMMA1.1 (LCvy). Ext

+ Ao-AusS+ CRTrans 3 u [Trans ( u ) A Vxo. 'x,

for any finite sequence of formulas among vo,..., v,.

ql,. . . , cpm of

rn

E u A (pi * cpi")] i=l

LfzF with free variables

PROOF(cf. Levy [1960], Theorem 2). A proof can be found in FELGNER [1971a], pp. 18f and KRIVINE[1971]. It should be noted that having derived the axiom of pairing (Corollary 1.l), one can easily get Lemma 1.1 with the additional clause a E u. THEOREM1.1 (LCvy).

+

Ext + Ao-AusS CRTrans ReplS.

k

PROOF(outline). Since we refer later to the proof of this theorem, we

REFLECTION PRINCIPLES AND INDESCRIBABILITY

283

indicate the main steps of the proof which uses an argument taken from the proof of Theorem 3 of LEVY [1960]: Let a,, . . ., a,, a, b be all the free variables occurring in cp(a,b) and let + ( d ) be the formula we wish to prove: $(d):=Vx,Y,z

(cP(x,Y)A q(&z)+y = Z ) + 3 u V v (v E u - 3 x E dcp(x,v).).

Put

1

qJ dX,Y = q (X,Y

1,

( o ~ ( x= ) $(x),

) = 3 w cp (x, w 1, q ~= 4 V x l * * *xnx +(x). cpI(X,Y

Applying Lemma 1.1 to these four formulas we obtain the existence of a transitive set c such that (i) (ii) (iii) (iv)

Vxl.*.xnx,y E c VXl

* *

'Xn,X

(P(X,.Y)-CP"(X,Y))

Ec

(3wcp(x,w) t-* (3 w cp(x,w))")

VX1***Xn,XE c

V X ~. ., . X,,X +(x)

(+(x)t-* +'(x))

-.

c-* ( V X ~ , .

9

XnJ

+(x))~*

The idea of the proof is as follows: Because of (iv), we need only show that the axiom of replacement (for cp as above) holds when relativized to c . Hence we assume that d and the parameters occurring in cp are elements of c. Because of (iii), we need only show $ ( d ) (instead of + ' ( d ) ) . So we may assume Vx,y,z (p(x,y) A cp(x,z)+ y = z ) , i.e. cp defines a function in the obvious sense. Since c is transitive, d c . (i) implies that cp is absolute with respect to c, and together with (ii) we see that c is closed under the function defined by cp. So we obtain: Vy ( 3 X E d cp(X,y) f* y E C

-

A 3X

E d cp (X,y)).

Hence by Ao-AusS there is a set d' such that V Y (Y E d '

3 x E d cp (X,Y 11,

and this establishes our claim $ ( d ) .

COROLLARY 1.1. Ext + Ao-AusS + CRTrans Pair.

PROOF. By Proposition 1.3(a) there is a set containing 0 and (0) as elements. By A,-AusS, {0,{0}} is a set. Now apply the axiom of replacement to the formula cp(c,d):=(c = 0 A d = u ) v ( c = { 0 }

d=b).

The range of the function defined by this formula on the set {0,{0}} is just {a,b}.

284

KLAUS GLOEDE

COROLLARY 1.2. Ext + A0-AusS + CR, that $(a)+Trans ( a ) .

PR$ for any formula 1(1 such

We have not yet made any use of the axiom of foundation. As in BERNAYS[1961], p. 143 we can easily derive the axiom schema of foundation Funds

cp(a)+3x (cp(x)

VY E x

A

7

cp(Y))

from the axiom of foundation, Fund, by using PRTrans. This is accomplished by a modification of Bernays' proof (note that we are not yet allowed to use class variables): PROPOSITION 1.4. Ext + A,-AusS

+ Fund + PRTrans

Funds.

PROOF. Let i,b denote the negation of the formula we wish to prove, i.e. cp(a)A VX [cp(X)+3Y (cp(Y) A Y E XI19

and let a, a l ,..., a, be a11 the free variables occurring in 1(1. Applying PRTrans, we obtain (i)

i,b(a)+3u [Trans ( u )A a, a l,..., a, E u

A

$"(a)].

By the recursive definition of relativization, we have: $ ' ( u ) t * c p b ( a ) ~Vx E b [cpb(x)+3y E b ( c p ' ( y ) ~y Ex)].

Using A,-AusS, there is a set c such that VX

Then

a Eb (ii)

3u (a E u

A

A

(X

EC

Eb

f*X

~ , ! ~ ' ( a ) +Eac

A

cpb(X)).

A

VX E c 3~ E c y E X ,

$ ( a ) )+3 u [ a E u

A

VX E u 3 y E u y E XI.

But the conclusion of (ii) contradicts the axiom of foundation, hence by contraposition and using (ii), we obtain $.

-

1.1.5. We summarize these results as follows, considering the following theories:

T:=EXt+ AO-AUSS+ PRTrans, T:=Ext+ AO-AUSS+ PRstrans, T;=EXt+AO-AusS+ CRTrans, Tb=EXt+ Ao-AUSS+ CRstrans, T,=TI+Fund

for i

=

1,...,4.

REFLECTION PRINCIPLES AND INDESCRIBABILITY

285

Then we have the following results:

T: Pair +Sum + Inf, T;

TI + RepiS,

.

T i F Ti + Pow, T i F T: + Pow + ReplS,

in particular, T4F ZF. (We shall see later that ZF T4,too.) The relationship between these theories may be illustrated by the following diagram (Ti +Ti means: Ti is an extension of Ti):

It can be shown that all these extensions are proper (assuming consistency, of course): LEVY-VAUGHT[1961] proved that T4(= ZF) is “essentially reflexive” over T2, in particular T4 is properly stronger than T,. Moreover, it can be proved (in ZFC or ZF + “The countable union of countable sets is countable”) that H ( o , ) ,the set of hereditarily-countable sets, is a model of T, (cf. GLOEDE[1970], p. 31), but obviously it does not satisfy the axiom of power set. From these results it is immediate that all the extensions depicted in the above diagram are proper (and neither T, is an extension of T3 nor conversely). In particular, the full axiom schema of replacement is not derivable in T, (assuming consistency). On the other hand, in TIwith PRTrans restricted to 2-formulas one obtains the axiom of replacement for 2-functions.’ Similarly, in Tz (possibly with a restricted PRs,,,.,) one can prove the 21(9)-replacement schema of KUNEN[ 19721, and hence Tz is quite a strong set theory (KUNEN [1972], pp. 589ff and LEVY-VAUGHT[ 19611, 02) provided the A,-Aussonderungsaxiom of T2is being replaced by a stronger schema (possibly the full schema of subsets). This suggests that the systems TI and T,, with reflection restricted to suitable classes of formulas, may be more reasonable theories than TI and Tz .

Reflection principles in Ackermann’s set theory have been investigated [1961], and REINHARDT by LEVY [1965] (Appendix C), L~vY-VAUGHT [1967], [1970]. 1.2. Introducing class terms The language .2zF of set theory has been introduced as the usual first-order language with two binary-relation symbols denoting equality

‘See, e.g. BARWISE [1969], Lemma 2.2. The Kripke-Platek theory KP of admissible sets is the theory obtained from T, by restricting the reflection schema to Sformulas and adding the axiom schema of foundation. The above mentioned result, however, does not require the latter schema.

286

KLAUS GLOEDE

and membership relation respectively. This is a particularly simple language as far as the metamathematics of set theory is concerned. On the other hand, for the development of set theory within this language, it is often more convenient to extend the basic language by introducing class terms (e.g. in the proof of Theorem 1 . 1 we already spoke of the “function” defined by a particular formula). This can be done in such a way that every formula of the extended language, possibly containing class terms, is equivalent to a formula of the basic language. As usual we proceed as follows: The basic language ZezF is extended by adding class terms of the form {x I...} and adding the following formation rules for formulas: If cp(a,...) is a formula of the extended language then so is the formula a E { x I qk...)}. Moreover, we add to the usual axioms and rules of the predicate calculus with equality the following schema: CH

a E{XIc

pk.

..) ) t * q ( a , . . . )

(Church)

by means of which class terms can be eliminated in an obvious way.’ Following QUINE[1963] (cf. also JENSEN [1967]), we may even extend the logical frame further by introducing syntactical variables for class terms. In order to distinguish notationally from the class variables A, B, ... to be introduced in the next chapter, we use A, B, . .. to denote class terms in the above sense. Note that a formula containing such a syntactical variable A denotes a schema of formulas of the extended language, in contradistinction to a formula containing a (formal) class variable A. With respect to these class terms, it is convenient to define equality and membership relation between class terms and (set) terms by the following schema of definitions:

tl=tz:f,VX(XEt,t*XEt2),

t l E t z : f ; , 3 X ( X = t l AX E t 2 )

if t l and t 2 are class terms or set variables, but not both set variables. It is now possible to rewrite the axiom schema of replacement in the following form (assuming the axiom of pairing): Ft ( F ) A 3 x ( x = dom ( F ) )- + 3 y ( y

= rng

(F))

where we have made use of the following definitions: Ft ( F ) :f;,Vz ( z E F + 3 x , y ( z = ( x , y ) ) )A Vx,y,z ( ( x , y )E F A ( x , z )E F + y

=z

) “ F is a function”,

’ As long as we have not introduced class variables, we use the word “class term” for class abstracts of the form {x I cp(x,. . .)} whereas BERNAYS[1961] uses the word “class term” to denote either a class variable or a class term of the form {x I cp(x,. . .)} (in a somewhat extended sense insofar as cp may contain class variables).

REFLECTION PRINCIPLES AND INDESCRIBABILITY

287

dom ( F ) := {x 13y ( ( x , y ) E F ) } "domain of F", rng ( F ) := {y I 3 x ( ( x , y ) E F ) } "range of F". For the following discussion let us assume the following convention: By

cp(ao,.. . , a,, A,,.. .,A,,) we denote the formula of the extended language

obtained from the formula cp(a,,..., a,, bo,..., b.) of the basic language 2ZzFby substituting Ai for bi for i = 0,. .. , n. We now extend the notion of relativization to formulas of the extended language by requiring that for any formula q(a, ,..., a,, b, ,...,b,,) of the basic language and any class terms Ao,. . .,A, Rel(a,cp(ao,..., amrAo,-..,An))

or

($(a,,..., am,Ao,*..,An)")

to be the formula Re1 (a,$) where $ is a formula of $PzF obtained from cp(ao,.. . , a,, Ao,. . . ,A,,) by eliminating the class terms occurring in this formula in some canonical way. Alternatively, we may proceed as follows: We define by recursion the relativization of formulas and class terms of the extended language as follows:

-

Re1 (a,cp)= cp if cp is atomic, Re1 ( a , cp) = Re1 (a,cp),similarly for the propositional connectives A , v ,+, and c*, Re1 ( a ,Vx cp) = Vx E a Re1 (a,cp), Re1 ( a , 3x cp) = 3 x E a Re1 (a,cp), Re1 (a,b E {x I cp}) = b E {x I Re1 (a,q)}, Re1 (a,{x 9)) = {x I cp)" = {x I Re1 (a,cp)).

I

Thus Re1 (a,cp(a,,..., a, A,,. .., A,,)) may also be defined by the formula cpa(ao,.. . , a,, A o a ,.. .,A:) which is obtained from q ( a o , .. .,a,, A,, . . . ,A,,) by relativizing the quantifiers occurring in this formula to a and substituting Ai" for Ai. Now suppose again that p(ao,..., a,,Ao ,..., A,) is a formula of the extended language. Then the partial reflection principle PR, applied to such a formula and particular class terms (or, more exactly, to the equivalent formula of the basic language) yields (+>

p(ao ,..., a,,,,Ao,..., A,,) + 3 u [ $ ( u ) A u,..., ~ am E U A

cpU(ao,.. . , a,,Ao",...,A,")].

(We assume that the free variables of cp(ao,.. ., a,, Ao,. ..,A,) are among a,, ..., a,-note that the class terms may contain free variables.) If $(a)+Trans ( a ) ,we may replace A: by A: n u in (+) (here we need

288

KLAUS GLOEDE

the assumption on the free variables).' However, in general A n a need not be equal to A n a (even if a is transitive); in fact, the principle 3 u (I,!J(U)AA" n u =A n u)

is a principle of complete reflection. Thus, in order to obtain the axiom of replacement, we either have to use a schema of complete reflection (cf. the proof of Theorem 1.1: we have made essential use of the fact that the formula defining a function is absolute with respect to some appropriate set c), or as in BERNAYS[1961], use a schema of partial reflection for formulas containing class variables which remain unchanged under the process of relativization. (It follows from Theorem 1.2 below that the latter principle implies the schema of complete reflection C RTrans.)However, even without introducing free class variables, within the logical frame of ZF set theory, modified by the use of syntactical variables for class terms, we can treat these variables like the free class variables in Bernays' schema and consider the following principle: PRII,O

p(ao,..., a,,Ao ,..., A , ) + 3 u [Trans(u) A

p"(ao,..., %,Ao n u ,..., A, n u ) ]

(where p(ao,.. ., a,, Ao,.. . , A,) is a formula of the extended language in the sense explained above). Using an argument from Remark 1.1 it can easily be seen that we may add the clause a o E u A . . . A a, E u as a conjunctive member within the scope of 3 u in the principle above, and in this form A in u may be replaced by Ai. LEMMA1.2.

Ext

+ Ao-AusS+ PRII,OFPair.

PROOF. Apply PRII," to the formula 3 x (x

= ao) A

3 x (x

= a').

One can now use the method of BERNAYS[1961], pp. 133ff. to show that PRII," (together with the axioms mentioned in Lemma 1.10) implies the axiom schema of replacement (using a class term F instead of a free class variable F as in Bernays' paper). Alternatively, we can prove the following THEOREM1.2. Ext +Ao-AusS

+ PRII,OF CRTrans.

PROOF. Let p(ao,..., a,) to be a formula of ZZF with free variables as 'Thus, we could also define the relativization of a class term Re1 (a,{x I cp}) to be E a A Re1 (a,cp)},provided a is transitive and a,, . ..,a, E a where ao,...,a, are the free variables of cp.

{x I x

REFLECTION PRINCIPLES A N D INDESCRIBABILITY

289

indicated and define A : = { y ) 3 X o . * . X n ( y = ( X o ,..., X n ) A ~p(Xo,..., X"))}, I,!J(A,u):= IX ( a = X ) A V X , 32 ~ ( 2 = (x,y)) A V & . . . x , ( 3 y (y =(Xo, ..., X n ) h Y E A ) (p(x0,. ..,Xn)).

-

By Lemma 1.2, + ( A , a ) is provable in the theory considered, hence by PRII," there is a transitive set b such that

aEb

V x o - . . x nE b (3y E b (y =(xO,..., xn)

A

Vx,y E b ( x , y ) E b

A

y € A n b)-pb(Xo ,..., L)),

therefore,

a f b

A

A

V x O * * * x , E b ( q ( ~,..., o x,)-~~(xO

,..., X n ) ) .

Let T, denote the theory Ext + Ao-AusS + Fund + Pow + PRII,". Then we have the following

COROLLARY1.3. T,

1 T4 and

T,

1 ZF

We shall see later that ZF T, and, moreover, PRII,O can be further strengthened to yield a principle of complete reflection which is related to PRII," like CRTransto PRTrans.

1.3. Derivation of reflection principles in ZF set theory

In this section we shall make further use of class terms in deriving principles of reflection in ZF set theory. For this purpose it is convenient to use the concept of normal functions: Ordinals are defined as usual in such a way that they are well'ordered by the €-relation: ord ( a ) :-Trans

( a ) A Vx,y E a (x E Y v x A

V.2 ( 2

c U +

=Y v Y E X )

2 =oV3X E Z

Vy

E X (YeZ))

a is an ordinal

On := {x I ord (x)} is the class of ordinals, a + 1 := a u {a},lim(a) :-ord(a) A a # O A Vx E a 3 y E a (x ~ y ) . Assuming the axiom of foundation, the last conjunctive member in the definition of an ordinal is redundant, and in this case ord (Q)is defined by a 8,-formula. We use lowercase greek letters to denote ordinals. A normal function is one defined on the class of all ordinals, with range included in the class of ordinals and which is strictly increasing and

290

KLAUS GLOEDE

continuous: Ft ( F ) A dom ( F ) = On A rng ( F ) g On Nft ( F ) :o A Vx,y E On (x E y + F ( x ) E F ( y ) ) A V X( L i m ( x ) + F ( x ) = U { F ( y ) l y E x } ) . For the remaining part of this section we assume the usual axioms of ZF. It is easy to see that the composition of normal functions is again a normal function and that every normal function F has arbitrary large critical points (i.e. ordinals a such that a = F(a)). Using transfinite induction one can prove that there is a sequence of sets (V, I x E On) satisfying the recursive conditions v,,=@,

v,+,=~(V,),

Lim(a)+V,=~{v, I x ~ a } ,

and V = U{V,Ix €On} by the axiom of foundation. (V, is $(a) in BERNAYS[1961], p. 146.) METATHEOREM 1.1 (Scott-Scarpellini). For every formula cp (vo,. . . ,v,) of gZF with free variables as indicated there is a normal function F (depending on cp) such that F ( a ) = a + V x o . . . x , E V, (cp(xo,..., xn)f*Rel(V,,cp(xo,..., x,))). [1969], p. 23 where the PROOF. A proof can be found e.g. in MOSTOWSKI proof is given in Quine-Morse set theory. The above version, however, which uses the concept of relativization rather than a formalized version of satisfaction for classes (which is not definable in ZF) can be proved in ZF along the lines of Mostowski’s proof. (See also KRIVINE[1971], Ch. IV.) Since the V,’s are supertransitive, we obtain:

COROLLARY 1.4. ZF 1 CRStrans,and hence ZF 1 T4. Metatheorem 1.1 has a further corollary for class terms: COROLLR1.5. For every class term A with free variables among vo,. . ., v, there is a normal function F such that F ( a ) = a +Vxo**.x,E V, (A n V,

= Rel(V,,A) n

V,).

Combining Metatheorem 1.1 and Corollary 1.4, and using the fact that the composition of finitely many normal functions F , , ... ,Fk is again a normal function whose critical points are critical points of each Fi we obtain: COROLLARY1.6. For every formula q ( a o,..., a,,,, bo,..., b , ) of ~ Z and every class terms Ao, ...,A, such that the free variables of

F

REFLECTION PRINCIPLES AND INDESCRIBABILITY

29 1

q ( a o,..., a,,Ao,. . ., A , ) are among ao,..., a,,, there is a normal function F such that

F ( ( Y )= (Y + V X ~ * * E *X V a~[ ~ ( x o. ., ,. x,, Ao,. .. , An) n V,, . . .,A, n Va)]. COROLLARY1.7. ZF

PRII;,

and hence ZF

q V a ( X ~.,.,. X m A o

Ts.

As a further consequence of Corollary 1.6, the principle PRII," of partial reflection can be strengthened in ZF to the following principle of complete reflection: CRK,"

3 u [Trans(u) A V x o . . ~ x ,Eu(rp(xo,. ,, f* rp"(xo,..., x m , A on u ,..., A, n u))]

where rp and A o . .. .,A,, are as in Corollary .6. (In addition, Trans ( u ) may be replaced by Strans ( u ) . ) This possibility of strengthening a principle of partial reflection to the corresponding principle of complete reflection is a peculiar feature of PRII," (possibly with Strans ( u ) in place of Trans ( u ) ) and the schema PRIIo' to be introduced later, since in general the principle of complete reflection C R(2) is stronger than the corresponding principle of partial reflection PR(2) for the same set of formulas 2. For example cf. the end of §1.1, 42.1, and remark (2), p. 305. 2 . From class terms to class variables

2.1. Bernays' set theory B In the preceding chapter we introduced a version of ZF set theory which allows for the use of class terms and even has syntactical variables A , B, ... for arbitrary class terms. Replacing these syntactical variables by formal class variables A, B,. . . we obtain Bernays' set theory B (as in his 1958 monograph). The language .%of B is an extension of the basic language LfZFof ZF which has in addition free-class variables A, B, ..., but no bound-class variables. The formulas of this language will be denoted by capital greek letters @, Y,.... For simplicity, we assume that the language of B has atomic formulas only of the form u E v, u = v, and u E A. We use the same symbol of equality "=" in formulas of the form t = s where t and s are set or class variables; in case at least one of t, s is a class variable, t = s is defined according to the principle of extensionality (cf. 91.2). Thus, we write a = A for Rep(A,a) (BERNAYS [1958]) and a = A (BERNAYS [1961]). Similarly we define A E t : - 3 x ( x = A A x E t ) if t is a

292

KLAUS GLOEDE

class or a set variable (cf. 4 1.2). We further assume that formulas of B do not contain any class terms (apart from class variables); formulas of the form a E {x I @(x,...)} are regarded as “abbreviations” of @(a,.. .). (Alternatively we may assume that class terms of the above kind are eliminated from formulas of B by means of the (extended) Church schema.) We assume a suitable system of logical axioms and rules including the schema of equality u = b. + .4,(u,. . .) +@ ( b , ...). (In particular, we have a = b.+.a E A + b E A . ) Axioms for B are Al-A8 of BERNAYS[1958]; we may also use the following axioms for B: Ext, Pair, Sum, Pow, Inf, and Fund (as in the case of ZF),

and finally Repl’

Ft (F)A 3 x ( x = dom ( F ) ) + 3 y ( y

= rng

(F)),

which is obtained from the axiom schema of replacement of ZF (as defined in 41.2) by replacing the syntactical variable F by a free-class variable F. We might call a formula of 2, a A,-formula iff it is equivalent to a formula in which all quantifiers are restricted to set variables (as in the sense of § 1.1) and consider the following schema:

8 o - A sS’ ~

3 X v y ( y EX e y E a

A

@ ( y , ...))

where @ is any Ao-formulaof ZB.However, since a E A is a A,-formula, this schema immediately gives the full Aussonderungsaxiom (axiom of subsets) of B:

Au s‘

3 x Vy ( y E x + + y E a

A

y€A)

We have excluded class terms from our formal language in order to simplify the notion of relativization for formulas of B: If 4, is such a formula then @a (or Rel(a,@)) is the formula obtained from @ by restricting any quantifiers occurring in @ (note that there are only quantifiers on set variables) to a (as in the case of a formula of 5fzF). Again, if @ contains any defined symbol or class terms then Re1 (a,@)is defined to be Re1 (a,W) where is obtained from @, by eliminating the defined symbols and the class terms occurring in CD. Let us now consider the schema of partial reflection for formulas of the language of B: @(a,,.. . , a,, Ao,...,A,) + 3 u [Trans ( u ) A CDU(ao,. .., a,, A, n u,... , A, n u ) ]

P RHO’

where

4,

is any formula of ZBwith free variables as indicated.

REFLECTION PRINCIPLES AND INDESCRIBABILITY

293

REMARK. As in the case of PRII,", one may add in PRIIo' the clause a . E u A . .* A a , E u as a conjunctive member within the scope of 3u and replace Ai n u by Ai. THEOREM2.1.

Ext

+ Aus' + PRII,'t-Pair + Sum + Inf + Repl'.

The axiom of pairing is proved as in Lemma 1.2. The proof of the remaining axioms is given in Bernays [1961], pp. 127, 128, and 133ff. REMARK. Thiele's version of the axiom of replacement (Formula (191, p. 133 of BERNAYS [1961]) is in fact equivalent to (18) in B using the von Neumann rank of a set in a manner due to SCOTT[19551. COROLLARY 2.1.

Ext + Aus' + Pow + Fund + PRIIo' 1 B.

Conversely, one can prove by the method of 0 1.3 (Metatheorem 1.1): METATHEOREM 2.1. For every formula @(ao,..., a,, A. ,..., A,) o f LtB with free variables as indicated, there is a normal function F (more precisely, a class term F defined by a formula of LfB which depends on @ such that B Nft ( F ) ) such that one can prove in B:

+

F(cx)=cx+~/x~ E *V*, *(@(x, x , ,..., X ~ , A ~ , . . . , A , ) @ @"a(x0,...,x,, A. n V,,. .. , A, n V , ) ) .

Thus again PRIIo' can be strengthened in B to give a schema of complete reflection CRII,' as in the case of PRII2; similarly, if in PRIIo' and CRIIo', we replace Trans ( u ) by Strans ( u ) , in this case the stronger principles imply the axiom of power set (cf. Proposition 1.2). Note that the proof of Metatheorem 2.1 is similar to the proof of Metatheorem 1.1 rather than Corollary 1.6, since a € A is now an atomic formula (whereas a € A need not be equivalent to an atomic formula in ZF). In fact, Corollary 1.6 can be proved along the lines of the proof of Metatheorem 1.1 by treating a E A as an atomic formula. Thus we have the following results (as usual we write TI = T, to mean that TI and T, have the same theorems):

+ +

Z F = Ext + Ao-AusS Pow + Fund + PRII,", B = Ext Ao-AusS' Pow Fund + PRIIol,

+

+

and we can delete the axioms of power set on the right-hand sides if the respective schemata of partial reflection assert the existence of a "reflecting" set a such that Strans ( a ) rather than Trans ( a ) . 2.2. Montague- Vaught 's theorem Before

presenting

various

results

from

MONTAGUE-VAUGHT

[ 19591, we recall some definitions (throughout this section we assume the

294

KLAUS GLOEDE

axioms of ZF): w is the least infinite ordinal (and the set of natural numbers), Osq cf) :- Ft cf) A dom ( f ) E On A rng ( f ) C On “ f is a sequence of ordinals”,

AOsq (f) :* Osq ( f 1A VX,Y( x E Y

A

Y E dom ( f 1-+f ( x )E f ( y ))

“f is an ascending sequence of ordinals”. If f is a function such that a dom ( f ) , then sup+f ( x ):= u { f ( x )+ 1 I x E a}.

sup f ( x ):= u c f ( x ) I x E a } ,

x E”

X E O

Nf t (f,a):t,AOsq ( f ) A dom c f ) = a

A

rng c f ) C a

A

V x E a (lim ( x)

a

= supt s(x)),

--j

f ( x )= SUP f (Y 1) YE*

“f is a normal function on a”, conf ( a $ ) :-3s

(AOsq (s)

A

dom (s)

=p A

cf ( Q ) := p x conf ( a , x )

xe@

( p x q ( x ) denotes the least ordinal x such that cp ( x ) if there is such an x, 0 otherwise), reg(a):t,cf(a)=a “ a is regular”,

sing ( a ):++

reg ( a )

“a is singular”,

in ( a ) :c*a > w

A

Vf (Ft cf) A dom (f) E V,

A

rng cf) C V, +rng (f) E V,)

“ a is (strongly) inaccessible”. Any infinite regular ordinal is a limit number, an inaccessible ordinal is regular. Assuming the axiom of choice AC, inaccessibles can also be characterized as follows (MONTAGUE-VAUGHT[ 19591, Lemma 6.4): (ZFC)

in ( a )c,a > w

A

reg ( a ) A V x E a

(I P ( x ) 1 E a ) ,

where la1 denotes the cardinal number of a (assuming AC). The formal notion of satisfaction for first-order formulas in a structure which is a set can be extended for second-order formulas of set theory (which are obtained from the language of B by admitting even bound-

REFLECTION PRINCIPLES AND INDESCRIBABILITY

295

class variables VXi and 3Xi) in several ways. Here we decide to reduce the notion of satisfaction for the latter class of formulas in a structure of the form (a, E ) to the satisfaction of first-order formulas in the structure (P(u),E)as follows: As in 91.1.2 one can assign to each secondorder set-theoretical formula @ a term r@l (the Godel number of @) and define in ZF a set Fml’ which intuitively represents the set of all such Godel numbers such that ZF r@l E Fml’ for each second-order formula @. Moreover, one can define in ZF a function * which assigns in a suitable way to each e E Fm1’ a set e* E FmlZF. Instead of giving a formal definition of * (this would presuppose an explicit definition of FmlzFand Fml’ as well as several syntactical concepts) we describe it informally. If @(vo,. .., urn,A o , .. ., A,,)is a second-order formula of set theory, @* is the formula of YZFobtained as follows: replace the second-order variables Ao,...,A, by the first-order variables urn+’,..., urn+,+2(renaming bound variables in @, if there are any among urntI,..., urn+,,+2),replace any first-order quantifier Vx (3x) occurring in @ by Vx E urn+1(3x E urn+1 resp.) and any second-order quantifier V X (3X) by a first-order quantifier Vx (3x resp.) for some suitably chosen set variable x. We then define (a,€) I= @‘[a$]

for a formula @ as above, and a sequence a of m sequence b of n + 1 elements of ??(a) by ( P ( a ) Z )I=

where c is the sequence of n

+ 1 elements of a, and a

@*[Cl

+ m + 3 elements

a(O>,...,a(m), a, Wh..., b(n)

This means that we interpret class variables in a structure (a,€) as ranging over the set of subsets of a. We denote by (a,€) I=’ e[a;bI, or (though notationally not quite correct),

if n and m are suitably chosen, the satisfaction predicate for secondorder formulas of set theory formalized within ZF set theory in a manner described above informally. (For more details the reader may consult [ 19691 or BOWEN[1972]. In Mostowski’s book, however, MOSTOWSKI this notion of satisfaction is defined for structures ( A , € )(where A may be a proper class) in Quine-Morse set theory; the same definition, restricted to structures which are sets, can be defined in ZF set theory).

296

KLAUS GLOEDE

As usual, we write (a,E)+=2r@Tf o r ( a , E ) p ‘ V x , - - - x ,V X , . - - X ,

@I,

or, equivalently, VX~**E *X a ,V y o * * . y L n a ((a,E)t=zr@-[xo ,..., x m ; y o ,..., y n l )

if @ is a second-order formula with free first-order variables among vo,..., v, and free second-order (class) variables among Ao,.. . ,A,. Also (a,€) kZ E

and

(a,€)

‘@,

+

Q2’

for a schema or a set of second-order formulas E and formulas @,, Q2 is assumed to be defined as usual. Finally, ( a , € )< (b,E) is a formal version (definable in ZF) of the predicate “(a,€) is an elementary subsystem (with respect to first-order formulas, i.e. with respect to formulas of ZZF) of (b,€)” in the sense of TARSKI-VAUGHT [1957]. In order to simplify notation, we often identify a formula with its Godel number (similarly in the case of a set of formulas) and write e.g. ( a , € ) Ext or ( a , € ) B, since in these cases the use of the symbols or indicates that we are dealing with the formal counterparts of the respective (metamathematical) satisfaction relations. Using these definitions it can be shown (cf. MONTAGUE-VAUGHT [1959], p. 234):

v

LEMMA2.1. If a # 0 and a # 0 , then

6) (ii) (iii) (iv) (v) ( 4

Trans ( a ) + ( a , E ) Ext + Fund, l i m ( a ) + ( V a , € ) k Pair+Sum+Pow, w Ea+(Vu,E)~lnf,

( V U , GFB f,in ( a 1, V,,E) B + ( V,,E) in (a)+reg (a).

v

+ ZF,

The converse of (v) is not true. This is a consequence of the following: THEOREM2.2 (Montague-Vaught). ( V,,E)

vB

+

36 < a (V.,E)< ( Vu,E).

PROOF. We indicate a proof of the informal version of this theorem using the method of MOSTOWSKI [1969], pp. 25f. Let (cp,ln < w ) be an such that the free variables of (P, are enumeration of the formulas of TZF among vo,..., v,. Since V, is a model of ZF, by Scott-Scarpellini’s Metatheorem 1.1 there exists for each n a function f n (definable in V,)

REFLECTION PRINCIPLES AND INDESCRIBABILITY

such that Nft (f.,a) and

V.$< a Lf,,(()

297

-

= .$ +

V X ~ * - -EXV,E (Re1 (Va,cpn(Xo,..., x , ) ) Re1 (Vt,Pn(xO,...,xn)))l.

Define a function f on a by recursion on P < a such that f ( 0 )= 1, f ( P + l ) = m a x c f ( P ) , supf.(P))+l, n ( 0

Lim ( A )

A

h < a +f(h)

= sup f(.$). ,
(Note that f and the sequence (f. In < w ) are definable in ZF but not in V,.) Obviously, f is strictly increasing and continuous. Any critical point of f is a critical point of each f,,, hence if f(P) = p then for all n < o:

V X O..,. ,X n E V p (Re1 (V,,cp, ( X O , ... ,xn1) t-,Re1 ( Vp,cp,,( X O , ...,X . ))I. Finally, it remains to show that f has a critical point. In fact, if reg ( a )A a > w (in particular, if V , is a model of B by Lemma 2.1 (iv)), then we have Nftcf,a) and even Nft(g,a) where g is the function enumerating the critical points of f in increasing order. A subset a a is called closed unbounded in a iff it is the range of a normal function on a :

c

clo unb ( a , a ) :e 3f (Nft cf,a) A a = rng cf)). Incidentally, we have thus proved the following stronger result:

COROLLARY2.2. reg ( a ) ~ ( V , , € ) k Z F + 3 x a(clounb(x,a) A x c (5 < I ( V,,E)< ( vU*aH, in particular (Montague-Vaught): i n ( a ) + 3 . $ < a ( ( V E , E ) < ( V , , E ) ~ c f ( . $ )w=) . COROLLARY 2.3. (V,,E)

k26 +35 < a ( V . , E )

ZF.

The same result can be expressed as follows:

COROLLARY 2.4. (V,,E)

PRIlo’+35 < a( V,,E)

PRn2.

We have seen in 02.1 (Metatheorem 2.1) that the principle PRII,’ can be strengthened in the theory B, in fact, if (V,,E) is a model of B then for each formula @(aO ,..., a,,,, Ao,..., A,) of 5!?Bwith free variables as indicated and any bo,..., b. C V,, the set {(
-

~Vxo-~~xmEV,(Rel(V,,@(xo ,..., xm,bo,..., 6 , )

Re1 ( V , , @ ( X .~.,,.x m , bo,. . ., bn1)))

298

KLAUS GLOEDE

contains a set which is closed unbounded in a. Combining this result with Corollary 2.2 and the fact that for any regular a > w the closedunbounded subsets of a generate a filter in the power set of a, we obtain the following result:

-

If ( V,,E) is a model of B then ( V,,E) satisfies the following principles: (+)

36 [ ( V , , E ) F Z F AV X ~ ’ * * XE, V~(@(XO, ...,x,,Ao,..., A,) Re1 ( V6,@(xo,. . . , x,, Ao,...,A,,)))I,

in particular, (++)

@(ao ,..., a,,Ao ,...,A , ) + 3 u [Strans(u)A ( u , E ) k Z F A A

Re1 (u,@(ao,. . . , a,, Ao,.. . , A,))]

(where in both cases, @ is any formula of 2ZB with free variables as indicated). This result should be compared with schema (llg) of BERNAYS [1961], p. 141 and LCvy’s principle ((I) of BERNAYS [1961], p. 122, cf. LEVY[1960]). These latter principles have in place of the formula Strans ( u ) A ( u , E )F ZF the formula Strans ( u ) A Mod ( u ) (Bernays), and ScmzF( u ) (Levy), respectively. In ZF, S t r a n s ( a ) ~M o d ( a ) t * S t r a n s ( a ) ~( a , E ) p B c* Strans ( a )A ScmZF( a ) , and Montague-Vaught’s theorem shows that ScmzF( a ) and Trans ( a )A Mod ( a ) both are stronger than Trans ( a )A ( a , € ) ZF (even if we add “strans (a)” in both cases), and in fact the latter principles are stronger than (++) and even (+) (assuming consistency). On the other hand, (+ +) is a schema which is derivable in Quine-Morse set theory (cf. p. 306) and can be used as a motivation for LCvy’s principle. Finally, as it is known that the theories 6 and ZF are equiconsistent, (++) cannot be proved in B (again assuming that B is consistent). 3. Reflection principles for second-order languages

3.1. Second-order set theory The language 2Z* of second-order set theory is obtained from the first-order language 2’zF of ZF set theory by introducing second-order (class) variables Xi(i < w ) (as in the case of set variables, we often use capital letters from the beginning of the alphabet for free variables and letters from the end for bound variables). Again, as in the case of Bernays’ set theory 6, we assume that we have only atomic formulas of

REFLECTION PRINCIPLES AND INDESCRIBABILITY

299

the form u = v, ; E and u E X and that our formal language does not contain class terms (apart from class variables). t = s :-Vx (x E t - x E s) (if t and s are variables, but not both set variables) and X E t :-3x (x = X A x E t ) (for a set or class variable t ) is defined as in $4. In a suitably extended language, {x I Q,(x)}denotes the class of all sets x such that @(x). The axioms of von Neumann-Bernays' set theory VNB consist of the axioms of Bernays' set theory B: Ext, Pair, Sum, Pow, Inf, Fund and Repl', and the following schema of (predicative) class formation : II,'-Comp

3 X V y (y E X - @ ( y , a o , . . . , am,Ao, ..., A,,)),

where Q, is any formula of 9 'containing no bound-class variables (i.e. ' of VNB will also be denoted by any formula of Lf6). The language 2 =%'VN~.

If we replace in VNB the IIo'-comprehension schema by the (impredicative) schema of comprehension

3x v y ( y E x WY,. . .I>,

IIo2-Comp

where

Q,

f,

is any second-order formula, we obtain Quine-Morse set theory

QM.

LEMMA3.1 (ZF).

(a,€)

+*IT,'-Comp

for any set a # 0.

PROOF. This fact is due to the particular definition of satisfaction+' which interprets classes in (a,€) as subsets of a, and the fact that ( a , € )k' e [x,. . .] is a formula of ZF set theory. COROLLARY 3.1 (ZF).

+' B

( Va,E)

-

( V A k*VNB * ( V a , E ) QM in (a). f ,

From the results of the previous chapter, we obtain:

+ +

VNB = Ext + AUS' + Pow + Fund + IIo'-Comp PRII,', QM = Ext AUS' + Pow Fund II2-Comp PRIIo'. LCvy's classification of the formulas of ZF set theory (LEVY[1965]) can be extended to formulas of VNB and QM set theory as follows: A II,'-formula @ is a formula of ZVNB which does not contain any boundclass variable (i.e. a formula of ZB), Xo' = no'.A II!,+l(Z!,+I)-formulaQ, is a formula of LZvNeof the form VXW (or 3XW resp.) for a %'(IT,')-formula W. Formulas of =%'vNaare also called IIoZ-formulas. Let Q, be any formula of LZvN6. Then Q, has a prenex normal form

+

(1)

+

Q I t 1 - * - Q,t,

+

$ ''

300

KLAUS GLOEDE

where each ti is a set or a class variable, Qi is V or 3 and W does not contain any quantifier. Q l t l .=.Q,t, is called the prefix of the formula (1). Clearly, (1) is equivalent to a ll,l-formula for some rn. We can obtain another classification of the second-order formulas by “ignoring” the bounded-set variables even in front of bounded-class variables: The reduced form of (1) is obtained from (1) by deleting any set quantifier in the prefix of (1). A fi,’-formula ($,‘-formula) is a formula in prenex normal form such that the reduced form of it is a II,’-formula (2,’-formula). A A,’-formula is a formula which is equivalent to both a n,’- and a 2,’-formula, similarly €or A,,’. We sometimes use the symbol Q to denote any of the symbols ll, 2, A, fi, 2, 6. We show that under suitable assumptions the ll,,’- and the fin’classifications agree and hence obtain a simple method for classifying a according to its bound variables. For the remaining part formula of ZvNB of this section, if not mentioned otherwise, we assume the axioms of VNB. First, note that any two (and hence any finite number of) adjacent quantifiers of the same kind can be reduced to a single one using ordered pairs; in the case of class quantifiers, one may use a coding of finitely many classes into a single class (cf. ROBINSON[1945]).For example we may define the ordered pair of two classes A, B by ((0)x A ) u ({ l} X B ) , hence we have in VNB:

v x V Y Q,(X,Y , ...) vz W Z , ...), f-*

where W(Z,...) is obtained from Q,(X,Y,...) by replacing any occurrence of X and Y of the form u E X by (0,u) E 2 and u E Y by (1,u) E 2 resp. (where Z is assumed to be a variable which does not occur in a). Now, let Q, again be any formula of ZvNB written in prenex normal form (1)-with alternating quantifiers. Obviously, if Q, is a IT,,’-formula, then it is a II,’-formula. In order to prove the converse, we have to show that any set quantifier occurring in the prefix of (1) can be moved over a class quantifier from left to right without introducing new bound-class variables. We generalize the ordered pair of classes by introducing the following coding for sequences of classes:

DEFINITION.A, := A ” { a }= {xI(a,x)

E A}.

(Though this notation might conflict with the metamathematical use of indices for class variables (e.g. in the notation @(ao,..., a,, A. ,..., A,)) we hope that the reader will always discover the right meaning from this context.)

REFLECTION PRINCIPLES AND INDESCRIBABILITY

30 1

In order to advance a set quantifier over a class quantifier in a suitable way, it seems necessary to assume some kind of axiom of choice. We consider the following forms:

V x E a ( x # 0)+3f [Ft cf) A dom (f) = a A V X E a f ( x )E x ] (set form of the axiom of choice)

AC SAC

3 F [Ft ( F ) A dom ( F ) = v

A

v x ( x # o+F(X) E X ) ] (strong axiom o f choice)

v x 3Y @(x,Y , ...)+3Y v x @(x,Yx,.. .),

Qn'-AC

where @ is any Q,'-formula.

LEMMA 3.2.

(a) VNB + IIol-AC SAC.

(b) In VNB, the II,'-AC is equivalent to each of the following schemata : (9

2;+i-AC,

vx 3~ @(x,Y,...I - 3 V~x @(x,Y,,...), @ a 2L+1-forrnula, (ii) 3 x V Y @(x,Y,.. .) -VY 3 x @(x,Y,,...), @a II!,+l-formula. (iii) PROOF. (a) Let @ ( a , A )be the IIo'-formula a = O V A € a , i.e. -3x (x E a ) v 3 x (x E a A V y ( y E x - y E A ) ) . Applying the IIol-ACto the formula V x 3 Y @(x,Y ) which is derivable in

VNB, we obtain the existence of a class A such that

V x (x#O+A, E x ) . Then F := {x,z Ix # 0 A z = A,} u {(0,0)}is a function as required. In order to prove (b) we have to show that II,'-AC implies Z;+l-AC. Thus, assume the former and let

@(a,A,...) = 3 X * ( a , A , X,...) be a 2!,+1-formula,i.e. 9 a II,,'-formula. Assume

v x 3Y 3X*(x, Y , X ,...). We have to show, 3 Y V x 3 X T ( x , Y,,X,. ..). Contracting the existential quantifiers, we have

v x 13Y 3x *(x, Y,x,...) cf 32 *(x, ZO,Zl,.. .)I

where Z,, Z 1are defined as on p. 300, and from V x 32 *(x,Zo,Z1,...), we obtain by applying II,,'-AC: 32 v x V x , (ZO),, (Zl),,. ..) 3 Y v x 3x *(x, Y,, x,...). 3 Y 3x v x *(x, Y,,x,,...),

302

KLAUS GLOEDE

[1971] shows that in several REMARK. Theorem 3c of MOSCHOVAKIS cases the application of IIo'-AC can be eliminated.

With respect to structures which are sets we have the following

LEMMA 3.3 (ZF + AC). (a,€) k2II,'-AC for each n < w and any set a # 0 which is transitive and closed under pairs. PROOF. Let @(ao,..., ak, a, A. ,..., A,, A ) be any III,'-formula with free variables as indicated and assume ( a , E ) ~ 2 V ~ k + 1 3 Y m + l,..., @ [ aa ko; bo,..., b m l

for any ao,.. ., a k E a, bo,.. ., bm C a. Then v x E a 3 y C a (a,€) kza[&,.. . , ak, X; bo,. . ., b m , y].

Since this is a formula of ZF, by A C there is a function f such that dom (f) = a, rng cf) C P ( a ) and Vx E a (a,€) 'F' @[ao,..., ak, x ; bo,. . . , b m , f(x>I.

Define B := {x,y I x E a a class as required.

A

y Ef(x)}. Then Vx E a B ,

= f(x)

and B C a is

THEOREM 3.1. For any l?L+l-forrnula CP there is a II;+,-forrnula@' with the same free variables such that VNB

+ III,'-AC

@

f-,

a'.

REMARK. a' can be obtained from @ in a simple and uniform manner, e.g. there is a primitive recursive set function (in the sense of JENSEN-KARP [1971]) F such that Fro' = '@I1. In fact, @' can be defined by recursion as follows: @' = @ if @ is a n,'-formula for some n,

Y,. . .)I' = 3 Y v x *'(x, Yx,...), [3x V Y *(x, Y,. ..)I' = V Y 3x *'(x, Yx,...). COROLLARY3.2 (ZF +AC). If @ is a fi,,'-formula, then there is a [VX 3 Y *(x,

II,'-fomufa @ with the same free variables such that for any transitive set a # 0 which i s closed under pairs : (a,€)

+*@

f*

@'.

Finally, we extend the operation of relativizing a second-order formula to a set a by defining Re1 (a,@)(or W ) to be the formula obtained of 5?veN from @ by replacing any quantifier of the form Vx (3x) by Vx E a(3x E a resp.) (as in the case of a first-order formula) and any quantifier of the

303

REFLECTION PRINCIPLES AND INDESCRIBABILITY

form V X ( 3 X )by Vx a (3x C a resp.) for a suitably chosen variable x. Again it is tacitly understood that defined symbols (and possibly class terms other than class variables) have been eliminated from @ before performing the process of relativizing. This definition is in agreement with in fact our definition of satisfaction in a structure ( a , € )by the predicate one can prove in ZF:

+*;

(a,€)~'e[a;b]t,Rel(a,@(a(O),..., a ( m ) , b ( O,..., ) b(n))) for any formula @ of LfVw with Godel number e such that the free variables of @ are among vo,. .., v,, Ao,.. ., A, and any sequences a of rn + 1 elements from a and any sequence b of n + 1 subsets of a (cf. (b) of 0 1.1.2). 3.2. Second-order reflection principles We now consider generalizations of the partial-reflection principles for second-order formulas: PRQn'

@(a0,..., ak, AO,..., Am)+3u(Trans ( u ) A @"(ao,. ,ak, A. n u,. ..,A, n u ) )

..

where @ is any (2,'-formula with free variables as indicated. Similarly, PRII; is the full second-order reflection principle of BERNAYS[ 19611. REMARK3.1. Using ordered k-tupels and a suitable coding of finite sequences of classes as in 93.1, and assuming some basic axioms (e.g. the axioms of pairing, union, AUS' and IIo'-Comp), the schema PRQ,' can be replaced by the following schema: @(A)+3u (Trans ( u ) A @."(An u ) ) where @(A)is any Q.'-formula with only A as its free variable. Similarly, as in Ch. 11, one may add in PRQn' the formula a. E u A * A , E u as a conjunction within the scope of " 3 u ", and Ai n u may be replaced by Ai. LEMMA3.4. PRII,'

1 PR z!,,

and hence both schemata are equivalent.

PROOF. Let 3 X * ( X , . ..) be a 2i+,-formula, i.e. *(A,. ..) a II,'-forrnula, and assume 3 X * ( X ,...). Then there is a class A such that T(A,...). Applying PRII"', there is a transitive set a such that *"(A n a,. ..), hence

( 3 X T(X, ...))".

Let BL, be the theory Ext + Fund + AUS' + PRII,' From 92.1 we recall that BL,

Pair+ Sum + Inf + Repl'.

(Bernays-LCvy).

304

KLAUS GLOEDE

BERNAYS[1961], pp. 138ff proves: B L I k P o w , hence B L 1 k B , since B = BLo + POW. PROPOSITION 3.1.

BL,,,

+ AC k II,'-AC.

PROOF. Let Q, be a III,'-formula and suppose that the instance of II,'-AC for such a Q, does not hold, i.e. v X 3 Y @ ( X , Y,...) A v Y 3 X - @ ( X , Y x

,...).

Obviously this formula is equivalent to a Ili+z-formulaq.(For simplicity, we assume that Q,(a,A) has no free variables other than a and A.) Applying PRIIA+zto q A Vx, y 32 (z = (x,y)) we obtain the existence of a transitive set b which is closed under pairs such that '$'.

However, this contradicts Lemma 3.3. COROLLARY 3.3.

BLntZ+ AC k SAC.

PROOF. Use Lemma 3.2(a). In fact, Corollary 3.3 can be improved to THEOREM 3.2. BL, +AC 1 SAC. PROOF. A proof is given in BERNAYS[1961], pp. 141ff. Taking a closer look at the proofs given in BERNAYS[1961], one easily sees that all his results only require BLI rather than BL = BLo+ PRII; = VNB + PRII: with one exception: Specker's result that the full impredicative schema of comprehension, IIo2-Cornp, holds in BL. If we introduce the following schemata: Q "'-Co m p

3 Y v x (x E Y -Q,(x,...)),

where Q, is any Q,'-formula (such that Y is not bound in method shows:

k

a), Specker's

k

THEOREM 3.3. (i) BL1 IIo'-Comp, hence BL1 VNB, (ii) (iii) (iv) (v)

BL,,, II,'-Comp, X,'-Comp, BL,+I k A,'-Comp, B L + PRfiA+l fi,'-Cornp, z:-Cornp, BLo+ PRfi,,' 6,I-Comp.

PROOF. Suppose QnL-Compdoes not hold for some Q,'-formula - 3 Y v x (x E Y -a.(x, ...)).

Q,, i.e.

REFLECTION PRINCIPLES AND INDESCRIBABILITY

305

Now, one only has to verify that in each of the cases (i)-(v) this formula is equivalent to a formula q to which the reflection principle of the respective theories applies. Hence there is a transitive set a # 0 such that T holds in (a,€) (under some appropriate assignment of the free variables of in (a) which contradicts the fact that (a,€) k2II$-Comp (cf. Lemma 3.1). Parts (iv) and (v) are from BOWEN[1972], Lemma 2.1.5. COROLLARY 3.4 (Specker). BL k II:-Comp,

and hence BL k QM.

Remarks and additions. (1) BOWEN[1972] denotes by BLntl the theory (which by Theorem 3.3 (i) and the Remark following below is equivalent to) VNB + PRII:?+l,let us denote it by BE,+1; B"L0= VNB + PRfIo'. Then (in our notation) BL,, 1BL,, and by Theorem 3.2: BL,+' +II,'-AC 1BL,,+'. (Here II,'-AC can also be replaced by IIt+,-Comp + AC.) Therefore, assuming II,'-AC, both theories are equivalent. Note that BL,+z + AC k II,,'-AC%y Proposition 3.1. We do not know whether BL,,+' and BL,+, are equivalent without assuming some kind of axiom of choice. Of course, assuming ZF +AC (for the metatheory) by Lemma 3.3 and Theorem 3.1, both have the same transitive standard models (with respect to the satisfaction relation k').In particular, (V,,E)

k2PRfI"'

-

(V,,E)

k' PRII,'.

In any case, even without the AC, we clearly have BL 1BLn for each n. Finally, it should be noted that BL is consistent iff the theories BL, are consistent for each n. (2) In BERNAYS[1961], pp. 138ff it is shown that in the theory BL,, the reflection principle PRIIk+I can be strengthened to a schema which asserts the existence of a "reflecting" set of the form V, for some inaccessible a (cf. Theorem 3.4). We cannot use this method in order to strengthen PRIIo', "since by Lemma 2.1 (iv), and the fact that B = Ext + AUS' + Fund + Pow + PRII,,' (§2.1), we have:

-

in (a) (V&)

kzB

f-,

(V,,E)

k2PRIIo',

and therefore one cannot prove in B (nor in VNB) the existence of inaccessible cardinals (assuming consistency). So one has to use the method of 02.1, and in fact Metatheorem 2.1 shows that PRIIo', too, can be strengthened to assert the existence of a "reflecting" set of the form V, (though a need not be inaccessible). One can also use the method of 02.2 (which is an extension of the methods yielding Metatheorem 2.1) to obtain a strengthening of PRrIo' (and similarly for PRIIA,,) as follows: Let T be the theory

VNB

+ All-Comp, i.e. BLo+Pow + Fund + A,'-Comp.

306

KLAUSGLOEDE

In T one can define a satisfaction relation Sat, ( A , a )for the class V of all sets with respect to formulas of TBsuch that Sat, is a Al'-formula and one can prove in T: Sat, ( A x { j } , r @ ( A j )t7,)@ ( A ) for any class A and any II,'-formula @(ai)with Ai as its free variable (cf. MOSTOWSKI [1969], BOWEN[1972], Lemma 2.1.14). This allows us to use the method of proof for Theorem 2.2 and Corollaries 2.2-2.4 replacing V, B (which becomes by V and deleting the assumption that (V,,E) redundant since V is a "model" of T and B C T). Thus in T one can prove, e.g. 35(V,,E) < (V,E), and in T the schema PRII,' can be strengthened to yield the schemata (+) and (++) of 02.2. Similarly, though by a somewhat different method, it is shown in BOWEN [1972], Corollary 2.1.19 that the following schema of complete reflection is provable in BL,,,,,: CRfi,,'

35 [a E V,

A

Vx E V,(@(x,A)

-

Re1 (V,,@(x,A n V*)>>I

where @ ( a , A )is any fi,'-formula with free variables as indicated (and similarly for formulas with finitely many free variables, cf. Remark 3.1). By the same method one can prove: BL,, + A;+,-Cornp CRII,' (cf. also Theorem 4.6). This is the best possible result insofar as in BL, + A i ' Comp (BL,) one cannot prove CRII,' (CRfI,' resp.) (assuming consistency; cf. GLOEDE[1970], p. 75). For a different situation in the case of P R I I 2 and PRII,' cf. the end of 01.3 and Metatheorem 2.1. (3) One can easily prove that the theory BL is consistent with Godel's axiom of constructibility V = L (Godel [1940]), (see BOWEN[1972], Theorem 3.1.2). This result has been extended to systems containing reflection principles of any finite or even transfinite-type theory by TAKEUTI[1965]. Let Ref' (BOWEN[ 19721) be the (weak) second-order complete reflection principle 35 Vx E V, [@(x) Re1 (V,,@(x))]

-

where @ ( a )is any IIo2-formula with a as its only free variable (thus @ may not contain any second-order free variable) and BL' = BL + Ref'. THARP [1967] has shown that BL' + (V = L)

BL

(cf. BOWEN[1972], Theorem 3.2.2),

and by Bowen's result mentioned above, B L I BL'. Therefore BL is consistent iff BL' is consistent.

REFLECTION PRINCIPLES AND INDESCRIBABILITY

307

(4) Finally, BOWEN[1972], Ch. 111, has shown that the consistency of BL or even BLI implies the consistency of VNB with various axioms which imply V # L (e.g.: for almost all regular cardinals a,( L a , € )is an elementary submodel of ( L , E ) ) and which originally have only been obtained from axioms asserting the existence of certain large cardinals, the existence of which cannot be proved in BL. 3.3. Mahlo’s Principle In his 1961 paper, Bernays emphasizes the fact that the schemata of reflection for second-order languages are “self-strengthening” in a very strong way (the corresponding schemata of partial reflection for firstorder formulas only partly partake of this property). In particular, the schema PRII,’ can be strengthened to yield the existence of “reflecting” sets of the form Vn,where a is a cardinal which is very large in the Mahlo hierarchy. In this section, we only discuss some of the results which are related to Mahlo’s principle and to the problems which are relevant to the class terms and class variables distinction, as expressed by MontagueVaught’s theorem. For further results, we refer to GAIFMAN[1967] and GLOEDE[19711. THEOREM3.4. In the theory BL,,, = Ext + AUS’ + Fund + PRX,,, the schema PRIIA., can be strengthened to the following schema o f partial reflection : @(Ao,... . A,,,)+35 [in (5)A @“c(A0n Vc,.. . ,A,,, n Vc)], where @ is any IIA+,-formulawith free variables as indicated.

PROOF. BERNAYS[1961], p. 141 shows that PRJI!,,, can be strengthened to a schema which has in place of “Trans ( u ) ” the stronger property Strans ( u ) A Mod ( u ) . As mentioned at the end of 92.2, one can prove in ZF:

-

Strans(a) A Mod(a)t,(a,E)+’B

+’ VNB.

(a,€)

By a result of SHEPHERDSON [1952] (03.14) (cf. MONTAGUE-VAUGHT [1959], p. 228), any super-complete model ( a , € ) of VNB is of the form V, for some a : this follows from the absoluteness of the rank function and the axiom of Fundierung. Clearly, any such CY must be inaccessible by Lemma 2.1(iv). Note that we have a corresponding result for PRIIo’, provided we drop the requirement “in (t)”,cf. §3.2(2).

308

KLAUS GLOEDE

In particular, one can prove in BL, that there is a “large” class of inaccessibles. There are several ways of interpreting the concept of a “large” class of ordinals, e.g. we might call a class A of ordinals large, if it is unbounded or even closed unbounded (i.e. the range of a normal function defined for all the ordinals). Unfortunately, the class of inaccessibles is not closed, since the limit of an ascending sequence of inaccessibles need not be inaccessible (since it may be cofinal with 0 ) . A more suitable notion of a “large” class of ordinals is provided by the following definition: DEFINITION3.1. stationary (A) :-VF [Nft (F)+,3[ E A F(S)= 51, stationary (A,&):- V f [Nft (fp)+3[ E A f ( 5 ) = 51. Thus a class A is stationary (sometimes also called dense) iff it intersects each closed unbounded class of ordinals; “ A is stationary in a” is the corresponding concept relativized to V,. LEMMA3.5. lim (a)A A C a. + .( V,,E)

stationary ( X , ) [ A ]t* stationary ( A , a ) .

A further strengthening of PRIIA,, is provided by the following THEOREM3.5. BL,,,

((5 I in (5) A

@(Ao,.. . ,A,)+stationary

Qv6(A, n Vc,. . . ,A,,, n V.)})

f o r any II!,+,-formula @ with free variables as indicated. PROOF. Let F be any normal function. Applying Theorem 3.15 to the n!,+,-formula Nft ( F ) A @(An,.. .,A m ) ,

we obtain the existence of some ordinal a such that in(a)A N f t ( F n V,)”-

A

@‘-(Ann Va,..., A,,, n V a ) .

Clearly, lim ( a ) + [Nft ( F n V,)”-

t* a

= F(a)l,

and this proves the theorem. Let us define the following versions of Mahlo’s operation: DEFINITION 3.2.

H ( A ) := (5 I in (5) A stationary (A,()}, Ho(A) := (5 I reg (5) A stationary (A,[)}.

REFLECTION PRINCIPLES AND INDESCRIBABILITY

309

(These operations are roughly the duals of the operation M of KEISLERTARSKI[ 19641.) In BERNAYS[1961], p. 41 it is shown that one can define in ZF a predicate Ma (a,?) ("a is a Mahlo number of order y " ) by a formula of TeZF satisfying the following recursive conditions: Ma (a,O)f* a E H(On) (i.e. in ( a ) ) ,

Ma (a,? + 1) f* a E H({5 I Ma (5,~)}), Ma ( a , A ) f* VTJ< A Ma (a,q) for A alimit number. (This definition agrees with the notion of a hyper-inaccessible number o f type y as defined in LEVY[ 19601.) Replacing H by Hn, we obtain Mahlo's original definition of the T o u numbers ( T a n y is the a th ordinal among the rOr numbers, cf. MAHLO[1911-19131). COROLLARY 3.5. B L I E M , where M is the following formula (which might be called Mahlo 's Principle):

stationary ( A ) + stationary (H(A)).

M

PROOF. Apply Theorem 3.5 to the H,'-formula stationary (A) and use Lemma 3.5. Although the intersection of two stationary classes may be empty, one can prove in BLI the following stronger Mahlo diagonalization principle : M*

VTJ(stationary (A,,)) + stationary ({Elin (6) A VTJ <[ stationary (A,,,()}):

COROLLARY 3.6.

BL,

M*.

PROOF. Let YP(A,TJ) be the H,'-formula stationary (A,,) (where A,, = {x ( ( T J ,E ~ A}). ) Applying Theorem 3.5 to the formula VTJV(A,r)) which clearly is equivalent to a H,'-formula, we have:

B = (5 1 in (5)

A

VTJ< ( Y"<(A n V<,TJ)} is stationary.

COROLLARY 3.7. BL, 1 stationary ((6 1 Ma (&y)}). PROOF. By induction on y using Corollary 3.5 for the case y = 0 (On is stationary) and the successor stages, Corollary 3.6 for the limit stages. We note that the result obtained in Corollary 3.7 can already be derived from the principle M* rather than PRnl1, and in fact even from the principle M * obtained from M* by replacing the class variable A by a syntactical variable A ranging over class terms (of the language of ZF) only. While (the universal closure of) M* is a Hzl-statement,each instance BL1.Then of M * is a Z2'-formula. Let a be an ordinal such that (V,,E)

+*

310

KLAUS GLOEDE

(V,,E) kzPR C2' by Lemma 3.4. Therefore, each instance of M * being satisfied in V, is already satisfied in some V, for some /3 < a.Moreover, since there are only countably, many formulas of 2?= (and hence countably many instances of M * ) , an argument due to LEVY[1971] (Corollary 10b) shows that there is an inaccessible /3 < (Y such that V, satisfies each instance of M * . On the other hand, JENSEN[1972] (Theorem 6.1) has shown that (V,,E)

kzPRIL'

reg (a)A Vx C a [stationary (x,a) +35 < a stationary (x,t)l f*

assuming Godel's axiom V = L. In particular (in ZF + (V = L)): ( v,,E)

kzP

R ~++' ( v,,E)

k2B + M*,

whereas hence

(V,,E)k2 PRlIl'+3(


(V,,E)+zB+bl*-+3t < a ( V c , E ) k 2 Z F + M * as mentioned above. This result might be compared with MontagueVaught's theorem, since B and ZF are related to each other like M * is related to M". Some applications of the concept of Mahlo numbers can be found in SCHMERL [ 19721 and SCHMERL-SHELAH [ 19723. For recursive analogues of Mahlo numbers cf. ACZEL-RICHTER[ 19701. 4. Indescribable cardinals 4.1. Higher-order languages

Having introduced class variables, which are called variables of type 2, whereas set variables are regarded as variables of type 1, it is natural to extend the language of 2fVNBfurther by adding variables of any finite (and possibly even transfinite) type. Variables of type n will also be denoted by X", Y " ... , . For formulas of this language (with atomic formulas of the form s = t and s E t for any kind of variables s, t ) we extend Lkvy's classification of first-order formulas as follows (cf. HANF-SCOTT[ 19611):'

' The definition that follows does not completely agree with the notation introduced in the which are not previous sections since we have used I,' to denote the set of formulas of 2vNs allowed to contain free variables of type 3, but only set and class variables. In most cases we consider only formulas which contain free variables of type at most 2 (i.e. class variables). Thus, if there is any risk of confusion, we give additional information on the free variables which are allowed to occur in the formulas under consideration.

REFLECTION PRINCIPLES AND INDESCRIBABILITY

311

A IIom-formula is a formula in which every bounded variable is of type rn and every free variable is of type G rn + 1, Zom= nom. A K+Iformula (Z~+,-formula)is a formula of the form VX"'"-v ( 3 X m + ' * resp.) where !P is a Cn"-fomula (II," -formula resp.). A A,"-fomula is a formula which (in some appropriate type theory) is equivalent to both a n,,"- and a 2,"'-formula. Thus the IIol-formulasare just those of Bernays' set theory B (as in the 1958 monograph), and the IIt-formulas containing free variables of type at most 2 are the formulas of VNB set theory (disregarding the fact that for some technical reasons, we have taken u = v, u E v and u E A as the only atomic formulas of these languages). It is debatable whether the language of finite- or transfinite-type theory is more adequate for set theory than the languages considered in the previous chapter (for a discussion, cf. TAKEUTI[1965] and [19691), in any case the idea of investigating the type-theoretical classification of various definitions and propositions with respect to a particular structure has turned out to be fundamental for a large number of results (cf. 01.3 for applications within the scope of this paper). Therefore, we do not introduce a logical system for type theory (this could be done along the lines sketched in 003.1 and 3.2 for the second-order language), but consider only particular models for certain axioms in the language of type theory (like the reflection principles for this language). Taking this approach we avoid taking into account logical axioms for type theory like comprehension axioms and various kinds of choice principles (like those introduced in 03.1) since these are satisfied in any model under consideration (cf. Lemmas 3.1, 3.3 and Corollary 3.2 for the case of second-order language). Throughout this chapter we assume the axioms of ZF + A C for our metatheory (the use of the axiom of choice could be avoided in most cases by some simple modifications). For the following we again require a suitable formalization of satisfaction in a structure of the form (a,€) for formulas of finite-type theory. In particular, we assume that a set is assigned in a suitable manner to each formula @ of finite-type theory. Then a formal satisfaction predicate for IIom+'-formulascan be defined in ZF set theory (e.g. by reduction to the case of satisfaction of a IIol-formula in the structure ( P ' " ( a ) , E ) (where P " ( a ) is the result of applying the power set operation to a m-times, Po"()= a ) in a manner indicated in 02.2 for the case m = 1). We let G

r@7

( a , € ) +"'+I

e [ b , . . . , ak);(bO,...,b n ) I ,

(though somewhat imprecise we also use the notation

(a,€>I="'+'e[ao,.. . , ak ;h,... b,l)

312

KLAUS GLOEDE

be a suitable formalization in ZF set theory of the corresponding metamathematical statement " e is the Godel number of a IIom+l-formula@ with free set variables vo,.. . , vr, free class variables A o , .. . , A,, ao,.. . , a, E a, bo,..., b,, c a , and @ is true in ( a , € ) if ai(bi) is assigned to v, (Ai resp.) for i s k, j s n, and bound variables X'" are interpreted to range over P ' ( a ) " . is of 91.1.2, kzis the same relation as defined in 92.2.) A formal definition of satisfaction as required can also be given without referring to satisfaction for IIo'-formulas, details are given in BOWEN [1972], pp. 42ff, for the case m = 1 (and this definition can easily be generalized for any finite m ) . The requirements on the formal definition of satisfaction which we need in the sequel are the following: (+I

4.1.1. Uniform definability. For each m > O , n 2 0 there is a A l m formula v',,,( v o ,Xom+l,. ..,X,"'+l) with free variables as indicated such that for every IIo"-formula @(XOmfl,. . . , X,,"+') with free variables as indicated, for any inaccessible a and all bo,.. . , b, E 8"(V,): (V,,E)+"+l

'O.'[bo,..., b , ] * ( V , , € ) ~ = " + r'Pm7[r@7;bo,..., ' b,].

(v',,, is just the formula defining satisfaction in (V,E): ( V , E ) ~ " ''@."Al*q,n('@.',A), where A is a class coding finitely many classes Ao,.. . , A,,.) As a consequence (cf. LEVY [19711, Theorem 8): 4.1.2. Absoluteness. For each n,m > O there is a n,"-formula W(vo,Ao) with the only free variables uo and A , which is universal for the set of IInm-formulas @(Ao)which contain only the free class variable Ao, i.e. for any such formula @(Ao),every inaccessible a and every A c V, : (V,

E)

km+' '07[A]

(V,,E) +"+I

t ,

T[T,A].

(To obtain v' from the A,"-formula v'" of 34.1.1 consider the formula

QXnmYrma(v0, A , , X , ",..., X,") where qma is the HI"' (2,")-formula which is equivalent to v', if Q is V (or 3 resp.). Now, contract the quantifiers QX,," and the first quantifier Q Y" of PmQinto a single quantifier QXnm(as described in §3.1 for the case m = 1) and substitute g ( v o )for v o where g('VXlm QX," @?) = -@- for any II,,"-formula VXI"* . * QX," @(Ao,X I m ,..., X,"), g ( v o )= 0 other-

wise.) (The requirement that a be inaccessible can be weakened in §§4.1.1 and 4.1.2.)

REFLECTION PRINCIPLES A N D INDESCRIBABILITY

313

4.1.3. Adequacy. We extend the process of relativization to nnmformulas as follows: W’ is the formula obtained from @ by replacing any quantifier of the form Vx by V x E a , VX by Vx E P ( a ) , VX“’ by Vx EP‘(a), and similarly for existential quantifiers (where in the last two cases x is a suitably chosen first-order variable). Then we have, as in the case of a first-order formula, the following schema which expresses the material adequacy of the satisfaction relation:

For any nmm-forrnula@(vo ,..., vk, A. ,..., A t ) with free variables as indicated and any elements a, ,..., ak E a, b, ,..., b, E p ( a ) : ( a , E ) F ” ” ‘O,”U~ ,..., a r ; b , ,..., b,]*@’”(ao,..., U k , bo ,..., bL). Since we shall consider only structures of the form ( a , € ) for transitive sets a, any free variable of type i may also be considered as a variable of any higher type. Therefore, it is sufficient to define satisfaction for formulas which contain only free variables of the same (highest) type. We have separated the first-order free variables from the higher-order free variables only for technical reasons, in connection with the reflection principles which are considered later.

REMARK.

4.2. Indescribable cardinals

By P W we denote the partidreflection principle for formulas in which contain free variables of type at most 2: DEFINITION.Let

r

r be either II,’”, X n m , or An’”. Then:

P R r @(u,,..., U k , A, ,..., A i ) + 3 ~[Trans ( U ) A

(PU(ao,. . . , a,, A , n u,. .., A, n u)l,

where @ is any r-formula with free variables as indicated.

I’-indesc ( a ):t,a > 0 A (V,,E) +’”+’PRT “ a is I?-indescribable”, r-desc ( a ):o-r-indesc ( a ) “a is r-describable”.

REMARK4.1. (1) Note that for each set of formulas l‘ as considered above the predicate “ a is r-indescribable” is defined by a formula of ZZF. (2) If m > 0 , we may also replace the principle PRII,”’ by the same principle restricted to llnm-formulas @ ( A )with the free-class variable A only (cf. Remark 3.1). (3) Intuitively, “CY is rInm-indescribable” means that for each K‘“-

3 14

KLAUS GLOEDE

formula @(uo,... , uk,A",.. . , A1)(with free variables as indicated) and any sets a. ,..., ak E V,, bo,..., b, C V,, if ( V,,E)

+"'+I

@[a,, . .. , ak ; bo,. . ., bil,

then there is a transitive set a E V, such that ao,.. . , ak E a (if m > 0 or n > 1) and ( v , , E ) ~ m + ' R e l ( ~ ~ + l,..., , ~ )Uk,a; [ u o b o n a , ...bl n u ] , i.e. by §4.1.3 Re1 (V,,Rel (a,@(a,.. . , ak,bo n a,. .. , b , n a ) ) ) which again is equivalent to

, bl n )) Re1 (a,@(&,. .., a,, bon a,. _. (since a n V,

= a),

and again by 4.1.3 this is equivalent to

( a , E ) ~ ' " + ' @ ,..., [ a o a k ; b o n a,..., b l n a l .

Moreover, if n > 0 or m > 0 then, by Theorem 3.4 (and the remark at the end of the proof for the case of PRII,') a can be assumed to be of the form V, for some p < a, and /3 can be assumed to be inaccessible if n > 0 and m > O . (Note that we not only identify a formula with its Godel number, but also use the same letters to denote numerals (in the metamathematical sense) and natural numbers (in the formal sense). We hope that the reader will discover the right meaning from the context as a correct distinction will render the notation unnecessarily complicated.) (4) Some authors define the notion of r-indescribability by using instead of I?-formulas which contain free second-order variables A,,, . .. ,Ak, the corresponding formulas which have unary relation constants Aiin place of A iand replacing the structures (V,,E) by structures ( V,, E, bo,..., b k ) where the bi's are subsets of V, interpreting Ai (whereas b, is assigned to Aiin our notation), and similarly for structures (a,€) for arbitrary sets a (cf. LEVY [1971]). In this chapter we are only concerned with the case m > O which implies that every indescribable ordinal is a cardinal, in fact inaccessible:

-

PROPOSITION 4.1. &'-indesc (a) in ( a )t,(V,,€) V B . PROOF. Immediate from the fact that Pow + PRIIoi (§2.1), (94) and Lemma 2.1(iv).

Similarly:

-

PROPOSITION 4.2. rIIa-indesc (a) { V,,E)

B = Ext + Fund + AuS'

BL.

+

REFLECTION PRINCIPLES A N D JNDESCRIBABILITY

315

PROPOSITION 4.3. n,'-indesc (a)t-,C!,+,-index(a). PROOF. Lemma 3.4. DEFINITION4.1. rrnm:= px(II,"-indesc (x)), cnm := px(X,,"-indesc 7,"

(x)),

:= px(IInm-index (x) A

2,"'-index (x)).

Since the universal closure of each instance of PRII,' is (equivalent to) a II!,+'-sentence, it follows from 4.1.2 and the fact that the clause "for all II,'-formulas " can be expressed by a universal quantifier ranging over the set of Godel numbers of II"' formulas that for n > O there is a II!,+'sentence W such that ( VwE) I='PRJJn' t-,(Vm,E)

'I?,

and hence n-,' is II!,+l-describable(if it exists).' The last statement is true even for the case n = 0, for ( V,,E)

PR&'

(V,,E)

t-,

kzVNB

by Proposition 4.1, and the universal closure of the conjunction of the finitely many axioms, by means of which VNB is axiomatizable, is a 1111-sentence.Alternatively, one may use the fact that there is a 111'sentence 9 such that in (a)f* (V,,E) kzT.Therefore, we obtain: 4.1 (HANF-SCOTT[1961], LEVY [1971], Corollary 17). THEOREM (i) (ii)

n-,,I

= v!,+' = rn',

7,'

< n-!,+' < n-:.

(With the understanding that pxcp(x) < px+(x) is read as follows: if 35+(5>,then Xcp(5) and pxcp(x)< pxlCl(x).) We now turn to the case m > 1. The situation is completely different for this case for the following reason: In our definition of rInmindescribability, we have restricted the II," -formulas to those containing free variables of type at most 2, and for the case m > 1, this is a true restriction which, however, is necessary: REMARK4.2. If in our definition of K"-indescribability, we allow any IInm-formula@ (containing variables of type m + 1) to occur in PRII,", then we obtain (for m > 0) the following wider notion of indescribability: a is II,"-indescribable in the wider sense iff for each II,"'-formula

' BY " / L x ( P ( xexists" ) we mean "3[cp([)and / L X ( P ( X ) is the least ordinal x such that ( ~ ( x ) " .

316

KLAUS GLOEDE

@(AO"+', .. .,Akm+')with free variables as indicated and any bo,...,bk E 8"(V,): if (V,,E) +"+I @ [ b o ,..., b , ] , then there is a transitive set a E V, such that ( a , € )+"+I @[bon P m - ' ( U ) , . . . ,bk n g m - * ( U ) ] .

If m = 1, this agrees with our previous definition, however, if rn > 1 then there are no n,"'-indescribable cardinals in the wider sense: Consider the III,z-formula3X (X= Ad). Then clearly for each a:

3 X (X= Ad)[ V, 1.

( V,,E)

So, if there is a transitive a E V,, such that (a,€)

3 X ( X = Ad)[V,

n9(a)l,

then there exists some b C a such that

b

=

V,

n9(a).

On the other hand, V, n 9 ( a ) = 9 ( a ) (since a C V,), hence 9 ( a )C a, a contradiction. Now returning to our original definition of H," -indescribability, let us assume m > 1 and let us consider some instance of PRII," (corresponding to some n,"-formula @ with the free variable A only). This is, in some suitable type theory, equivalent to a &,"-formula *(A), and its universal closure is of the form VXW(X).By the standard arguments of type theory the quantifier VX can be moved across the quantifiers of higher type in the Z,"-formula V, and hence VXT(X) is equivalent to a C,"-formula. Now, as in the case rn = 1, the statement (V,,E)k"+' PRII," introduces only an additional set quantifier ("for all n,"formulas"), and therefore we have a X,"-sentence Tosuch that (V,,E) +'"+'PRII,"

t*

(V,,E)

+"'+I

'Pa,

is C,"-describable, if it exists. Similarly, anmis also and hence rnm II,"-describable, (if it exists). Thus we obtain: THEOREM 4.2 (HANF-SCOTT[1961], LEVY [1971], Corollary 17). rn", ~

Therefore

n

"

< 7," < r;+i <

rnm+' # anm+' for mtl

r,

< ~ ; + i < n om + l

(m> O ) .

each m,n, but it is unknown whether or


anm+' < rnm+*.

However, Moschovakis has recently shown that the second alternative holds if one assumes V = L. The inequalities of Theorems 4.l(ii) and 4.2 are very strong, e.g. one can prove:

REFLECTION PRINCIPLES AND INDESCRIBABILITY

317

THEOREM4.3 (cf. BOWEN[1972], 02.1.15, THARP[1967], LEVY [19711, Theorem 19h). HA+,-indesc( a )-+ stationary ((6 1 HI,'-indesc ( ( ) } , a ) . PROOF. Let a be IIk+l-indescribable, and let T be the IIi+l-sentence (described in the proof of Theorem 4.1)such that for any inaccessible p : { V&)

t*

II,,'-index (p).

In particular we have: (V,,E)pU, since a is II,'-indescribable and inaccessible. Hence by Theorem 3.5 and Lemma 3.5

(6 I in (5)A ( V& f=' U}is stationary in a, whence the result follows. Similarly: THEOREM4.4 (cf. LEVY [1971], Theorem 19i). For rn > 0 : if indesc ( a ) or C.,",l-indesc ( a ) ,then stationary ((8 1 IT,"' -indesc (()},a), and

stationary ((6 1 Z."-indesc ( ( ) } , a ) .

Additions. (1) LEVY [I9711 investigates the following weaker form of indescribability (where r again denotes IInmor S,,"'): w-r-indesc ( a ):- a > 0 A (a,<) km+l PRT, " a is weakly r-indescribable". The main difference between both notions of indescribability results from a technical problem: If a is a limit ordinal, then V, is closed under ordered pairs and these can be used, e.g. for contracting quantifiers of the same kind. In case of the structure (a,<),however, one has to use an ordinal-pairing function (like Godel's function P : On X On -+ On which is one-to-one and onto). If a is closed under P, e.g. if a is regular, P can be used instead of the operation of ordered pairs. Second, whereas for limit ordinals a, V, is closed under the power-set operation this is not true for a. Thus, one can only prove:

w-IIo'-indesc ( a )++reg( a )A a > o (compare with Proposition 4.1), and any weakly H,'-indescribable cardinal is weakly inaccessible, but need not be (strongly) inaccessible. On the other hand, if a is strongly inaccessible then I V, 1 = la I (by the AC). This implies that for inaccessible

318

KLAUS GLOEDE

a and any rn, n such that m > 1 or m

=

1 and n > 1:

w-II,"-indesc ( a )t,nnm-indesc( a ) (LEVY[1971] Theorem 2.1). Indescribability in the weak sense has also been investigated by KUNEN [1968], 016. (2) Whereas LCvy's definition of r-indescribable ordinals is obtained from our definition by replacing the structures (V,,E) by (a,<),there is a recursive analogue of the r-indescribable cardinals which results from the former (roughly) by considering the structures ( L , , E ) in place of (V,,E). For more details we refer to ACZEL-RICHTER [1970]. 4.3. Large cardinals and indescribability In this section we review some indescribability results for large cardinals. We do not include proofs since in almost all cases a particular method of proof is required. DEFINITION 4.2. A cardinal K > o is called measurable iff there is a K-complete (which means closure under < K members) non-principal prime ideal P(K);it is called w-measurable iff there is a non-principal R,-complete prime ideal in P(K).The least w-measurable cardinal is measurable, and hence the least-measurable cardinal is %'-describable, since it is the least o-measurable cardinal and is > XI).On the other hand: If K is measurable then K is II1'-indescribable (HANF-SCOTT[1961]). (The proof uses an ultrapower construction and an extension of Los' Theorem, cf. SILVER[1971], Theorem 1.10.) VAUGHT[1963] has shown that the converse of the above is not true, this also follows from Silver's results described below. DEFINITION 4.3. (ERDOS-HAJNAL[1958]). If f is a function and the domain of f is K < " ' ( K ( ~ ) ) ,the set of finite (n-element resp.) subsets of K , then a set a K is called homogeneous for f iff f ( x ) = f ( y ) for all finite subsets x , y a of the same cardinality (of cardinality n resp.).

c c

K

K

c

[Ft (f) A dom (f) = K<" A rng (f) 2 + 3 x ( X a A x has order type a A x is homogeneous for f)], +(a)" : e V f [Ft (f) A dom (f) = K ( " ) A rng c f ) c2 + 3 x ( x C a A x has order type a A x is homogeneous for f)]. +(a)<@:-Vf

If K is measurable, then K + ( K ) < ~ (Erdos-Hajnal), and the least K such that K + ( K ) < * is obviously II,'-describable, hence by Hanf-Scott it is less than the first measurable cardinal (providing such a cardinal exists). STATEMENT 4.1.

K

+( K ) '

II,'-indesc ( K ) .

t ,

REFLECTION PRINCIPLES AND INDESCRIBABILITY

319

A proof of this statement and other properties equivalent to II,'indescribability can be found in SILVER[1971], Theorem 1.13 (cf. also BOWEN[1972], 42.2).

STATEMENT 4.2. p x ( x + (x)*) < p x ( x + (0)'"') < p x ( x + (x)'"); in fact, if K + (o)'", then there is a A < K such that A is II,,"'-indescribable for each n, m < w (SILVER [1966],Corollary 4.4; see also REINHARDT-SILVER [1965] and 04.16 below). On the other hand, if K = p x ( x +(a)'"') and a < K , then K is clearly IIl'-describable (assuming that a is a limit ordinal). STATEMENT 4.3. The notion of indescribability can be extended by using the language of transfinite type theory, viz. the partial-reflection principle PRII,' where I,' is a straightforward generalization of IInmto transfinite-type theory. (Having gone so far, one could also allow n to take transfinite values, i.e. introducing infinitary languages.) In order to define indescribability for transfinite-type theory in ZF set theory there are two approaches which are not equivalent since they are based on different formalizations of satisfaction for transfinite-type theory. For the case of finite m one can reduce satisfaction of IIom"-statements in (V,,E) to satisfaction of IIoi-formulas containing additional unary relation constants in the structure (V,+,, €, V,, V,+l,.... Va+m-l), and since V,, . . ., V,+,-, are (first-order) definable in (V,+,,E), again, this can be reduced to satisfaction of II,'-statements in (V,+,,E) (cf. 04.1). However, for transfinite y, V,+, need no longer be first-order definable in V,+, for 6 < y, so the latter reduction is not possible in the case of transfinite-type theory. Of course, one could still generalize the first reduction for transfinite type theory; however, we shall take the following approach due to REINHARDT[1970], Definition 4.7 and JENSEN [1967a], DEFINITION.a is K-indescribable iff for each IIo'-formula cp ( X o , . . , X , ) with free variables as indicated and all A,, ..., A, C V,: if (V,+,,E) k cp [A,, . . ., A,] then there is a p < a such that,

( V o + , , E ) k ~ [ A o nVfiy*.*,An

Val.

We let K-indesc ( a )be the corresponding formal statement (expressible by a formula $ ( ~ , aof) 9zF using the formalized notion of satisfaction). Clearly we have (by the result of 44.1 to which we referred above):

-

IIom+l-indesc ( a )++ m-indesc (a) for each finite m, in particular: 0-indesc ( a ) IIo'-indesc ( a )f* in (a)f, (V,,E) BL. 1-indesc ( a )t* II2-indesc (a)f* (V,,E)

k*B,

320

KLAUS GLOEDE

Then Silver's result mentioned in 84.2 can be strengthened as follows: If K is the least y such that y - + ( o ) ' ~then {( < K I K-indesc (6)) is stationary in K.

(Cf. JENSEN[1967a], Satz 4, p. 110, and GLOEDE[1972], Theorem 10.7.) If Y < a s p then we define (REINHARDT [1970], Definition 4.6) in ZF ( V A < (VB,E) to mean (informally) ( V,+,,E)

I=

q [ a o , ... , a,]

c*

(vB+,,€) t= q [ a o , .. . , G I

for every a. ,..., a, E V, and every 6Pz,-formula q ( v o,..., v,) with free variables as indicated. Similarly, we define (V&,A

n Va) < (VB,€,A> (forA G V p ) .

Thus (VwE)

3 (VB,E)c*(Va,E)<(Vp,E),

and ( V,,E) < (V,,E) means that (V,,E) is a second-order elementary extension of (V,,E). THEOREM4.5 (REINHARDT[1970], Theorem 4.8). If K is indescribable for some y < K , then there is some (Y < K such that

y

+ 1-

(V&) < (V&. (Unfortunately, for y = 0 this does not imply Montague-Vaught's Theorem 2.2.) In particular, if K is no""*-indescribable, i.e. ( m + 1)indescribable, then ( V,,E) satisfies the principle of complete reflection CRIIo"'+' (which is defined in analogy to CRn,' in 83.2. Using Reinhardt's method, this latter result can be improved as follows: THEOREM4.6 (GLOEDE [1970], Theorem 12.8). Assuming PRIIl"+' + Alm+'-Comp,there is some (Y such that

B+

(Va,E,A n Va) i m (V,E,A),

in particular, if IIlm+l-indesc( K ) , then (V,,E)

+"'+I

PRII,,"+'.

PROOF. By 4.1.1 there is a Alm+'-formulaT(vo,Xo)such that for any given class A :

(i) Vx (( V,E) +"'+' @[x ;A 1 .++ W ( @ , x ) , A 11, for each Ilom+'-formulacD(v,,Xo)with free variables as indicated, and by

REFLECTION PRINCIPLES AND INDESCRIBABILITY

32 1

-

4.1.2, v' is absolute with respect to V , if p is inaccessible:

W v o , A o ) [ ( @ , x )n; AV,l (ii) V x E VB((V,,E)Fm+* (V,,E) +"'+I @ [ x ; A n V , ] ) for each IIom+~-formula @(vo,Ao). By AI""-Comp there is some class B such that V y ( y E B t t v ' ( y , A ) ) ,i.e. B

= {@,x

-

1 @ is a IIom+'-formulaA ( V , E )

+"'+I

@[x ;A]}.

Define Qo(Ao,A1) := Vx (x E A . 'P(x,Ao)).Obviously, a0 is a I l l m + ' formula. Hence by PRI1,"'" there is some inaccessible cr such that QOV-(B n V,,A n V , ) , i.e. by (ii) (iii) V x ( x E B n V, - ( V , , E ) F " + " P [ X ; A n V , ] ) .

-

Now let @(uo,Ao)be any IIom+'-formulaand a E V,. Then we obtain, using (i) and (iii): (V,E) Fm+' @ [ a: A ]

hence

(V,,E) +"'+I

";A

n

Val,

References ACZEL, P. and RICHTER, W. (1970) Inductive definitions and analogues of large cardinals. In Conference in Math. Logic, London, '70, W. Hodges (Ed.), pp. 1-9. Springer Lecture Notes in Mathematics, Vol. 255. Springer, Berlin. 351 pp. BAR-HILLEL, Y.,POZNANSKI,E. I. J., RABIN, M. O., and ROBINSON,A. (Eds.) (1961) Essays on the foundations of mathematics. (Dedicated to A. A. Fraenkel.) Magnes Press, Jerusalem. 351 pp. BARWISE,J. (1969) Infinitary logic and admissible sets. J.S.L. 34, 226-252. BERNAYS,P. (1958) Axiomatic set theory (with a historical introduction by A. A. Fraenkel). North-Holland, Amsterdam. 226 pp. (2nd ed. (1968), 235 pp.) BERNAYS,P. (1961) Zur Frage der Unendlichkeitsschemata in der axiomatischen Mengenlehre. In BAR-HILLEL et al. (1961), pp. 3-49. This volume. (English translation-revised version.) BOWEN,K. A. (1972) The relative consistency of some consequences of the existence of measurable cardinal numbers. Dissertationes Math. Warszawa. 63 pp. BULOFF, J. J., HOLYOKE, T. C., and HAHN, S. W. (Eds.). (1969) Foundations of mathematics. (Symposium papers commemorating the sixtieth birthday of Kurt Godel.) Springer, Berlin. 195 p. ERDOS,P. and HAJNAL,A. (1958) On the structure of set mappings. Acta Math. Acad. Sci. Hung. 9, 111-131.

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ERDOS,P. and HAJNAL,A. (1962) Some remarks concerning our paper “On the structure of set mappings”. Ibid. 13, 223-226.

FELGNER, U. (197la) Models of ZF-set theory. Springer Lecture Notes in Mathematics, Vol. 223. Springer, Berlin. 173 pp. GAIFMAN, H. (1967) A generalization of Mahlo’s method for obtaining large cardinal numbers. Israel J. Math. 5 , 188-201. GLOEDE,K. (1970) Reflection principles and large cardinals. Ph.D. thesis. Heidelberg. 129 PP. GLOEDE,K. (1971) Filters closed under Mahlo’s and Gaifman’s operation. In Proceedings of the Cambridge Summer School in Mathematical Logic, August ’71. A. R. D. Mathias and H. Rogers (Eds.), pp. 495-530. Springer Lecture Notes in Mathematics. Vol. 337. Springer, Berlin. 660 pp. GLOEDE, K. (1972) Ordinals with partition properties and the constructible hierarchy. Zeitschr. f . math. Logik 18, 135-164. GODEL, K. (1940) The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory. Annals of Math. Studies, Vol. 3. Princeton University Press, Princeton, N.J., 66 pp. (7th printing (1966).) HANF, W. and SCOTT,D. (1961) Classifying inaccessible cardinals (abstract). Notices A.M.S. 8, 445. JENSEN, R. B. (1967) Modelk der Mengenlehre. Springer Lecture Notes in Mathematics. Vol. 37. Berlin. 176 pp. JENSEN, R. B. (1967a) GroBe Kardinalzahlen. Notes of lectures given at Oberwolfach (Germany). 121 pp. JENSEN, R. B. (1972) The fine structure of the constructible hierarchy. AnnaLs o f Math. Logic 4, 229-308. JENSEN,R. B. and KARP, C. (1971) Primitive recursive set functions. In SCOTT(1971), pp. 143-1 76. KEISLER, H. J. and ROWBOTTOM,F. (1965) Constructible sets and weakly compact cardinals (abstract). Notices A.M.S. 12, 373-374. KEISLER, H. J. and TARSKI,A. (1964) From accessible to inaccessible cardinals. Fund. Math. 53, 225-308. (Corrections, ibid. 57, 119 (1965).) KRIVINE,J . L. (1971) Introduction to axiomatic set theory. Reidel, Dordrecht. 98 pp. KUNEN,K. (1968) Inaccessibility properties of cardinals. Doctoral Dissertation. Stanford. 117 pp. KUNEN,K. (1972) The Hanf number of second-order logic. J.S.L. 37, 588-594. LEVY, A. (1958) Contributions to the metamathematics of set theory. Ph.D. thesis. Jerusalem. LEVY, A. (1960) Axiom schemata of strong infinity in axiomatic set theory. Pacific J. Math. 10, 223-238. LEVY, A. (1965) A hierarchy of formulas in set theory. Memoirs A.M.S. 57, 76 pp. LEVY, A. (1971) The size of the indescribable cardinals. In SCOTT(1971), pp. 205-218.

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LEVY, A. and VAUGHT,R. L. (1961) Principles of partial reflection in the set theories of Zermelo and Ackermann. Pacific J. Math. 11, 1045-1062. MAHLO,P. (191I ) Uber lineare transfinite Mengen. Berichte iiber die Verhundlungen der Koniglich Siichsischen Akademie der Wissenschaften zu Leipzig, Math.-Phys. Klasse 63, 187-225. MAHLO, P. (1912-1913) Zur Theorie und Anwendung der p,-Zahlen, ibid., I: 64, 108-112 (1912); 11: 65, 268-282 (1913). MONTAGUE,R. and VAUGHT,R. L. (1959) Natural models of set theories. Fund. Math. 47, 219-242. MOSCHOVAKIS,Y. N. (1971) Predicative classes. In SCOTT(1971), pp. 247-264. MOSTOWSKI,A. (1969) Constructible sets with applications. North-Holland, Amsterdam. 269 PP. QUINE, W. V. (1963) Set theory and its logic. Belknap Press, Cambridge, Mass. 359 pp. (Revised ed., 361 pp.). REINHARDT, W. N. (1967) Topics in the metamathematics of set theory, Doctoral Dissertation, 85 pp. W. N. (1970) Ackermann’s set theory equals ZF. Ann. Math. Logic 2,189-249 REINHARDT, REINHARDT, W. N. and SILVER, J. (1965) On some problems of Erdos and Hajnal (abstract). Notices A.M.S. 12, 723. ROBINSON,R. M. (1945) On finite sequences of classes. J.S.L. 10, 125-126. SCHMERL,J. H. (1972) An elementary sentence which has ordered models. J.S.L. 37, 521-530. SCHMERL,J. H. and SHELAH, S. (1972) On power-like models for hyperinaccessible cardinals. J.S.L. 37, 531-537. SCOrT,

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SILVER,J. (1966) Some applications of model theory in set theory. Doctoral Dissertation, Berkeley, 110 pp. (Published (with the deletion of 04) in SILVER (1971)). SILVER,J. (1971) Some applications of model theory in set theory. Ann. of Math. Logic 3 , 45-1 10. TAKEUTI,G . (1965) On the axiom of constructibility. Proceedings of Symposia of Logic, Computability and Automata at Rome. Mimeographed Notes, New York. TAKEUTI, G. (1969) The universe of set theory, pp. 74128. In BULOFF et al. (1969). TARSKI, A. and VAUGHT, R. L. (1957) Arithmetical extensions of relational systems. Cornpositio Math. 18, 81-102. THARP,L. (1967) On a set theory of Bernays. J.S.L. 32, 319-321. VAUGHT, R. L. (1963) Indescribable cardinals (abstract). Notices A.M.S. 10, 126.