Set Theoretic Aspects of Atr0

if it consists of a string of

k

zt

(respectively

set quantifiers beginning with an existen-

tial (respectively 'universal) one, followed by an arithmetical formula. I

I

Zk- TI

(respectively TI - TI O) we mean the formal system consisting of O k plus the transfinite induction scheme restricted to formulas

u,v,w,x,y, . • . .

E,

and

lye employ without comment a number of

abbreviations which are familiar from textbooks on

ZF set theory.

Consider the

following set theoretic axioms: 1.

Axiom of extensionality:

Vu(uEx

2.

Axiom of regularity:

3.

Axiom of infinity:

4.

Axioms asserting that the universe is closed under primitive recursive

x t q,

~

+-+

uEy)

3uEx(u n x

~

x

=

y.

= q,).

3x(q,Ex & VuEx(u U [u ] E x)).

set functions (Jensen/Karp [22]). 5. Vx(xt¢

A set ~

r

of ordered pairs is said to be regular if

3uExVvEx {v,u} I r).

Our key axiom asserts that if

r

is regular then

S.G. SIMPSON

258 there exists a function

f

with fielder)

C

domain(f)

and, for all

u E domain(f),

(v,u) E r },

feu) = {f(v) 6.

Axiom of choice:

7.

Axiom of countability:

Our basic system

ATR

s

Vx3r

(r

is a well ordering of

Vx (x

is countable).

consists of axioms

o

x).

1

through

Axioms

5.

6

and

7

are

regarded as optional extra axioms.

Actually, our main interest is in the full through 7, i.e. ATRs plus the axiom of count-

1

system consisting of axioms

o

ability. ATR

ATR s

Note that

s

o

is a subsystem of

is finitely axiomatizable.

o

Hence

ZF. ATR~

We shall begin §3 by showing that plus the axiom of countability is

finitely axiomatizable.

§3.

Conservative extension result.

In this section we present some basic results about lationship between

and

ATR

Keep in mind that

ATR~

ATR~

O. ATR is a system of second order arithmetic. O

theory while

3.1

ATR~

is finitely axiomatizable.

~.

Proof.

and about the reis a system of set

We claim that the scheme of closure under primitive recursive set

functions can be replaced by an axiom asserting closure under rudimentary functions

Fo-F

sitive set

a

(Jensen [.21] p. 239), plus an axiom asserting that for any trant

and ordinal 5

the constructible hierarchy

Vx3t(xEt & t

exists, plus an axiom first that axiom

a,

tive recursive ordinal functions.

3 2

is transitive).

(applied to linear orderings

La(t)

starting at

t

To prove the claim, note

r)

yields closure under primi-

Now apply Theorem 2.5 of Jensen/Karp [22].

We remark on some alternative simplified axiomatizations of s ATR If this is done, then axiom 5 O O. can be replaced by its special case in which r is a linear ordering. (The spec-

s ATR •

~.

One may add the axiom of choice to

ial case just says that every well ordering is isomorphic to an ordinal.) the axiom of countability implies the axiom of choice.

axiom of countability, a different simplification is possibLe: La(t)

can be replaced by its special case in which

proof of these remarks.

a

Clearly

In the presence of the

= wand

the axiom about t

=~.

We omit

I'e conjecture that either of the mentioned simplifica-

tions can be made even without the axiom of choice. In any case we have:

3.3

Lemma. s ATR ~

a model of

o

In any model of

ATR~, the hereditarily countable sets form

plus the axiom of countability.

Set theoretic aspects of ATR o Proof.

Obvious.

259

(A set is said to be hereditarily countable if the smallest

transitive set containing it is countable.) We now exhibit a close relationship of mutual interpretability between

ATR

and ATR Assume that the language of second order arithmetic has been interO' preted into the set theoretic language in the usual way: number variables range over

set variables range over subsets of

00,

3.4

Each axiom of

Proof.

Let

~

formula, and let

-<,

{m: m-< n}.

that

S along m~

-<

up to

n,

Thus

n & S(j,{(i,k) :


k~m &

Y is

says

(i,k) E Y})}.

Using the axiom of regularity, prove by induction on

contains a set

3.5

be an arithmetical

s

of

to

S(j,X)

let t be the transitive closure of {<}, let a be the O' and for each n let Jnl < a be the ordinal of the restriction

ATR

ordinal of

-<

let

00,

be an arithmetical formula which asserts that

Y = {(j,m) : Reasoning in

We then have:

is a theorem of

be a well ordering of

~(n,Y)

the result of iterating

etc.

00,

s

o

Y such that

Any model of

~.

In/

We omit details.


can be expanded to a model of

the axiom of countability. Proof. empty set s

~

we make the following definitions. A tree is a nonATR O of (codes for) finite sequences of natural numbers such that

Within T

t &t E T

~

sET.

A tree

i.e. there is no function Trees

T and

T'

f

T

is said to be well founded if

such that

Vn f[n]ET

where

are said to be isomorphic, written

T

T has no path,

f[n]

~

T',

( f (0), ... ,

exists an isomorphism between them, i.e. an order preserving bijection of T'.

If

sand

catenation of

are finite sequences of natural numbers,

t

s

followed by

T = {t: s ""'t E T}. s for all sET, i f

A tree

t. T

s""'{m) E T

If

T

is a tree and

T and

jection of

Tonto

able trees is to mean using the

T'

s~ t

sET,

Tonto

is the con-

we write

is said to be suitable i f i t is well founded and, and

s""'{n) E T

Clearly the class of suitable trees is that if

f(n~l».

if there

and

Ts""'{m)~

Ts""'{n) then

m = n,

TI~. The point of the definition is

are suitable then there is at most one order preserving biT'.

~~ on rr~.

Hence the relation If

3n«n>E T' & T~T' (n»'

T

and

T'

T

~

T'

of isomorphism between suit-

are suitable trees we write

The relation

'Y

t

is again

~~ axiom of choice, a consequence of ATR O'

1

~l

on

1

TIl'

T E T' We are

S.G. SIMPSON

260

We interpret the set theoretic language into the language of second order arithmetic as follows. suitable trees.

Set theoretic variables are interpreted as ranging over

The equality relation = between set theoretic variables is in-

terpreted as

~,

founded tree

T

is interpreted as

and

E.

The idea here is that a well

is to be identified with a hereditarity countable set

The restriction to suitable trees is for convenience only. trees we have if

ITI = IT' I

T ~ T',

if and only if

T '" E T'.

and

Note that for suitable

ITi E IT'I

if and only

We must verify that the suitable tree interpretations of axioms 7

are theorems of

ATR

Recall that a set theoretic formula is

O' built up using only bounded quantifiers formula

6i

~(xI, ••• ,xn)

on the

ITt

VuEx, 3uEx.

I

6

through if it is

0 Note that for each

6 0 the corresponding suitable tree formula ~(TI, .•• ,Tn) is

class of suitable trees.

Hence we may apply

6i

comprehension

(a consequence of

ATR to get closure under rudimentary functions. FurtherO) more, given suitable trees corresponding to a transitive set t and an ordinal a,

we can use arithmetical transfinite recursion along a well ordering of type to define a suitable tree corresponding to

tion is routine.

The rest of the verifica-

L (t).

a

The above discussion implies that any model M of ATR can be expanded to O s plus the axiom of countability. The elements of MS Ms 0 f ATRO are the equivalence classes of suitable trees in M under _ in M. This coma modeI

pletes the proof of Lemma 3.5. Combining Lemmas 3.3, 3.4 and 3.5 we obtain immediately the following conservative extension result (one may compare Theorem 4.6 of Feferman [6]): 3.6 metic.

~.

Let

cr be a sentence in the language of second order arith-

The following are equivalent.

(i) ATR~ (ii) ATR~

plus the axiom of countability proves proves

§4.

Models of

Let

M = (IMI, EM)

cr;

cr;

ATR

s O' and

N

(INI,E

is an E-transitive submodel of

J1

b E INI, a EM b i f and only i f

a E INI

if

N)

be models of

ATR

and, for all

INI ~ 1M: and a EN b.

s O'

We say that a E IMI

and

The purpose of this sec-

N

Set theoretic aspects of ATR o ATR~

tion is to prove that every model of ATR~.

which is again a model of

261

has a proper E-transitive submodel

This answers a question which was raised by

McAloon and Ressayre [23J. The main part of our argument consists in showing that the proofs of certain well known theorems from hyperarithmetic theory can be pushed through in Our notation for hyperarithmetic theory is as in §3 of [13] . e E OY such that

Y if there exists

hyperarithmetic in

We say that

ATRO'

X is

Y• H e

X is recursive in

is equivalent to the assertion that The principal axiom of ATR O Y Y VY Ve(eEO ~ H exists). e We say that

X is

Zl

in

1

X

~l

is

in

1

Y if both

4.1

if

X

such that

co \ X

are

Zl

Vm(mEX

Zl

1

in

X is hyperarithmetic in

formula

1

~ ~(m,Y».

~(m,Y),

with

We say that

Y. A well known theorem i f and only i f

Y

X

is

The following lemma entails that Kleene's theorem is provable in

Y.

Ve(e E OY

Y,

X and

Kleene [17J asserts that

if there exists a

Y

no free set variables other than

Lemma.

The following

~ H~ exists).

~i

is

in

~

provable in

Then for all

ACA '

O

Let

Y be a set such that

X, X is hyperarithmetic in

Y

if and only

Y.

Recall (from §3 of [13]) that there is an arithmetical formula Y e E OY then H is defined as the unique Z such that e H(Y,e,Z). Suppose first that X is hyperarithmetic in Y. Then X = (HY). for e 1 some e E OY and some i. Thus we have Proof.

H(Y,e,Z)

such that i f

m EX

3Z(H(Y,e,Z) & m

~

VZ(H(Y,e,Z) so

X

is

in

rr 1

af

~i in Y. In ACA alone we can prove O (although we cannot prove that OY exists as a

X is Y

Hence we can find a recursive function f such that Vm(m EX ~ f (m) E OY). OY write lilY ~ Ij r Y to mean that there exists an order isomorphism l l. Suc h an i soY 1.} onto a proper 1n . it iaI segment a f {e: e
set). For

m E (Z)i)

Y.

Conversely, suppose that OY is complete l in

that

~

i,j

morphism is called a

comparison~.

Under the given hypothesis on

Y,

we can

ACA that either lilY ~ IjlY or IjlY ~ lilY since the appropriate O comparison map is recursive in HY We claim that there exists e E OY such that i' Y If(m) ,Y ~ lel for all m E X. If not, then for all i we have that i E OY Y). i f and only i f i OY and 3m(mEX & lilY ~ If(m)I Hence OY is Zl in Y,

prove in

+

contradicting the fact that

OY

is complete

rri

in

Y.

1

This proves the claim.

262

S.G. SIMPSON

We now see that for all

IelY

m. m E X i f and only i f If (m)1 Y:: Y• R Thus X is arithmetical in

map Which is recursive in X is hyperarithmetic in

Y.

e

via a comparison

Y. R

It follows that

e

This completes the proof of Lemma 4.1.

l{e now proceed to show that the proof of a result of Gandy. Kreisel. and Tait

[14] can be pushed through in X E Y to mean that 4.2

~.

such that

3i(X

=

ATR

(Y)i)

For sets of integers X and O' where (Y)i = {m (m.i) Y}.

The following is provable in

A is not hyperarithmetic in

no free set variables other than

X and

Y. Y.

ATR

O'

!&.t

~ ~(X.Y) If

A and

Y be sets

~i

be a

3~(X,Y)

Y write

then

formula with 3X(~(X.Y)

& A ; X). Proof.

Let

1 l:l-AC

O

axiom of choice. i.e.

be the Vi3X8(i.X)

~

3YVi8(i'(Y)i)

for arithmetical 8. We shall make use of the result of Friedman [7], [11] that 1 proves ~.-ACO' O

ATR

If

X.Y E Z and if

arithmetic. write ~(X.Y).

i.e.

Feo ~ (X. Y)

Z

to mean that

Z encodes a countable co-mode I of

is true when the bound set variables in it are interpreted

~(X.Y)

as ranging over

is a formula in the language of second order

~(X.Y)

{(Z)i

i E eo}. A formula is said to be essentially

~i

if it

is in the smallest class of formulas containing the arithmetical formulas and closed under existential set quantification and universal number quantification. 1 In view of ~l-ACO it is easy to see that ATR proves the following instance O of an w-model reflection principle: for essentially formulas ~(X.Y). if

Li

~(X.Y)

is true then

32(2 I~ ACA

O

+ ~(X.Y»).

With these observations in mind. we now proceed to the proof of Lemma 4.2. Let eo

f.g.h •.•• into

eo.

be function variables intended to range over unary functions from

We assume that such variables have been introduced into the language

of second order arithmetic in the usual way.

We write

(f)i(m)

=

f«m,i))

and

and the Kleene normal form theorem f[n] = (f(O) .... •f(n-l). In view of for ~l formulas. it will suffice to prove the following assertion in ATR O: 1 ~

g

be a function which is not hyperarithmetic in

Y.

Let

8(t.Y)

arithmetical formula with no free set (or function) variables other than 3fVn8(f(n].Y)

then

3f(Vn8(f[nJ.Y)

be an

Y.

If

& Vi g ¥ (f)i)' 1

Assume the hypotheses. The following true statements are essentially ~l: Y Y Ve(eEO ~ R exists); g is not hyperarithmetic in Y; 3fVn8(f[nJ.Y). We can e

therefore find 2 such that sequence t is good if

2

Feo

ACA

O

+ these statements.

Say that a finite

Set theoretic aspects of ATR o

Z

FOl

Clearly the empty sequence is good. we can find a good m and

n

t

~

s

such that

we would have that

& (t)i(m)

n).

follow that

Hence

Fol

Z

g

Z

s f[th(t»)

:3f(Vn8(f[n) ,Y)

Fol

= t).

We claim that if

s

is good then for all

g[th«t)i») # (t)i'

g (m)

n

is

iiI 1

g

263

i f and only i f

i

If not, then for all Z

Fol :3

good

(t ~ s

t

-

in

Y.

Hence by Lemma 4.1 it would

is hyperarithmetic in

Y.

This contradiction proves the

claim. Now standing outside

Z and applying the claim repeatedly, we can find good

sequences

to ~ t l ~ ••. ~ t i ~ ••• so that for all i, g[th«ti)i») # (ti)i' (The sequence of good sequences to,t ••• is recursive in the satisfaction set l, for the Ol-model Z.) Putting f = U t we get Vn8(f[n),Y) and Vi g # (f)i' i Eol i This completes the proof of Lemma 4.2. Write XfllY 4.3

{2n

nEX}

U {2n+l

Lemma. The following is provable in

such that

A is not hyperarithmetic in

no free set variables other than not hyperarithmetic in Proof.

Let

Y.

X and

Y.

Let A and Y ~ O' (j) (X,Y) be a Zl formula with 1 :3X(j)(X,Y) then :3X[(j) (X,Y) & A

ATR

Let If

X $ Y).

\jt(X,Y,Z)

be. a

formula saying that

X,Y E Z and exists).

Since the statement in question is essentially

Ai. Z and, for the given Y, \jt(X,Y,Z)

that

arithmetic in metic in 4.4

X

X fll Y

@

Y.

Lemma.

-+

W E Z).

Thus we have

The following is provable in

Vi[(A)i

hyperarithmetic in Proof.

(j)(X,Y)

and

A is not hyperarith-

This proves Lemma 4.3.

with no free set variables other than such that

1 Zl' we can find X and Z such holds. Then clearly VW(W hyper-

X and

is not hyperarithmetic in X

$

ATR O'

Y.

Y),

If

Let (j)(X,Y) be a :3X(j) (X, Y)

then

formula

A is

:3X[(j)(X,Y} & Vi[(A)i

not

Y)).

Straightforward generalization of the proofs of Lemmas 4.2 and 4.3.

Note that one cannot strengthen the conclusion to say that for all exists

zi

~

X such that

hyperarithmetic in

(j)(X,Y) and Y).

Vi[(A)i

hyperarithmetic in

X

A there $

Y

-+

(A)i

S.G. SIMPSON

264

is a all

and let N be a submodel of M. We say that N Let M be a model of ATR O Zl elementary submodel of M i f N has the same integers as M and, for 1 Zl formulas with parameters from N, N satisfies ~ if and only if M 1

satisfies

Note that any such

~.

axiomatized by TI 12

4.5

N is again a model of

be a countable model of

II

Let

~.

such that M F \li[(A\

is not hyperarithmetic in

elementary submodel

such that

Proof.

N

an enumeration of the integers of theorem.

YEN

and

~(e,X,Y) be a universal Z~

Let

ATR

sentences.

M.

Fix

and let

ATRO'

Y] .

\Ii (A).

1.

Then

is

A. Y E M be Zl 1

i N.

formula and let

O

{en: nEw}

be

A,Y E M as in the hypothesis of the

Use Lemma 4.4 repeatedly to define a sequence of sets

XO,X ... , l' is not hyperarithmetic

Corollary.

n {N

Let

Proef.

M be a countable model of Zl

: N is a

{A : MFA

4.7

ATR

M has a

X .. E M such that X = Y and for all n , M F \Ii [(A) i n" o in X fll ... fll X and i f M F 3X ~(en'X,XO (f) '" fll X then n) o n], X. Let N be the submodel of II consisting of {X nEw}. n: M satisfies the desired conclusions.

4.6

since

O'

1

elementary submodel of

ATR

is such an n It is clear that

Then

M}

is hyperarithmetic}.

Immediate from the previous theorem.

Remark.

An w-model

M is said to be a

~-model i f

tary submodel of the w-model consisting of all subsets of Corollary 4.6 is that if

n

O'

X .H

M is a

~-model

{N

N is a

{A

A is hyperarithmetic}.

~.

M is a

Zl

1

elemen-

A special case of

then

elementary submodel of

M}

This special case had been proved earlier by Simpson [25].

4.8

Corollary.

elementary submodel Proof. that

ATR

O

Let

N

M be a countable model of

such that

N~

M.

Immediate from the previous corollary in view of the well known fact proves

3A (A

is not hyperarithmetic).

Set theoretic aspects of ATR o 4.9

Remark.

265

A consequence of Corollary 4.8 is that every w-model of

ATR

O O proved earlier by Quinsey (Chapter 6 of [24]) using a completely different method. ATR '

has a proper oo-submodel which is again a model of

Actually Quinsey proved the following generalization.

This result had been

Let

T

~

ATR

O

be a recur-

sively axiomatizable theory in the language of second order arithmetic. oo-model of

T has a proper w-submodel which is again a model of

~i

models produced by Quinsey [24] are not in general

T.

Then any

The w-sub-

elementary.

We now use the results of §3 to reformulate Theorem 4.5 in set theoretic terms.

~. Let M = (IMI,EM) be a countable model of ATR~. ~ A

4.10 and

Y be sets of integers in

metic in

Y].

Then

YEN, \/i(A)i Proof.

that since

N,

to

S

N

•

such that

M

F\/i

E to] (A) i

is not hyperarithN) N=(INI,E such that

M has an E-transitive submodel and

N is again a model of

s ATR '

O

In view of Theorem 3.3 we may assume that

M

F

\/x

(x

is countable).

N is a ~l

1

M

is suitable if and only if check that

H

FT

is suitable.

N is an E-transitive submodel of

Using this remark it is easy to

M and is canonically isomorphic

By the proof of Lemma 3.5 it follows that

N

F

ATR~.

The next corollary answers a question of McAloon and Ressayre [23]. 4.11

Corollary.

Let

M

proper E-transitive submodel Proof.

be a countable model of

s

ATR '

Then O s N which is again a model of ATR ' O

M has a

Immediate from Theorem 4.10.

The next corollary is anticipated by Friedman [8].

4.12.

Corollary.

Proof.

Let

countable sets. founded model CK 001 •

ATR~

has a well founded model of height

CK 001 •

M be the well founded model consisting of all hereditarily

Apply Theorem 4.10 with Y = ¢, A = Kleene's N of ATRs which does not contain O. Hence

o

O. N

We get a well has height

S.G. SIMPSON

266

4.13

The well known fact that

~.

L

is provable in

s

Therefore, in view of Corollary 4.6, it is natural to conO' jecture that for every countable M F ATR~.

n

ATR

is a hyperarithmetic suitable tree}

={ITI:T

CK till

{N : N is an E-transitive submodel of

M and

N 1= ATR~}

=

{x : M

F

x EL

We have been unable to prove this conjecture, even in the special case when well-founded of height §5. ZF,

Comparison with

namely

we take

KP

KP. ATR~

to another, much better known, fragment of

Kripke/Platek set theory. which we take to include the axiom of

=

For background material on KP

M is

til~K

In this section we compare infinity.

CrJ·

wI

to consist of axioms

1

KP

see Barwise [I], [2].

through

4

For our purposes

(see §2 above) plus the

~O

collection scheme Vu3v~(u,v) ~ Vx3yVuEx3vEy~(u,v)

where

~

is

plus the foundation scheme

~O'

Vu(VvEut(v)

~

t(u))

~

Vut(u)

t is arbitrary. A ~O formula is by definition a set theoretic formula in which all quantifiers are bounded, t .e , of the form VuEx or 3aEx.

where

Let

KP~

be

foundation for

KP

minus the foundation scheme.

formulas, and

~O

viz. the admissible sets [1], [2].

KP~

Of course

implies

KP~

has the same well founded models as

will playa role in our work, so we shall insist on the distinction between and

KP.

We shall also consider intermediate systems such as

tion (respectively

KP;

stricted to formulas

t

+ TI k

KP,

However, models which are not well founded KP~

+ L

KP~

founda-

k foundation) in which the foundation scheme is re-

which are L (respectively TI A set theoretic k k). L (respectively TI if it' consists of a string of k k k) quantifiers beginning with an existential (respectively universal) one, followed formula is said to be

by a

~O

formula.

We begin by pointing out that the'l"e exist well founded models of are not models of

KP~, and vice versa. nal not recursive in X.

5.1

~.

In particular, if

s

-

For

X 5:. til

let

X:

ATRs

0

which

till be the least ordi-

plus KP together prove that til~ exists for all X 5:. til. O O M is a well founded mOdel of ATR s + KP~, then X E M o

ATR

Set theoretic aspects of ATR o X

implies

' \ E M.

ATRs + KPGiven O' O enumeration of all binary relations ~ on co Proof.

and

-<

267

xc

We reason in

is a linear ordering of its field.

yew

such that

nesses

WO(-
For all

{-< mX :

let

co

such that

-<

mEw}

be an

is recursive in

m E w,

X

either there exists

Y witnesses "-WO (-
Hence

exists since it is just

wI X

woH m) ) } . Let

fEy & 3m E w(f witnesses

{range(f)

This completes the proof. wCK = w

the least non recursive ordinal. It is well known that CK has well founded models of height (Indeed, L is the smallest well "\ wCK 1

1

1

KP~.)

founded model of

We have also seen (in Corollary 4.12 above) that

has well founded models of height

wCK 1 •

ATR~

The next theorem was anticipated by

Simpson [25].

5.2

~.

model of KP~. s ATR O'

ATR

Any well founded model of

Any well founded model of

KP~

s

o

of height

of height

OK

wI

CK is not a wI is not a model of

of

Proof.

Immediate from the previous lemma.

Next we present a model theoretic argument showing that than

is stronger

KP~.

5.3

founda----

proves the existence of an w-model of

~.

tion. Proof. ACA

Reasoning in ATR let M and M ~ M be countable w-models of O l O O' as constructed in the proof of Corollary 4.2 (ii) of [27]. Form an

O w-structure for the set theoretic language as follows.

variables as ranging over trees

T EM l pret set theoretic equality = as _ in (See the proof of Lemma 3.5 above.) tion gives a model of 5.4

~.

retic viewpoint.

KP~

+ TIl

Interpret set theoretic

M ~ T is suitable. InterO and interpret as in MI'

such that

E

M l, It is not hard to see that this interpreta-

foundation.

KP and related systems have been studied from a proof theoIt is known from Howard [15], [16] and Jager [20] that

proves the same arithmetical sentences as

TIl - TI 00

0

COO U

TIl-TI) kEw k 0

or

KP equiva-

268

S.G. SIMPSON

ID [5] or equivalently ACA + parameterless ni-CAo' I O plus comprehension for ni formulas with no free set variables.

lently Feferman's system i.e.

ACA

O

These results can also be proved model theoretically (cf. Friedman [8]). case, it follows that the proof theoretic ordinal of

In any

KP is the Howard ordinal

°.

88 + Let KP- be KP~ plus full induction on the natural numbers. It is 2 1 known from Friedman (unpublished, but see [7], [9], and footnote 8 of [6]) and Jager [19] and Can t LnI [3] that the proof theoretic ordinal of

KP + TIl founda[13] Theorem 3.6 above, other hand, by §4 of together with 880°· On the s we know that the proof theoretic ordinal of ATR is f = 820. Thus it emerges O s ° is intermediate in strength between KP and KP + TIl foundation. that ATR O tion is

In the rest of this section we study what many would consider the canonical s KP; into ATR It is well known that the smallest O' s well founded model of KP~ is we cannot prove that L CK exL CK' In ATR or obvious interpretation of

O

"'1 "'1 ists as a set (see Corollary 4.12 above) but we can interpret the formula

x E L CK

as an abbreviation for

"'1

We then have: 5.5

(L CK F KP~). In other words, "'1 relativized to the transitive class L CK,

~. ATR~

axioms of

KP;

1-

P2:,oves the

"'1

Proof.

It is well known and easy to see that

LCK={IT!:T

is a hyper-

"'1

arithmetic suitable tree}.

s O' bounding principle

The usual proof of this fact goes through in

collection for L CK Then 6 0 '" 1 which is provable in ATR O

reduces to the well known

Zl

1

ATR

In a similar vein we have: 5.6

~.

(i) (ii) (iii)

ATRs + a ATR

s 0

(L CK

F

KP~ + Zl foundation).

+ TIl-TI O ~ (L CK "'1

F

KP; + TIl foundation).

F

KP) .

Proof.

In

t-

"'1

1

ATR s + 0

metic sets is

!~i -TI O

TI~-TIo f-

ATR~

TIi

(L CK "'1

one can prove as usual [2] that a property of hyperarith-

if and only if it is

Zl

over

L CK. "'1

This gives (i) and (ii),

Set theoretic aspects of ATR o

269

and (iii) is also clear. 5.7 KP~

Corollary.

KP~

does not prove

6 1

does not prove

w(O)&VkE~(w(k) ~

for

6

or

1

IT

l

set theoretic formulas

~

IT

foundation.

l

W(k+l»

In fact,

~ VkE~(k)

w.

It follows from the proof of Corollary 2.10 of [271 that

Proof.

duction on the natural numbers. induction, interpreted in

As in Theorem 5.6 these failures of

L CK, ~l

become failures of

6

and

1

ATR

IT~ and

consistent with the failure of some parameterless instances of

IT

O

is

6i

in-

IT~ and

foundation

l

respectively.

5.8

Corollarx.

a model of 1

6

Proof.

--

1

There is an w-model of

KP- + IT

foundation.

l

By the proof of Corollary 4.3 of [271 let

foundation which is not

M

be an

of

~-model

1

6 - TI in which there is a failure of some parameterless instance of ITl-TI As l O O S in the proof of the previous Corollary, the L CK of M satisfies IT foundal ~l foundation. tion but not

We finish with some inconclusive remarks concerning the formalization of

An occasionally useful theorem of KP is the 6 re1 The usual proof of 6 recursion is formalizable in 1 foundation but apparently not in KP~ + IT foundation. It would be in-

ordinal recursion theory.

cursion theorem ([1],[2]). KP~ + 6

1 l teresting to know how much of the foundation scheme is needed to carry out the

metarecursive priority arguments of Kreisel/Sacks [18] and Driscoll [4].

(For

that matter, how much ordinary induction is needed for ordinary priority arguments on

w?)

The most usual form of a metarecursive priority construction is that one G CK A = U{A : a<~l} where the binary re-

defines a metarecursively enumerable set lation KP;. field.

G

~ EA

is explicitly primitive recursive.

One then proves by induction on

nEw

This can be carried out in

that the

nth

requirement is satis-

This step seems to require instances of induction up to

Corollary 5.7 are not provable in

KP~.

n

which by

It may be necessary to use a nonrecursive

indexing of requirements by integers less than some fixed (nonstandard) integer n.

S.G. SIMPSON

270

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J. Barwise, Infinitary logic and admissible sets, J. Symb. Logic 34 (1969) pp. 226-252.

[2]

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