Generalized Recursion Theory

GENERALIZED RECURSION THEORY PROCEEDINGS OF THE 1972 OSLO SYMPOSIUM Edited by J . E . FENSTAD University of Oslo and ...

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GENERALIZED RECURSION THEORY PROCEEDINGS OF THE 1972 OSLO SYMPOSIUM

Edited by

J . E . FENSTAD University of Oslo and

P. G. H I N M A N University of Oslo and University of Michigan, Ann Arbor

1974

N O R T H - H O L L A N D PUBLISHING C O M P A N Y

-

AMSTERDAM

LONDON

AMERICAN ELSEVIER PUBLISHING C O M P A N Y , INC. - NEW Y O R K

0 NORTH-HOLLAND PUBLISHING COMPANY

-

1974

All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the Copyright owner.

Libraty of Congress Catalog Card Number 73-81531 North-Holland ISBN for the series 0 7204 22000 for this volume 0 7204 22760 American Elsevier ISBN 0 444 10545 X

Published by North-Holland Publishing Company

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Amsterdam

North-Holland Publishing Company, Ltd.

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London

Sole distributors for the U.S.A. and Canada American Elsevier Publishing Company, Inc.

52 Vanderbilt Avenue

New York, N.Y., 10017

PRINTED IN THE NETHERLANDS

PREFACE The Symposium on Generalized Recursion Theory was held at the University of Oslo, June 12- 16, 1972. The Symposium received generous financial support from the Norwegian Research Council and from the University of Oslo. About 50 persons attended the meeting. This volume contains 12 of the papers presented at the meeting. Of the five remaining papers the contribution of Y.N. Moschovakis replaces the one originally presented, which will be published by North-Holland in 1973 under the title Elementary Induction on Abstract Structures. The paper “Post’s problem for admissible sets” by S. Simpson is a later addition. The Editors asked K. Devlin to write a survey paper on the Jensen theory of the fine structure of the constructible hierarchy. The two remaining papers, by S. Aanderaa and L. Harrington, solve important problems left open at the end of the Symposium, and we are happy to include these papers in the Proceedings. We should finally note that the authors have been free to revise their papers after the Symposium, which in some cases has led to extensions of the results as originally reported. We hope that the inclusion of a bibliography of papers on generalized recursion theory will increase the usefulness of the present volume. The participants of the Symposium agreed that a bibliography of the field would be useful, and the preparation of it was taken over by Gerald Sacks, who received extensive assistance from Leo Harrington. The Editors are grateful to them for their valuable work. The reader will note that the bibliography carries the disclaimer “uncritical”. This is to emphasize that the purpose was not to present a comprehensive and scholarly bibliography of works relevant to generalized recursion theory, but to provide a useful list of some of the basic papers. The Symposium was intended to present a broad view of methods and results in generalized recursion theory. We believe that the meeting acheved some measure of success toward this goal so that the published Proceedings also can serve as an introduction for the beginning research student who wants to specialize in this rich and fascinating branch of logic. The Editors V

PART I RECURSION IN OBJECTS OF FINITE TYPE

J E.Fenstad, I? G.Hinman (eds), Generalized Recursion Theory @ North-Holland PubL CornD., 1974

RECURSION IN THE SUPERJUMP Peter ACZEL University of Manchester

and Peter G. HINMAN University of Oslo and University of Michigan

The ordinary jump operator of recursion theory is a function OJ : w w + w w defined by 0, if { m ) ( a ) $ ;

oJ(a)(m)= 1, otherwise

.

By treating OJ as a (type-2) function: w w X w + w and coding the two arguments into one, we obtain from the schemata of [Kl] a notion of recursion relative to oJ. It is well-known that OJ is of the same degree as 2E, that a set A of natural numbers is recursive in oJ just in case it is hyperarithmetic, and that wyJ, the least ordinal not recursive in oJ, is just wl, the least non-recursive ordinal and the second admissible ordinal. The same procedure can, of course, be applied to any function Gump) J . For example, W defined by 0, if { m } ( a , o J ) $ ; 1, otherwise,

is recursively equivalent to the hyperiump and of the same degree as E l . The sets of natural numbers recursive in hJ are the recursive analogue of the Csets of descriptive set theory [Hi], and w t J is the least recursively inaccessible ordinal. 3

4

P. ACZEL and P.G. HINMAN

The superjump S as defined in [Gal is a type-3 jump defined by

In particular, S(oJ) = h J . By coding arguments as above we may consider S as a function: ( W o ) w + w . T h u s from [Kl] we have a notion of recursion relative to S. In this paper we study some properties of this notion. In $5 1 and 2 we discuss a hierarchy of jump operators, due to Platek, obtained by iteratingS over a set of ordinal notations. $3 contains some results concerning the size of as,the least ordinal not recursive in S. In $ 4 we extend Platek's hierarchy to one with the property that a set of natural numbers is recursive in S iff it is recursive in some jump operator occurring in the hierarchy. Finally, in § 5 we discuss several other type-3 functionals which are in some sense equivalent to s. In ## 1 and 2 we assume familiarity with [Pl] and conform for the most part to his notation. The rest of the paper does not have this prerequisite, but we recommend to the reader the clear general discussion of [PI, 257-2631 as background.

§ 1. Platek's hierarchy Modern mathematics must be considered more an art-form than a science, but it is perhaps a harder master than most of the arts: mathematics must not only be beautiful, it must also be correct. Sad to say, even the beautiful can be false and such an occurrance is the starting point of this paper. In [PI], Platek constructs a hierarchy of jumpsJf indexed by elementsa of a set 0' of ordinal notations. Jf = oJ, J ; = h J , and, roughly speaking, the hierarchy is obtained by iterating S over the set of notations. The construction is closely parallel to that of Kleene's setsH, for a E 6 , a set of notations for recursive ordinals, and even more closely parallel to that of Shoenfield's sets H c for a E O F , a set of notations for the ordinals recursive in the type-2 function F [Sh] . Since a set A of numbers is recursive in oJ iff it is recursive in some Ha (a E S ) ,and, for any F in which oJ is recursive, A is recursive in F iff it is recursive in s o m e H r (a E a"), it would be elegant and satisfying if also A were recursive in S iff it is recursive in some J: (a E 0'). The main theorem

RECURSION IN THE SUPERJUMP

5

of [HIassets that this is true - unfortunately it is not. Before describing the counterexample to Platek’s theorem, we point out where his argument breaks down. In the sentences beginning at the bottom of p. 265 of [Pl] and continuing at the top of p. 266 he applies the boundedness lemma to a function @(@(a) = so that for a E 6,’ H{ is recursive in J,$,,) to obtain a d E 0’ such that e a c h H F is recursive i n J i . The function H i s defined by the recursion theorem over 0 and thus @ is partial recursive with domain including O F . The boundedness lemma applies to total functions E 32, so to use it here we would need to find such a $ which agreed with @ on O F . The natural to choose would be

’

+

+

However, there is no apparent way to find aJ-number for (the characteristic function of) 6 ’ and thus show this $ E 92.In fact we shall see that for the functional K below, to which this step of the argument would have to apply, 6 does nor have a J-number and there is no d E 6’ such that all Hf (a E o K , are recursive in J:. Our counterexample consists in defining a jump K with the following three properties: (1) K is recursive in S; ( 2 ) a E as-+~f is recursive in K ; (3) for any A 5 o, if A is recursive in K , then for some a E 6’, A is recursive in J ,S. From (1) follows that S ( K ) is also recursive in S and hence that there existA C o recursive in S but not recursive in K - for example, { a : {a}(K).l}. Hence from (2) we have immediately that there are A recursive in S but in no (a E 0’). Property (3) completes the picture to show that the jumpsJ: (a E Cis) provide a natural hierarchy for the sets recursive in K . Of course, it also follows easily from (2) that O K is not recursive in any Jf (a E 8 ’) and that there is no d E 8 such that all A recursive in K are recursive in J:. In terms of ordinals, the uniqueness theorem for 0 [Pl, p. 2631 ensures that for every u < 16‘1 = sup { la I : a E O’}, u is recursive in some Jf and thus by (2) recursive in K . Hence 101 ’ 5 of. Conversely, if u is an ordinal

Ji

’


6

JS recursive in K , then by (3) there is some a E 0' such that CJ < ola < 101 '. The last inequality may be shown by constructing an order-preserving partial recursive map of oJlinto 0' and obtaining an upperbound by use ofJ&. Thus w f = 1 0 ' 1 . Of course O K is recursive in S ( K ) so of< < of. We turn now to the definition of K and the proof of properties (1) -(3). K is based on the same idea as the jump T of [Pl, Th. VI] .

Definition 1.1. (a) For any y : w X o + 2 we write p _< ,4 if y ( p , q ) = 0, let Fld(y) = (r, : p I , p } and l e t p < ,q i f p _< ,q andp f q ; (b) W i s the set of y : o X w -+ 2 such that < well-orders Fld (7); (c) for y €94,ll yll is the order type of < ,; (d) for any y andr Eo,

,

Note that for y E W a n d r E o,y r r E W a n d IIy' I rll = 0 if r B Fld(y) while llr r rli < llyll i f r E Fld(y).

Definition 1.2. (a) for y € W a n d llyll= 0, K , =oJ; (b) for y E W a n d llyll> 0, K , = Xa .S(K,r,(o))(a+), where o;'(m) = a(m + 1); K,(a), if 7"; = (c) K ( 7 , Am 0. otherwise.

-

Theorem 1.3. K is recursive in S.


I

then G is partial recursive in S. By the recursion theorem there exists an F such that G(Z, m , y,a) = {Z}’(m, y, a). If F = {F}S, we claim that for all a, y, and m :

Note that the only properties of S used in the preceding theorem are that oJ and hJ are recursive in S. The next lemma records the other (very general) properties of S that are needed in this section. Thus our methods apply in many other situations. For example, they provide an alternative proof to the result of [Mo, 8 Ilthat Kleene’s proposed hierarchy for 3E fails to exhaust even the sets of numbers recursive in 3E.

Lemma 1.4.

(a) There exists a primitive rekursive $J such that for all J and a,

44 = #J(G(J));

(b) there exists a primitive recursive f such that for any e, J, and J ’ , i f J is recursive in J’ with index e, then S(J) is recursive in S ( J ‘ )with index f (e).

Proof. (a) is proved in two lines just as the corresponding fact one type down with OJ in place of S. For (b), suppose that for all 0 and q

It suffices to show that there is a primitive recursive g such that for any a, m = ( m o , ..., mkPl),and a € w w ,

as then

g is defined by the recursion theorem and by cases depending on the index a.

8


The only difficult case is when

where b and q are coded into a. Here we define g(a, e) to be the “natural” index such that

It is straightforward to show for this case that { a } (m, a ,J ) = n

+

{g(a,e)}(m, a, J ’ ) N n .

Conversely, if { g ( a , e ) } ( m , a , J ’ = ) n , then by virtue of the first term in the definition, Xp { g ( b , e ) } ( p , m , a , J ’ )is total and a l l computations of its values are subcomputations of the computation of {g(a,e)}( m, a ,J ’ ) . Hence, the induction hypothesis guarantees that h p * { b }( p , m,a, J)is the same total function and thus that ( a } (m, a, J)is defined with value n . a, J) we take g(a,e ) to be the In the case {a}(b,m,a, J ) = { b }(m, ‘‘natural’’ index c , computed from an index f o r g , such that

-

Corollary 1.5. There exist primitive recursive g, and h such that for any y €9’ and any p , q : (a) If llyll> 0 then S(KYlp)is recursive in K , with index f(p); (b) p < ,q + KrrP is recursive in KYr4 with index g ( p , q ) ; (c) ifllyll= Ily rpll + 1, then K7 is recursive in S(KYrp)and y with index h(p).

r

Proof. (a) is immediate from the definition. When p < ,q, y r p = (y r q ) p so(b)followsfrom(a)and 1.4(a).SupposeIIyll=Ilyrpll + l . I f r < , p , t h e n KYr, may be computed from KrrP by (b). If r Q: , p , then either r = p , so that KYr, is trivially recursive in Krrp, or IIy rrll = 0 and K,,, = OJ so again is recursive in KYrp (with index computable from 7). Hence using y we may compute eachKYrruniformly from KrrP. By 1.4(b) the same is true ofS(Kyrr)

9


and S ( K T r p )and from the definition of K , it is clear that this is sufficient to establish (c). 0 To obtain result (2) it is much more convenient to work with a hierarchy slightly different from that of [Pl] obtained by introducing an ordering relation < and at limit stages 3' 5 e requiring that Am { e }( m ,J,") ascend in the ordering < s. The construction is entirely parallel to that of [Sh] . Let bs and7: denote the set of notations and jumps thus obtained. The new system is a subsystem of Platek's and it is an easy exercise to prove.

-

Lemma 1.6. There exist partial recursive f and g such that for all a E 6', f ( a ) E 6' and Jf is recursive in Tf(a)with index g(a). For each a E

us,let yo be defined by

r

Then y, € W a n d llyall = lal. Note that if b < S a , then yo b = yb.

Lemma 1 -7. There exist partial recursive f and g such that fur all a E 0'. (a) y is recursive in 7: with index f ( a ) ; (b) :7 is recursive in K , with index g(a). Proof. (a) is proved by a straightforward induction over 6 '. At limit stages one uses the fact that oJ is recursive in 7,". For (b) we define g by the recursion theorem over 6s as follows. For a = 1,7,"= KTa = oJ. If a = 2b, the induction hypothesis yields that J i is recursive in K Y b = KrQrbwith index g(b). Then by using 1.5(a) and 1.4(b) we may compute an indexg(a) o f T f = S ( y i ) from K,,. If a = 3 5 e , let @(m)= { e ) ( m , T f ) .Since b < S a we can find as in the previous case an index of 7: and hence of @ from KYQrband then by l.S(a) from K,,. Similarly, for each m we can compute an index ofTi(,) from KY,. Putting these together, we obtain an index g(a) for 7; from K,,. 0

-s ,J, -s is reTheorem 1.8. There is a partial recursive h such that for all a E 6 cursive in K with index h(a).

10


a'.

Proof. We again define h by the recursion theorem over For a = 1,Yf = oJ = Aa K(hm l , a ) , so it is clear how to pick h( 1). F o r a = 2b, using h ( b ) and (a) of the previous lemma we can compute an index for yb from K . By 1.7(b) we have an index of 7; from K and yo. But y, is obviously recursive in yb so we have one from K alone. For a = 3b * 5 e , let @(m)= { e } ( r n , T f ) . Using h ( b ) we can find an index of @ from K and using h ( @ ( m ) )we can find an index of Ti(m)from K . Hence we can find an index of 3; from K . 0

-

Result (2) now follows immediately from 1.6 and 1.8. Lemma 1.9. There exist primitive recursive f and g such that for any d E 0' and any y Ecklrecursive in with index e, f ( d , e ) E 6' and K , is recursive in Jf(d,,) with index g ( d , e ) .

Ji

Proof. Let d , e , and y be as described. We first define functions @ and $ recursive in with indices from depending only on e , such that for all p , @ ( p )E 6' and K r r P is recursive in J&, wjth index $ ( p ) . Once this is done we can compute from the index of @ from by the boundedness lemma a c E O S s u c h t h a t l d l < l c l a n d V p - I@(p)I 0. If not, set @ ( p )= d (a trick useful below) and $ ( p ) any index of K r r p = oJ from@. If so, we assume as induction hypothesis that f o r q a n d r such that IIy f q l l < IIy r rll< IIy rpll, @ ( q )and @(r)are defined and 1@(4)1< I@(r)l.If p has an immediate predecessor q in the ordering w, any b, and any indexed set of jumps { J , : n E w } such that for all n , J , is K -effective with index {b},(n), if J = ham * J(,)o(a) ( ( m ) l ) ,then J is K-effective with index f2(b).If b < w, then also f2(b) < w. K


18

Proof. If {a},

r w E w w , then

for an appropriate c 2 . Thus it suffices to set f 2 ( b )= Sb,(c,,b).

0

Proof of Theorem 2.7. (a) follows immediately from Lemmas 2.5 and 2.7. For (b), let = fj w . Then and f 2 are (ordinary) primitive recursive. By the (ordinary) recursion theorem there exists a primitive recursive function f with index f such that:

3. r

fi

where h is a primitive recursive function such that for any admissible

K

withg as in Lemma 2.8. It is now straightforward to prove by induction on 6" that for a E O S ,if K is I a I -recursively inaccessible, then Jf is K-effective with index f ( a ) . 0

$3.S and the first recursively Mahlo ordinal In this section we extend the ideas of $ 2 to obtain a bound for w f , the least ordinal not recursive in S. Definition 3.1. (i) For any ordinal K , K isrecursively Mahlo iff K is admissible and for any K -recursive function f from K to K , there is an admissible h < K which is closed under f; (ii) po is the least recursively Mahlo ordinal.


19

It is easy to see that po is po-recursively inaccessible and is not the first such ordinal. Hence 16’1 < p,. We shall show here that wf < p o . It has recently been shown by Leo Harrington that wf = p o ; his proof appears in his contribution to this volume. That of= po appeared as a theorem in [Ac], but its purported proof there depended on the fallacious results of [PI].

Definition 3.2. For any K > w and any e < K , S is K -effective with index e iff is admissible and for any J and d such that J is K-effective with index d , S ( J ) is K-effective with index S b l ( e , d ) .S is K-effective iff S is K-effective with some index. K

The next lemma is analogous to a weak form of Lemma 2.8.

Lemma 3.3. There exists a primitive ordinal recursive function g such that for any K and e such that S is K -effective with index e, any a < w and m = mo, ..., mk-l < w,any c = c o , ..., cI-l < K a n d a = a,, ...,alp, E such that ai= { c ; } ~ 1 w for i l,(m,c>= n. Proof. This is similar to that of Lemma 2.8 and we treat only the case

-

, and0 = hr { b l } ( r , m , a , S ) ,with b,, b , , where J = h y q { b o } ( q , m , a , yS) and p coded into a . Given that { a } ( m , a ,S) = n , it follows that J and 0 are both total objects with all of their computations “preceding” the given one. By the induction hypothesis we can compute from a , m and c an index d , such that J is K-effective with index d , and b , such that 0 = {b2IKr w . Then

Hence it suffices to choose g(a,e) such that

Corollary 3.4. For any

K,

ifS is K -effective, then wf

5K.


20

Proof. If S is K-effective, then by the preceding lemma any well-ordering of o recursive in S is also K-recursive, hence of order-type < K by Lemma 2.5. 0 Lemma 3.5. For any J, e, and K , if^ is recursively Mahlo and K E E f , , , then i s also a limit of ordinals h E E f J,e.

K

Proof. Suppose K is recursively Mahlo, K E Ef,,, and let Ef, be as in the proof of Lemma 2.10. Then for any admissible h with o,e < h < K , h E Ef J,e ++ h E Ef,. With the help of the uniform normal form theorem (A) we can define a primitive ordinal recursive relation R such that for such A,

Now let h, be any ordinal

If(xj)I 2 [xi[.So 13m * S e l 2 13n1*5e1(.

It is now possible to show that 172 1 is F-recursively Mahlo. The proof will only be sketched as it is precisely the same as Richter’s argument in [Ri]. In fact the 9 2 hierarchy (for F trivial) is essentially just Richter’s system of notations for po with the onerous 3(x7y) clause eliminated. Our first job is to show that ;% 1 and the %-admissible ordinals are in fact F-admissible. This should be clear for 72 -admissible ordinals which are not %-inaccessible - the first 92-admissible greater than In I is just wFHn.So let P be either an %-inaccessible ordinal or 1% 1. We will place a recursion theoretic structure on the hierarchy restricted to 92, as follows. 31

Definition. [el P(x) ~y means e = (eo,e,), {eo}(x) E 9ZP,and { e l } H { e o } ( x ) ( xz) y . n [el %P(x).l means [el %O(x) ‘ y for somey. If [el p(x)J. then I [el %@(x) I = I {eo}(x) I, otherwise I [ eynO(x) t= m. A partial function cp is partial 7Zp-recursive if for some e and all x, cp(x>s [el %~(x) . We now give a series of theorems which together imply that /3 is Fadmissible.

Lemma I . I f cp is a partial 7Zq-recursivefinction, and if the range of cp is a

THE SUPERJUMP AND THE FIRST RECURSIVELY MAHLO ORDINAL

49

subset of %, then there is an e such that for all x in the domain of cp, Icp(x)l I I {eI(X>I 2.

During the preparation of this paper the author was partially supported by a Sloan Foundation Fellowship and by a grant from the National Science Foundation.

53

Y.N.MOSCHOVAKIS

54

1. Preliminaries The set T o of objects of t y p e j over the integers is defined by the induction

A ppint

x = ( X I , ...,x , ) is a tuple of objects of finite type and the type of x is the largestj such that some x i is of type j . A (product) space is a Cartesian product

X = X , X ... xx, where each Xi is some T ( j )and the type of 31 is the largest j such that some Xi is T ( j ) .If x = ( x l , ...,x,) a n d y = (y,, ...,y,), we write ( x , y )= (x,, ...,x,, y l , ...,y,). Similarly, if 31 = X , X ... X X , and y = Yd X ... X Y,, then % X y = X I X ... X X , X Y , X ._.X Y,. A relation of v p e k > 0 is any subset R C 3c of a space of type k - 1. We also call these pointsets of type k and we write interchangeably R(x)oxER. Following Kleene, we let d,pi,y j vary over T ( j ) .It is also customary to reserve the variables n , k , m, ... for naming integers and the unsuperscripted Greek letters a, 8, y, ... for naming objects of type 1, or Peals. We assume that the reader is familiar with at least the basic definitions and facts of Kleene.[ 19591. In particular, it is defined there what it means for a (total) function

to be primitive recursive. This is by means of eight natural schemes and involves no indexing. A function

STRUCTURAL CHARACTERIZATIONS OF CLASSES OF RELATIONS

551

is primitive recursive if there is some primitive recursive

g : T(i)X x - + w such that

f ( x ) = haig(ai,x) . Similarly, f :

x + y =Yr,X...XY,

is primitive recursive, if there exist primitive recursive functions

Y

If 5X and are of type 0 by substitution,

-+

with some recursive partial g : T(m)X X -+ w . Similarly, a relation R is recursive in U if xR is recursive in U , and it is semirecursive in U if there is a partial function f : X -+ w , recursive in U , such that R =Domain (f). It is simple to check that a total functionf : 5X + w is recursive (in U ) if and only if the graph off, Graph (f) = {(x,n) : f(x) = n } is recursive (in U ) as a relation. This is not always true for partial functions, which is why it is not convenient in this context to identify partial functions with their graphs. Since an object F of type k > 0 is a function of type k , it makes sense to ask whether F is recursive in U. This relation is transitive. We agree that the objects of type 0, the integers, are recursive in every object. If F is recursive in G and G is recursive in F , we say that F and G are (recursively) equivalent


57

or that they have the same Kleene degree. It follows from one of Kleene's basic substitution results that if U and V are equivalent, then for every pointsetR,

R is recursive in U

R is recursive in V .

Also, if U and V are equivalent and of the same type, then R is semirecursive in U

R is semirecursive in V .

Following Kleene [ 19631, we define for each object U and each k 2 1, the k-section of U , k x ( U )=

{R : R is a pointset of type 5 k and R is recursive in U ) .

We also define the k-envelope of U as in the introduction, ken ( U ) = {R : R is a pointset of type 5 k

and R is semirecursive in U ) . The object mE ( m 2 2) representing quantification over T(m-2)is defined by mE((p-1) =

0 if (3p"-2)[orz-'(p"-2)

= 01,

1 otherwise.

We call an object U of type m normal i f m E is recursive in U . (All objects of type 0 or 1 are normal.) Almost all reasonable structure results that are known about recursion relative to an object U have been proved on the hypothesis that U is normal, and many are known to fail if U is not normal. Kleene's inductive definition of recursion naturally assigns to each pair e, a such that {e}(a)is d e f i e d an ordinal le, 01, the stage of the induction at which we first recognized that (e, a , n ) E X , for some n. The following theorem is the correct general version of Theorem 6 of Moschovakis I19671 and can be established by a variation of the proof given there: The Stage Comparison Theorem for Kleene Recursion. Let x,y vary over

Y.N. MOSCHOVAKIS

158

the spaces X,y respectively, both of type 5 m ( m 2 2). There is a recursive partial function f ( a m , e , x , z , y ) such that

{ e ) ( x )isdefinedand le,xl< lz,y I * f ( m E , e , x , z , y ) = 0 , {z} ( y )is defined and Iz , y

(Here le,x I

I < Ie ,x I *f (mE,e ,x, z ,y ) = 1.

I Iz,y I is true if { z } ( y )is not defined.)

This result is due to Candy [ 19621 for m = 2 and to Platek [ 19661 for m 2 3, independently of Moschovakis [ 19671. (The proof in Moschovakis [1967] is given only form = 3 and Grilliot [1967] extended it to all m.) It is the basic theorem about recursion relative to normal objects. One of the easy consequences of the Stage Comparison Theorem is the existence of selection operators: for each recursive partial function

f:wXX+w with type (X)5 m (m22) there is a recursive partial function g ( a m , x )such

that ( 3 n ) [ f ( n , x ) i sdefined] --g(mE,x)

is defined,

( 3 n ) [ f ( n , x )is defined] +f(g(mE,x),x)is defined. From this follows immediately that a partial function f : 35 + w with type ( X)5 m is recursive in a normal object U of type m if and only if the graph o f f is semirecursive in U . In these circumstances we often identify a partial function with its graph. A pointclass is any collection of pointsets. We call a pointclass I'of type k , or a k-pointclass, if every pointset in I' is of type 5 k, e.g. if r is a k-section or a k-envelope. A pointclass I' is closed under 3 J if whenever R 5 T ( j )X X is in l?, so is P defined by

P(x)

-

(3ai)R(ai,x).

Closure under V1,& and v is defined similarly. A k-pointclass I' is closed under primitive recursive substitution if it con-


59

tains all primitive recursive pointsets of type 1. Let HC be the set of hereditarily countable sets. The Kondo-Addison uniformization of Il: predicates of reals implies HC satisfies A. dependent choice. Consequently HC is an abstract I-section. I t suffices to show%(Aj) is a El substructure of HC. Let be a A. formula with parameters in%(Af) such that

a(.)

There exists an arithmetic predicate A( Y) such that [ Y i s a c o d e & A(Y)]+-+HCk9(mY).

a(.)

The set parameters occurring in correspond to Af codes occurring in A ( Y ) . The Kondo-Addison uniformization supplies a code Z such that 2 satisfies A( Y) and is A: in the A; codes occurring in A( Y). Since max ( i , j ) > 1, Z €A:. HenceW(Aj) k (Ex)g(x).

4. Generic type 2 objects Let K be a countable abstract. 1-section. Suppose F maps owinto w and is 0 off K . If F is generic in the sense of the following forcing relation, then the 1-section of (F, 2E) is K . Let p be a partial function from ww into w . p generates a hierarchy {T,} of reals as defined below. If p is total, then the T,’s are equivalent to the S,’s of section 2 when n = 0 and F = p . If p is not total, then there may be a CJ such that T,,+l has an index but is not total. Stage 0. To = { l}. 1 is an index for To. To is total. if T , has an index, T, is total, 2e 4 T,, Stage o t l . 2 e is an index for and { e } T u ( m )is defined for all m. ( { e } T o is the unique partial function from o into w recursive in T, via Godel number e.) TO+l is total if it has an index and p ( X m { e } T o ( m ) )is defined whenever 2e is an index for T,,+l. Tu+l is T , augmented by: all indices for Tu+l; all triples ( 3 e , m ,n ) such and all pairs ( 5 e , n )such that that { e } T o ( m )= n and 2 e is an index for Tu+l; p ( X m { e } T ~ ( r n )=) n and 2e is an index for Tu+l.

Im(=CJ means m is an index for T , .

88

G.E. SACKS

Srage h (limits). 7 e is an index for T A if 2e is an index for Ts+l for some 6 < X and hm{e}Ts(m)is the characteristic function of a Set R of indices such that h = sup {Iml(m E R ) .

T, is total if it has an index. Th is

u { T , 16 < A}

augmented by all indices for

TA.

p is said to gerierafe T , if To has an index and is total. Fact H is easily proved by induction on u. Fact H. If u < y and p generates T, and T y , then To has lowei Turing degree than T y . Suppose p is total and S is a real. The arguments of Shoenfield [4] show S is Turing reducible to some T, generated by p if and only if S is recursive in p , 2E in the sense of Kleene [ 2 ] . If Tu+lhas an index but is not total, then Tu+l is said to be the maximum o f p . p is afircitig cotidition if it meets two requirements. ( I ) p E77ZK and has a maximum. (2) X is in the domain of p if and only if X is Turing reducible to T, for some 6 < u, where To+, is the maximum o f p . Requirement (2) is not as limiting as it may appear, because Fact H implies that the generation of To by p utilizes the vaiue of p ( X ) only if X is Turing reducible to T6 for some 6 < u . Froin this point on p , q , r, ... denote forcing conditions. p is extended by q (in symbols p 3 q ) if the graph of p is contained in the graph of q . The language L ( K ) will be used t o define the desired generic F's. The individual constants of L ( K ) are: for each m € w ; f for each f E ww n W K ; _ u and . To for each ordinal u E 9 " K ; a n d 3 for each S E 2w nW K . The variabies o f k ( K )are: x,y,... (numbers); p , v, ... (ordinals); and Tp,T,, ._.(sets). The atomic formulas of L ( K ) are of the form: Ix I = p , Ifx I = p , p < v, and S < T,. The sentences of L ( K ) are built u p from the atomic formulas by substitution of appropriate individual constants for variables and by application of propositional connectives (& and -) and existential number and ordinal quantifiers. 9 is a ranked sentence of rank u if 9 contains n o ordinal quantifiers and u is the least ordinal greater than every ordinal occurring in 9 . Such an 9 is rruein { T 6 i 6 < o ] i f i t is true w h e n 6 < y-i s i n t e r p r e t e d a s 6 < ~ , l-m I =-6 as

m

THE 1-SECTION OF A TYPE n OBJECT

89

m is an index for T,, and S i f p 1 - 9 ( 3 for some u. (iv)plt- 9 & 9 i f p I t 9 a n d p I t - 9 . [ 4 IF 91 and 9 is not ranked. (v) p It-- 9 A sentence 9is said to be Elif it is in prenex normal form and contains no universal quantifiers.

-

Proposition 4.1. The relation p It-9, restricted to El g’s,is E, over%K. Proof. Suppose w € W K and is a partial function from w w into w . The set of To’s generated by w is El over-K uniformly in w. (This last is a consequence of the Z, admissibility of%K and the autonomous fashion in which indices are assigned to T , when h is a limit ordinal.) Thus if w has a maximum, then that maximum belongs t o W K . It follows that the set of forcing conditions is El over3flK. Let F map ww into w and be 0 off K . F satisfiesp (in symbds F Ep) if the graph of p is contained in the graph of F. F is generic if for each sentence 9of the language L(K) there is a p such that F E p and either p IF 9 or p1 1 9 .Generic F’s exist because there are only countably many sentences to be forced. Standard arguments [ 151 show: if F is generic, then each true statement about F (expressible in the language %(K)) is forced by some p satisfied by F.

-

Lemma 4.2. If F is generic, then K C 1sc(F, 2E).

Proof. Suppose S: w + w belongs to K. Fix p ; since F is generic it suffices to find a 4 C p such that

for some u. Since p has a maxirmm there is an e and a y such that p generates Tr, 2e is an index for Tvl and

90

G.E. SACKS

is undefined. Let hn le, be a recursive function such that

{en

= n t {e}*

for all X C w and n E w . Clearly { e n } X is total if and only if { e } X is. It follows 2en is an index for TV1 because 2e is. In addition

c IT%))

p(Xm e,

is undefined, because the domain of p is an initial segment of Turing degrees. Choose 4 C p so that the domain of 4 consists of all functions Turing reducible to T7, and so that

for all n E o.Thus q generates Ty+lbut not Te2, since q(TP1) is undefined.. And S is Turing reducible to TPl since

for all n and r.

Lemma 4.3. If F is generic, then sc(F, 2E) C K . Proof. Suppose S E sc(F, 2E) K for the sake of a reductio ad absurdum. Then S is Turing reducible to some To generated by F but not i n ' X K ; D @ W K since F is generic. Let a be the least ordinal not i n W K . Then F generates some T, with index 7 e . Thus 2e is an index for some Ts+l generated by F in%K, and {e}T6 is the characteristic function of a set R such that ~

a=sup{InJ(nER}. Let f € % K that:

enumerate R . Since F is generic there is a p satisfied by F such

THE I-SECTION OF A TYPE n OBJECT

(a) P It- (X)(EP) t IfX I = PI ; (b) p IF (P)(Ex)(Ev)[P < v &k Ifx - I =~ (c p Il- 8 L T, ; (4 p IF (Ell) [ IZ“ I = PI. (a) is equivalent to

-

91

1 ;

(a*> (m)(4)p>q(Er)q.,(Ea)[rI~ lfml -- = E l > and (b) is equivalent to (b*) (0)(4)p>q(EY>q.,(Em)(Ey), 2 and U is a type n object in which nE is recursive. Then there exists a type 2 object V such that

92

G.E. SACKS

and 2E is recursive in V. Corollary 4.6. Suppose a is a countable El admissible ordinal. Then there exists a type 2 object V such that

and 2E is recursive in V. Corollary 4.7. If min (i,j ) 2 1, then the set of all lightface A; reals is the 1section of some type 2 object in which =Eis recursive.

5. Further results The method of section 4 is applicable to the study of Gandy's superjump [ 7 ] .Theorems 5.1 and 5.2 are typical results of [ 141 and were inspired by some questions raised by P. Hinman at the 1969 Manchester Logic Colloquium. Let F and G be objects of type 2, G' the superjump of G, and El the superjump of 2E. scG is said to be closed under hyperjump if l s c ( E 1 , x ) c 1scG

for every X E scG.

Theorem 5.1. Suppose sc G is closed under hyperjump. Then there exists an F such that ]scC= ,sc(F').

Theorem 5.2. (Assume 2w = w1.) There exists an H such that (C)(EF)[HIG+F'=G]. The method of section 4 does not appear to suffice for the proof of the plus-one theorem when k > 1. A stability result is needed to overcome prob-

THE 1-SECTION OF A TYPE n OBJECT

93

lems caused by gaps in the hierarchy of section 2, gaps that fall between objects recursive in F,n’2E when n > 0. Call R subrecursive in F, n+2Eif R is recursive in some So (as defined in section 2) with an index of the form ( 2 e , r ) ,where r is a subindividual. The stability result in question says: each nonempty recursively enumerable (in F, n+2E)collection of subrecursive (in F,n+2E)sets must have a recursive (in F , n + 2 E ) set among its members. At this writing it is not known if there exists a decent notion of abstract k-section when k > 1. Decency requires that Theorem 4.4 remain true when; “I-section’’ is replaced by “k-section” and “2” by “ k t l ” .

References [ l ] G.E. Sacks, Recursion in objects of finite type, Proceedings of the International Congress of Mathematicians 1 (1970) 251-254. [ 2 ] S.C. Kleene, Recursive functionals and quantifiers of finite type, Trans. Amer. Math. SOC.91 (1959) 1-52; 108 (1963) 106-142. [3] G.E. Sacks, The k-section of a type n object, to appear. [ 4 ] J . Shoenfield, A hierarchy based on a type 2 object, Trans. Amer. Math. SOC.134 (1968) 103-108. [ 5 ] T. Grilliot, Hierarchies based on objects of finite type, Jour. Symb. Log. 34 (1969) 177-182. [ 6 ] G.E. Sacks, Higher Recursion Theory, Springer Verlag, to appear. [ 7 ] R. Gandy, General recursive functionals of finite type and hierarchies of functions, University of Clermont-Ferrand (1962). [ 8 ] R . Platek, Foundations of Recursion Theory, Ph.D. Thesis, Stanford (1966). [ 9 ] C. Spector, Recursive well-orderings, Jour. Symb. Log. 20 (1955) 151-163. [ 101 A. Levy, A hierarchy of formulas in set theory, Memoirs of the American Mathematical Society, Number 57 (1965). [ 111 K. Godel, The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis (Princeton University Press, Princeton, 1966). [ 121 G. Kreisel, Set theoretic problems suggested by the notion of potential totality, in: Infinitistic Methods (Pergamon Press, Oxford, and PWN, Warsaw, 1961) pp. 103140. [13] J . Shoenfield, The problem of predicativity, Essays on the Foundations of Mathematics (Magnes Press, Jerusalem, 1961 and North-Holland, Amsterdam, 1962). [ 141 G.E. Sacks, Inverting the supejump, to appear. [ 151 S. Feferman, Some applications of the notion of forcing and generic sets, Fund. Math. 56 (1965) 325-345. [ 161 Y. Moschovakis, Hyperanalytic predicates, Trans. Amer. Math. SOC.129 (1967) 249-282.

PART I1 SETS AND ORDINALS

J.F.Fenstad, P. G.Hinman (eds.), Generalized Recursion Theory @ North-Holland Publ. Comp., 1974

ADMISSIBLE SETS OVER MODELS OF SET THEORY K. Jon BARWISE University of Wisconsin, Madison and Stanford University

9 1. Introduction The addition of urelements gives a new dimension to the theory of admissible sets, a dimension which has applications in several parts of logic. To see why this addition is an obvious step to take we begin by reviewing the development of Zermelo-Fraenkel set theory, ZF, as it is usually presented (see for example Shoenfield [ 19671, 39.1). The fundamental tenet of set theory is that given a collection of mathematical objects, subcollections are themselves perfectly reasonable mathematical objects, as are collections of these new objects, and so on. Thus we begin with a collectionM of objects called urelements which we think of as being given outright. We construct sets on the collectionM in stages. At each stage a,we have available all urelements and all sets constructed at previous stages. A collection is a set if it is formed at some stage in this construction; the collection of all sets built onM is denoted by V,. Now it turns out that ifwe allow strong enough principles of construction at each stage a,and if we assume that there are enough stages, then the urelements become redundant in that all ordinary mathematical objects occur, up to isomorphism, in V , i.e. in V , for the empty collection M . It is for this reason that the axioms of ZF explicitly rule out the existence of urelements; the combination of the power set and replacement axioms are so strong as t o make urelements unnecessary. So formulated, ZF provides us with an extremely elegant way to organize existing mathematics. It does this at a cost, though. The principle of parsimony, historically of great importance in mathematics, is violated at almost every

'

Research for and preparation of this paper were supported by NSF GP 27633 and NSF GP 34091X, respectively.

97

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K.J. BARWISE

turn. And one of the main advantages of the axiomatic method is lost since ZF has so few recognizable models in which to interpret its theorems. For these reasons, and others familiar to anyone versed in generalized recursion theory, it eventually becomes profitable to look at set theories weaker than ZF, weaker in the principles of set existence which they allow us to use. The theory we have in mind here is the Kripke-Platek theory KP for admissible sets. We now come to the main point. As we weaken the principles allowed in the construction of sets (in the move from ZF to KP) we destroy the earlier justification for throwing out urelements. In this paper we put them back in by “weakening” KP to a theory KPU which does not rule out the existence of urelements. KP will be equivalent to the theory KPU + “there are no urelements”. The result is worth the trouble. There is a great deal of folk literature about admissible sets as well as about admissible sets with urelements. A large portion of our talk at Oslo was devoted to a review of this folk material. When it came to writing it soon became obvious that neither time nor space would permit a complete treatment in this paper. We are currently at work on such a treatment, however, and plan to publish it as a textbook on admissible sets. In this paper, then, we abandon once again any reader ignorant of the basics of admissible sets, and discuss the material from the last third of our Oslo talk: admissible covers of nonstandard models. We have chosen this topic because it offers nice examples of the new degree of freedom afforded by urelements in admissible sets, examples in recursion theory and in the model theory of set theory. Proofs not given here will be found in the book referred t o above.

52.The axioms of KPU Let L be a first order language and let 137 = ( M , ...> be a structure for the language L. We wish to form admissible sets which haveM as a collection of urelements; these admissible sets are the intended models of the theory KPU, other models of KPU being so called nonstandard admissible sets on M . The theory KPU is formulated relative to a language L* = L(€, ...) which extends L by adding a membership symbol E and, possibly, other function, relation and constant symbols. Rather than describe L* precisely, we describe

ADMISSIBLE SETS OVER MODELS OF SET THEORY

99

its class of structures, leaving it to the reader to formalize L* in a way that suits his prejudices.

2.1. Definition. A structure for L* consists of = (M, ...) for the language L, M = 6 being kept open as a (1) a structure possibility, (2) a nonempty setA disjoint fromM. (3) a relation E C ( M U A ) X A which interprets the symbol E, (4) other function, relation and constants on M U A to interpret any other symbols in L(E, ...). We denote such a structure by 'u,= (5JX ;A , E , ...). We use variables of L* subject to the following conventions: Given a structure % ~ = ( ~ J x ; A , E ,for . . .L*, )

p , q , r , p l , ... a,b,c,d,al, ... X,.Y,Z,

...

range overM (urelements) range overA (sets) range over M U A .

We use u,u, w to denote any kind of variable. This notation gives us an easy way to assert that something holds of sets, or of urelements. For example, Vp 3a Vx (x € a ++ x = p ) asserts that { p } exists for any urelement p , where as Vp 3aVq(q € a ++a=p) asserts that there is a set a whose intersection with the class of all urelements is { p } . The axioms of KPU are of three kinds. The axioms of extensionality and foundation concern the basic nature of sets. The axioms of pair, union and AO-separation deal with the principles of set construction available to us. The most powerful axiom, AO-collection,guarantees that there are enough stages in our construction process. In order to state the latter two axioms we need to define the notion of AO-formulaof L(E, ...), due to LCvy [ 19651.

2.2. Definition. The collection of Ao-formulas of a language L(E, ...) is the smallest collection Y containing the atomic formulas of L* and closed under (1) if @ is in Y then so is l@ (2) if @, $ are in Y so are (@ A J / ) and (@ v J / ) (3) if @ is in Y then so are V u E u @ and 3 u € u @

for all variables, u and u.

K.J. BARWISE

100

The importance of A O-formulasrests in the fact that many useful predicates can be defined by AO-formulas and that any predicate defined by a Aoformula is very absolute.

2.3. Definition. The theory KPU (relative to a language L(E, ...)) consists of the universal closures of the following formulas: Extensionality : V x ( x E a a x E b ) + a = b 3a @(a)-+ 3 a [ @ ( a )A V b € a l @ ( b ) for ] d l formulas Foundation : @(a)in which b does not occur free. 3a ( x E a A y € a ) Pair: 3bVyEaVxEy(xEb) Union: l Ao- Separation: 3 b V x ( x € b + + x E a h @ ( x ) ) f o r a l Aoformulasin which b does not occur free. Ao-Collection: V x E a 3 y @ ( x , y ) +3 b V x E a 3 y E b @ ( x , y )for all A. formulas in which b does not occur free.

2.4. Definition. 1 .

KPU' is KPU plus the axiom:

3aVx [ x E a ++ 3 p ( x = p ) ] which asserts that there is a set of all urelements. 2. KP is KPU plus the axiom

V x 3a(x=a) which asserts that there are no urelements. One word of caution. There are some axioms built into our definition of structure for L(E, ...). For example, the sentence

follows from 2.1.3, and

follows from 2.1.2.


101

In a systematic treatment one would now develop axiomatically a large part of elementary set theory in KPU. It is done a l m s t exactly as it is for KP, the only trouble being that there is no such axiomatic development in print for KP. We must therefore leave it to the reader to work most of this out for himself. In particular, he should verify that the following are provable in KPU. “There is a unique set a with no elements” “Given a, there is a unique set b = U a such that x E b iff 3y € a (x Ey).” “Given a, b there is a unique set c = a U b such that x E c iff x E a or x E b.” “Given a, b there is a unique set c = a n b such that x E c iff x E a and x E b.” We define, as usual, the ordered pair of x,y by

and prove that ( x , y ) = ( z , w )i f f x = z a n d y = w ,and then prove in KPU that “for all a, b there is a set c = a X b, the Cartesian product of a and b, such that c = { ( x , y ): x E b and y E b}.”

53.Some useful principles provable in KPU A E l formula is one of the form 3u @(u)where @ is a AO-formula.It turns out that a wide class of formulas are equivalent to X I formulas in KPU.

3.1. Definition. The class of 2: formulas is the smallest class of formulas Y containing the A. formulas and closed under 2.2.2, 2.2.3 and if 4 is in Y so is 3u@, for all variables u . Thus, for example, the predicate, “x is a set of urelements” can be written 3a(x=a) A V u E x q p ( x = p )

which is E but, as written, is not XI.We will show, however, that for every

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K.J. BARWISE

X formula I#J there is a XI formula 4' with the same free variables such that KPU k 4 +-+4'. Given a formula 4 and a variable w we write q5(w) for the result of replacing each unbounded quantifier in 9 by a bounded quantifier: 3u

by

3uEw

Vu

by

VuEw

for all variables u . Thus @ ( w ) is a A, formula. If 4 is A, then @(w) = 4, since there are no unbounded quantifiers 4.

3.2. Lemma. For each X formula 4 the following are logically valid (i.e. true in all structures %w):

where u E u abbreviates the formula V x [x Eu + x f u ] . 3.3. 2 reflection principle. For all Z formulas @ the following is a theorem of

KPU:

(We assume u is any set variable not occuring free in 4, and stop making such assumptions explicit in the remainder of this paper.)

Proof. We know from the previous lemma that 3a@(') + 4 is just plain valid, so the axioms of KPU come in only in showing @ + 3a4('). The proof is by induction on 4, the case for A, formulas being trivial. We take the three most interesting cases, leaving the other two to the reader. Case (i). 4 is $ A 0 . Assume


and

KPU i-

e

-

103

3a@

as induction hypothesis and prove

KPU

I-

($ A

s)

--f

3a [ J /

A

el(") .

+

Let us work in K P U , assuming A 0 and proving 3a [I)'")A @(')I. Now there are a l , a 2 such that $("I), O('2) so let a = al U a2. Then Q(") and +(") hold by the above lemma. Case (ii). Q is Vu Eu +(u). Assume that

Again, working in KPU assume V u E u +(a) and prove 3aVu € u $(u)("). For each u E u there is an b such that +(u)@), so by AOcollection there is an a. such that Vu € u 3 b E a o $(u)@). Let a = Ua,. Now for every U E U3b Sa +(u)@') so Vu E u $(LA)(") by the above lemma. Case (iii). Q is 324 +(u). Assume +(u ) * 3b +(u)@)proved and suppose 32.4 J / ( u )true. We need an a such that 3u € a +(u)("). If +(u) holds, pick b so that +(a)@)and let a = b U {u}. Then u E a and $(u)(") by the above lemma. In his original development of admissible sets, Platek took the X reflection principles as one of the axioms, since it is more useful than Ao-collection.AO-collection,however, is usually easier to verify in a particular admissible set. We list below some of the consequences of the X refle'cfimprinciple.

3.4. Z-collection. Fbr every Z formula Q the following is a theorem of K P U :

Zf V x € a 3 y @ ( x , y )then there is a set b such that V x€ a 3y E b @(x,y) and V y Eb3xEaQ(x,y) 3.5. A-separation. For any two I:formulas @(x),$(x), the following is a

K.J. BARWISE

104

theorem of KPU: If for x

Ea, $(x) * 1$ ( x ) then

there is a set b ,

b = { x E a : $(x)}. 3.6. Lreplacement. For each Z formula $(x,y) the following is a theorem of KPU: If Vx E a 3 ! y $ ( x , y ) then there is a function f with domain a such that v x E a 4 6 ,f (XI).

The above is sometimes unusable because of the uniqueness condition in the hypothesis. In these situations it is 3.7 that often comes to the rescue.

3.7. Strong Lreplacement. For each Z formula $(x,y) the following is a theorem of KPU: If Vx E a 3y $(x,y ) then there is a function f with domain a such that for evely x E a

A set a is transitive if for all x E a and ally E x , y E a. Thus if a is a set of urelernents it is transitive. An urelement is never transitive since only sets are transitive. We can prove in KPU that for every x there is a unique transitive set a with x E a such that if b is any other transitive set containing x , then a C:b. This set a is called the transitive closure of x , TC(x). Using TC one can go on to justify recursive definitions over E. For example, the support function can be defined by

Thus, Sp (a) = { p : p E TC (a)}. A pure set is a set a with empty support, Sp ( a ) = 0. We will also need the second recursion theorem which for KPU takes the following form.


105

3.8. Second recursion theorem for KPU.Let @(x,y, R ) be a Z formula of L(E, ...,R), where x = xl... xfl ,y = y l... y k and R is an wary relation symbol occuringpositively in @. There is E l formula $ ( x , y )of L(E, ...) such that

Proof. We are using @(x,Y,$(* ,%)) to denote the result of replacing R(z l...zfl) by $(z l...z,,yl...yk) wherever it [email protected] we consider the case where n = k = 1. Let O(x,y,z) be

where Sat(z,ulu2u3) is the XI satisfaction relation for El formulasz of 3 = n t k + 1 variables (cf. L6vy [ 196.51). Let m be the Godel number of this formula B ( x , y , z ) and let $ ( x , y ) be O(x,y,m), or rather the Z l formula equivalent to it where m has been replaced by its definition. Then

$4. Admissible sets over M I t facilitates matters if we fix notation and let V, denote the most generous possible universe of sets built o n M so that our admissible sets onM will be substructures of V,. Thus, we define

V,(O)

=0

VM(at 1) = Power set ()@(,/I V,(h) = U,

UM

)

V,(a) if h is a limit ordinal

K.J. BARWISE

106

where the latter union is taken over all ordinals. If we need to keep things straight for some reason we subscript notions with a n M to denote their interpretations in VM. For examples, EM denotes the membership relation of VM where each p E M is taken as having no elements (even though in some other contextM might be a set of sets) and “a is transitiveM” means “x EM y EM a implies x EM a”. 4.1. Definition. A structure u ‘, = (%TI ; A ,E, ...) for L (E, ...) is admissibk if l!lm is a model of KPU, ifA is a transitiveM subset of V, (where !lx = ( M , ...)) and E is the restriction of € M to M U A .

If ordinary admissible sets are pictured as in fig. la, as they often are in informal discussions, then admissible sets with urelements should be pictured somewhat as in fig. 1b.

(a)

(b)

a) An admissible set A without urelements

b) An admissible set %m over M

Fig. 1.

The small cone in ’urn represents the pure sets of ,% , i.e. those a € A m with empty support. It is easy to verify that this collection of sets is an admissible set (without urelements).

4.2. Example. For any infinite cardinal K define H ( K )=~(YJI ; A ,E) where A = { a E VM : TCM(a)has cardinality ITCM(a)1 < K } . For any such K , H ( K ) ~ is admissible.H(K>m is a model of KPU’ iff K > IMI. We usually denote H ( u ) m by HFm since it consists of the hereditarily finite sets relative to

In.


107

A subset R of the admissible 'urn is C, if it is definable on by a El formula with parameters from A UM . R is Al if both R and its complement (A UM) - R are L1 on 2lm.

4.3. Example. If 9?l is an acceptable structure then a subset of 9?l is semisearch computable on 9?l iff it is El on HFw , the notations of acceptable and semi-search computable being those of Moschovakis. The result is due to Gordon. The ordinal of an admissible set is the least ordinal not in the admissible set.

4.4. Example. Given any structure 9?l there is a smallest admissible set over which is a model of KPU', i.e. where the set M itself is an element of the admissible set. Denote this admissible set by HYPm . The proofs in BanviseCandy-Moschovalus [ 197 11, if carried out in this setting, show that if m is acceptable then a relation S is inductive on 9?l iff it is C on HYP, and is hyperelementary on 93 iff it is an element of W m . The ordinal of HYPm is the closure ordinal of the class of first order positive inductive definitions over m. (We are using inductive and hyperelementary, as in Moschovakis [ 19731, for what was called semi-hyperprojective and hyperprojec tive in Barwise-Candy-Moschovakis [ 197 11 .) 4.5. Example. The results in infinitary logic of Barwise [ 1969al all go through in this more general setting without change. T o see how this may nevertheless be a significant extension, suppose am=( $ n ; A , E , , ...) is countable and admissible with ordinal w . (We will see many examples of such in the following sections.) Let TI, T, be theories of the admissible fragment LA of LU1,, C1 on am,such that every @ E T, is a pure set (and hence finitary). If every finite subset of T, is consistent with T, then T, U T, is consistent, even though T, may have infinitary sentences in it. The proof is a simple consequence of the compactness theorem for LA.

$5.Properties of the admissible cover Admissible sets am= (m ; A , € ,...) embody certain principles of set contribution, the ordinals a inA give us the stages, the sets of rank OL the principles of set formation available at stage a.What then is to be made of non-

108

K.J. BARWISE

standard models of KPU, KP or ZF.The results we discuss here shows that there is a hard core of admissibility in the heart of even the most non-standard models.

5.1. Definition. Given a model %I= ( M , E ,...>,where E is binary, the covering is the function which assigns to each x E M the set function CE for

xE

=

.

{ yE M : yEx}

The name “covering function” is new but the function itself is basic in the study of models of set theory. For example, if $%?=W , E ) is a substructure of (n = ( N , F ) then an x E M is fixed by (n if xE = xF ; otherwise x is enlarged by 9.If x E = x F for all x E M then % is an end extension of %I, written

9.KCend (n.

The covering function f o r m = ( M , E )maps elements o f M to subsets o f M Extensionality then this function is one-one. hence to elements of V,. If $%? There are many admissible ’urnwhich are admissible with respect to the covering function for (XQ. For example, anyH(K),m for K > ] M I . If %I satisfies enough axioms of set theory, however, there is ohe admissible 21n which really lives over m. This set is called the admissible cover of %I and is the object of study of this paper.

+

5.2. Theorem. Let T be some set theoiy containing KP and let p i = ( M , E ) be a model of T, standard or nonstandard, with covering function CE. There is an admissible set ’ 2 l over ~ 8, called the admissible cover of m, with Properties I-IX listed below. Property I. C, maps M into 21’, i.e. = (M;A,,E 1 A,,CE)

aM

and yiM is admissible with respect to C, ; is admissible.

This is equivalent to saying that (%I;A,,€,,CE) is admissible since E can be recovered from CE. Property 11. There is a function

* : A,

UM

p * = p forall p E M (u*)E = {b*:b E a ) for

u€A,.

-+

M satisfying:


109

One might call * an €-retraction of "2IM onto m. It is Property I1 which insures that the admissible cover of really lives over W. To be more precise: Property 111. a , is uniquely determined by Properties I and II. In fact, u', is contained in any admissible $3,' satisfying I and contains any admissible BM satishing II. Since I and I1 characterize the admissible cover of m, all other properties could be derived from them, but such a procedure would cause us to duplicate many steps in the proofs of I and 11. For example, the following is obvious from the proof of 11. Property IV. The cardinality of u',

is the same as that of M

The well founded part of a model m = ( M , E ) of KP, WF(m), is the transitive set (in the sense of V ) which is the range of the following collapsing function clpse. The domain of clpse is the set of all x E M for which E is well founded on TCm(x), so that the following makes sense: clpse ( x ) = {clpse ( y ) : y If

is well founded then clpse:

EX^}

m zz (WF(m),E).

Property V. The pure sets of the admissible cover of are exactly those sets in WF(SB), the well founded part of m. In particular, the ordinal of A , is just the ordinal of the well founded part of m. We will attempt a picture of m and its admissible cover at this point. The represents the point at which 93 becomes non-standard, the dotted line in lower portion being isomorphic to WF(S?I).

(b)

(a) a)

= ( M ,E ), a model of set theory

b) The admissible cover 2 Fig. 2.

l of~ (%I

K.J. BARWISE

110

This gives a hint of the new dimension now available to us. Consider, for example, a model !J?l of ZF with nonstandard integers. The admissible cover B has many infinite sets in it (aE for any infinite integer a, for example), of S but only the natural numbers for ordinals. This is in stark contrast to old fashioned admissible sets where ifA # HF then w € A . The next three properties are of a more recursion theoretic nature. We say that an inductive definition r on is a E inductive definition if the clause x E r ( R )is given by a C formula $ ( x , R )with R occuring positively in 4:

rrn

x E r(R)

f--f

$(x,R).

We use Z@ for the futed point of I?, as in Moschovakis [ 19731. A relation S of n variables is E inductively definable on !J?l if there is a E inductive definition I@of n + k variables, k 2 0, and x1... xk in %?such that

For example, the domain of the function clpse is E inductively defined on

(Bby the formula $ ( x , R ) :

is an admissible set then any C inducA theorem of Candy shows that if tively definable set on !J?l is El on !J?l. If SBis nonstandard, however,Io need as the domain of the function clpse not be even first order definable over shows.

in,

Property VI.A relation S on !)J is E inductively definable on m i f and only i f it is X on the admissible cover of m. The closure ordinal of a E inductive definition is at most the ordinal o f A M . In Barwise [ 1969b] we showed that the strict-IIi relations on an admissible set A coincide with the s.i.i.d. relations of Kunen [ 19683, and hence ifA is countable the strict l l i relations coincide with the I;, relations. The proofs of these results carry over verbatim to admissible sets with urelements. In Aczel [ 19701 it was announced that if is a countable nonstandard

Sn


111

coincide, not with the El relamodel of KP then the s-ni relations on tions, but with the x inductively definable relations on fm . This follows from Properties VI and VII, and the above paragraph.

is s-IIi on $I i f f it is s - IIf on the admis-

Property VII. A relation S on sible cover of m.

The non-trivial half of VII and half of 111 can be derived from VIII.

is the €-hard core (in the sense of Property VIII. The admissible cover of Kreisel) of the class of all models of KPU of the form

where '37 = ( N ,F ) is an end extension of 93 satisfying KP. In the next section we are going to discuss the use of admissible covers in constructing models of set theory. The first application uses only properties I and 11. In some applications, however, we need the following. Given m 5 % and Mo C M we write

m X l % [wrt M o ] to indicate that every x sentence with parameters from X has the same truth value in m as in % . IfM, = M we write m< '37 and if X = 8 we write

sm

91 . Let $I(= M , E ) and '37 = ( N , F )be models of KP with admissible covers 21M and 91N. I t follows from VIII that !Dl S e n d % iff%21,C.,1B In this case if x Eu', then x* has the same value regardless of whether the €-retraction El

* is taken in the sense of u',

or

aN.

Property IX. Given and '37 and % as above with k t A o = { x E A M :x* E M O } .Then

!DlE , u since ~ ) t, E

e,

e.

Let G ( u ) = U;=,gf(u), H(u)=h*(Uf=lgT(u)) = h*(G(u)),and K ( u ) = U" U C(u) U H(u). By hypothesis, G, H, K E C . By Lemma 1, let cp(y,x) be an €-formula equivalent to the formula

e.

Clearly, f*(u)=F8(([t,(K(u))] n [ H ( u ) X u n ] ) , u n ) E

FINE STRUCTURE OF THE CONSTRUCTIBLE HIERARCHY

133

LetG(u)= { ( z , y , ~ ) ( ( 3 u E y ) [ z E g ( u , ~ )x] E u }A. s a b o v e G E C . But f * ( u ) = F ~ ( G ( ~u”+’) ) , E e. Hence f rud +f* E C, for all f . Finally, let f be rud. We show that f E C?. Setf((z)) = f ( a ) , T ( y )= @ i nall other cases. ThusTis rud. So by the above, E C?. Let P ( x ) = {(x)}. Thus P E But look, f ( x )= {{f ( x ) } }= U U ( ~ { ( X ) } }= U U F ~ ( ~ ( P ( X ) ) , P ( E X))

7*

e.

e.

uu

As an immediate corollary of Lemmas 3 and 6 we have: Lemma 7 . Let A C V and define F9 by F9(x,y ) = A n x. Every function rud in A may be expressed as a composition of some of the (rud in A ) functions Fo, ..., F9.

We shall make immediate use of Lemma 7 in investigating the logical comz far suitable M.We assume, once and for all, plexity of the predicates kMn that we have a futed arithmetization of our language. Lemma 8. bgois uniformly Ey for transitive, rud closed M = ( M A ,> .

Proof. Let d: be the language consisting of: (i) variables wi,i E w . (ii) function symbols (binary) f o , ...,f9. We shall assume we have a futed arithmetization of %. We also assume that the reader understands what is meant by a “term” of L. Henceforth, let M = ( M ,A ) be arbitrary, transitive, and rud closed. We first define precisely how d: is to be interpreted in M. Let Q be the set of functionsp mapping a finite subset of {wili E a} into M . We may clearly assume Q is rud. Let C be the (rud) function which to each term r of d: assigns the set of all component terms of r , including variables. Let Vbl, be the rud predicate defining the set {wiI i E w } . Let P be the

K. J. DEVLIN

134

predicate

ThusP is rud in A . We may now define the interpretation of a term

y

=

P[ p ]

-

“T

T

of d: at a “point” p E Q by

is an L-term” A p E c A gg[P(C(T), g , p ) A g(7) = y ]

Hence the function if 7 is an &term and p E Q otherwise is (uniformly) ?2? (for transitive, rud closed M). Since M is rud closed we can use the above result to define kE0 as an M-predicate. Let cp E Frnl‘o. By Lemma 2, pM is rud in A . Hence the function r defined by I , if ( ~ ~ 1 x 1

Iyx) =

i

0, otherwise

is rud in A . So, by Lemma 7, we may assume r = T ~ where , T is a term of L, under the above interpretation (i.e. with Fiinterpretingfi for each i). In fact, we may clearly pick a recursive function u mapping FmlEo into the terms of d: so that whenever cp E Fml’o, pM[x] [ U ~ ) ] ~ [= X 1. ] But by our above result, this implies that kzo is (uniformly) ?2 (for transitive, rud closed M).

-

As an immediate consequence of this result, we have

Lemma 9. Let n 2 1. Then M=(M.A).

&‘is uniformly ?2:

for transitive,’ rud closed

We conclude this section with a few miscellaneous results of use later. The first two are technical, and will often be used without mention.

135


-

Lemma 10. Let M = ( M ,A ) be rud closed. If R C M is E,(M), there is a Xo(M) relation P such that R ( x ) 3x1Vx23x3 ... Q,x,P(x,xI, ...,x,).

Proof. SupposeR(x)

-

FM 3 0 ~ V 0 2 3 0 3... Q , o , ( P ( u , o..., ~ , v,)[x],

cp is a Co-formula. Using the rud functions (-,

-

..., -),

(-)?,

...,

where we

can easily obtain, via Lemma 1, a Co-formula J/ such that R ( x ) c--f bM 3ulVu2 ... Q,u,$(u,ul, ...,u,)[x]. ThenR(x) 3x,Vx, ... Qnxn [bM $ [ x , x ~ ..., , x,]], as required.

A ) be rud closed. If R C M is E,(M), there is a single Lemma 11. Let M = ( M , element p E M such that R is E ; ( (p}).

Proof. If R is Ey({pl, ..., p , } ) , thenR is also CF(((pl, ..., p , ) } ) . Let M = ( M A , ), n 2 0. Write X < x , M iff X C M and for every En formula cp and every x € X , ~ ( x , nX) A CP [XI

iff b~ CP [XI .

Clearly, i f X , M are transitive and X CM, we always have X < x o M. And for n>O,wehaveX<EnMiffXCMandforeveryPEEF(X),P#O-+

Pnxf0.

Recall that if ( X ,E) satisfies the axiom of extensionality, there is a unique isomorphism n : (X,E) 1 (W,E), where W is a unique transitive set. Furthermore, if Z C X is transitive, then n 1 Z = id Z . In fact, n is defined by € induction thus: n ( x ) = { n ( y )I y E x n X } for each x EX. The next result is of considerable importance.

r

Lemma 12. Let M be transitive and rud closed. Let X < x : , M. Then ( X ,A nX ) satisfies the axiom of extensionality and is rud closed. Let n : ( X , A n X ) S ( W , B )where , Wistransitive.Letf:Mn+MberudinA. Then for all z E X , n( f(z)) = f ( n ( z ) ) .

Proof. Since M is transitive, M satisfies the axiom of extensionality. Hence as X < q M, so does ( X , A n X ) . Similarly, by Lemma 5, ( X , A n X ) is rud closed. Hence, in particular, z G X + f ( z ) E X for f : M n + M rud in A . By induction on the (rud in A ) definition off, ~(f(z)) =f(n(%)) for each I EX.

136

K. J. DEVLIN

$3.Admissible sets Let M = ( M , A ) be non-empty and transitive. We say M is admissible iff M is rud closed and satisfies the X o-ReplacementAxiom: for all Co formulas cp and all u E M , l=M [ V ~ 3 y c+ p V U ~ V ( Eu) V X ( 3 y Eu) cp] [ a ] . In case A = !J in the above, we call M an admissible set. More generally, M is X,-admissible iff M is rud closed and satisfies the (analogous) C,-Replucement Axiom. Likewise a Enadmissible set. We prove below that M is admissible iff M is C , -admissible. All our results extend trivially from admissibility t o C,-admissibility, with “En” everywhere replacing XI,etc. Roughly speaking, an admissible set (or structure) behaves like the universe as far as El concepts are concerned. We give a few elementary results which set the tone for the rest of this exposition.

Convention: For the whole of this paper, we shall adopt the following abuse of notation. Suppose M is a structure, cp(v) is a formula, and x EM. We shall write FM~ ( xrather ) than FM~ ( V ) [ X ] .Clearly, this is purely a notational convenience. Firstly, we give the promised “stronger” form of the admissibility definition.

,

Lemma 13 ( C -Replacement). Let M be admissible, and let cp be a X -formula,

a E M . Then

Proof. Let $ be a E,,-formula such that

bMcp(x,y, a ) t--, 3 2 $ ( x , y , z , U ) .Then

Convention: The essentially superfluous role played by a in the above theorem


137

leads us to extend our previous convention slightly by allowing formulas to contain members of M as parameters. Again, this is clearly an avoidable convenience.

Lemma 14. L e t M be admissible. I f R ( x , y )is Xc,(M), so is (Vy €z)R(x,y).

-

Proof. Let cp be a Eo-formula with parameters from (“w.p.f.”) M such that W , y ) kM ~ w P ( ~ , YThen )-

So by Z0-Replacement,

which is z 1 ( M ) .

Lemma 15 (A l-Comprehension). Let M be admissible, P E A1(M). Then E M + P nU E M .

U

Proof. Let cp, $ be Eo-formulas w.p.f. M such that

So by Xl-Replacement there is u E M such that

138

K . J . DEVLIN

But M is rud closed (SO satisfies what might be called the Eo-Comprehension Axiom), and therefore we conclude t h a t P n u = { z E u I k M ( 3 y E u ) $ ( y , z ) } E M . The next result has nothing specificalIy to do with admissibility, but is of considerable value. Let f : C M -+ M mean that f : X .+ M for some X C M. Lemma 16. Let M bearbitray, f : C M - + M b eX,(M).Ifdom(f)is then in fact fand dom (f)are A (M).

n,(M),

Proof.

It was necessary to state the above result explicitly because we shall frequently have to deal with functions which, though definable, are not total functions. A particular case of the above theorem would of course occur when dom(f) EM, (whence dom(f) is Z,(M)). As usual, we shall use the notatiolif(x) =g(x) for partial functions, with its usual meaning(i.e. f(s)is defined iffg(x) is defined, in which case f(x> = 'dx)). Lemma 17. Let M be admissible, f : C M u C dom (f),then f " u E M .

+

M be X I(M). If u E M and

Proof. Since M is rud closed andf"u = ran(fr u ) , it suffices to prove that fl'uEM.Now,asuEM,fI'uisA1(M)byLemma 16.Letcp(x,y)bea X -formula w.p.f. M such that f(x) = y +-+ kMcp(x,y). Then kMVx 3 y [(x E u A cp(x,y)) V ( x e u ) ] , so byX I-Replacement there is u E M such that kM(Vx E u ) ( 3 y E u ) cp(x,y). Hence f r u C u X u. So, by A -Cornprehension, f 1II = (f u ) n (u X u ) E M . Theorem 18 (Recursion Theorem). Let M be admissible. Let h : Mn+' -+A4 bea L1(M)fiinctionsuch thatforallx E M , {(z,$)l z E h ( y , x ) }is weN-


139

founded. Let G = M n + 2 + M be El (M).Then there is a unique X (M)function F such that (i) ( y , x ) Edom(F) f3 {(z,x)lz E h ( y , x ) } C dom(F) (4 F ( y , x ) - G ( y , x , ( F ( z , x ) I z E h ( y , x ) ) ) . Proof. Let @ be the predicate

@(f,x)

“ f i s a function"^ (Vy E dom(f))(Vz E h ( y , x ) ) ( z E dom(f))

~ - - f

-

By Lemma 16, h , G are A1(M), so @ is Al(M). Let cp be a E1-formula w.p.f.M such that @ ( f , x ) p(f.x). Define a XI(M)predicate F by (using notation which will later be justified)

We verify (i) for this F. Suppose first that ( y , x )E dom ( F ) . Then, by definition, 3f[@(f,x)A y E dom (f)] . By definition of @, for such anf we must have(Vz Eh(y,x))(zEdom(f)). HencezEh(y,x)+(z,x)Edom(F). Now suppose that z E h ( y , x ) -+ ( z , x ) E dom (F). Note that a s M is transitive, h ( y , x ) C M . By our supposition,

so by Xl-Replacement,

Pick such a u . As @ is Al(M), by Al-Comprehension we see that w = u n {fl @(’,x)} EM. Hence u w EM.It is easily seen that @(UW,X).Noting that h ( y , x ) C dom(Uw), note that U w /‘h(y,x) EM. S e t f = U w 1h ( y , x ) U {(G(y,x, U w f h ( y , x ) ) , y ) } Clearly, . @ ( f , x ) so , ( y , x )E dom(F). Hence (i) holds for this F. We now show that F is a function and is unique. By (i), dom (F) is already uniquely determined, so for both of these it suffices to prove the following:

140

K. J. DEVLIN

To this end, suppose not. Then P = { y Iy E dom(f) n dom (f’) A f(y) # f ’ ( y ) }# 0. Let y o be an h-minimal element o f P . Since y o E P , f ( y o ) # f ‘ ( y o ) . But @ ( f , ~ )@ , ( f ’,x), so clearlyf(yO) =f’(yO) by the h-minimality o f y o EP. This contradiction suffices (and thus justifies our notation somewhat). Finally, it is trivial to note that (ii) must hold, virtually by definition. In view of the many set theoretic concepts defined by a recursion of the above type, it is clear that admissible sets play an important role in set theory. Say M is strongly admissible iff M is non-empty, transitive, rud closed, and satisfies the Strong CO-ReplacementAxiom: for all C o formulas cp w.p.f. M , kM Vu3u(VxEu)[3ycp(x,y)’(3y Eu)cp(x,y)].(Clearly, such a n M will also satisfy the “Strong El-Replacement Axiom”.) Strongly admissible structures M are (for reasons to be indicated later) also called rion-projectibleadmissible structures. The difference between admissibility and strong admissibility is closely connected with the difference between En predicates and An predicates, which is in turn closely connected with the difference between a function being partial and total. We shall have more to say on this matter later.

$4. The Jensen hierarchy Let X be a set. The rudimentary closure of X is the smallest set Y 3 X such that Y is rud closed. Lemma 19. If U is transitive, so is its rud closure.

Proof. Let W be the rud closure of U . Since rud functions are closed under composition, we clearly have W = {f(x)lx E U A f is rud}. An easy induction on the rud definition of any rudfshows that x E U + TC (f(x))C W . Hence W is transitive. (TC denotes the transitive closure function.) For U transitive, let rud(li) = the rud closure of U U { U ] . Of crucial importance is:


141

Lemma 20. L e t U be transitive. Then 9(u>n rud(U) = Zu(U). Proof. Clearly, P(U) n Z;,(UU { U } ) = X u ( U ) , so it suffices to show that

3 ( u ) n z , ( ~ u{ ~ ) ) = P ( ~ ) n n t d ( ~ ) . ~ e t ~ ~ P ({u}). ~)nz,(uu Then, exactly as in the proof of Lemma 2 , X E rud ( U ) (by induction on the Z o definitionofX).NowletX€P(U)nrud(U). T h e n X i s a X (rud(U)) subset of 0. By Lemma 1, we may in fact assume thatXisE:'gUd( )(UU { U } ) .

8

transitive, s o x i s

Also very relevant is:

Lemma 21. There is a rud function S such that whenever U is transitive, S(U) is transitive, U U { U } C S(U)and U,,,Sn(U)= rud(U). Proof. Set S(U) = ( U U { U } ) U (Ui8,,F;(UU Lemma 6.

{U})2).The result follows by

Lemma 22. There is a rud function Wo such that whenever r is a well-ordering of u, Wo (r, u ) is an end-extension of r which well-orders S(u). Proof. Define iu, j f , j ; by:iu(x)=t h e l e a s t i s 8 such t h a t ( 3 x 1 , x 2 E u ) [ F i ( x 1 , x 2 ) = x ] jy(x) = the r-least x1 E u such that (3x2 E u)[Fiu~,)(xl,x2) = x] jT(x) = the r-least x2 E u such that Fju(&f(x), x2) = x. Clearly, iu, j r , j ; are rud functions of u , x . Define Wo(r,u)= { ( x , y ) [ x , y E u~ x r y {}( x~, y ) l x ~ uA Y @ U } u{(x,y)lx@u

A ~ @ U [Ai u ( x ) < i u ( y )

viu(x)=

The Jensen hierarchy, (J,I a € OR), is defined as follows: J, =

b

J,+l = rud(J,)

J, =

u,,

J,,

if lim (A).

K.J. DEVLIN

142

Lemma 23. (i) Each J, is transitive. (ii) a Ip + J, C Jp (iii) rank (J,) = OR n J, = WLY. Proof. (i) By Lemma 19. (ii) Immediate. (iii) By induction: rank(J,+l)= rank(rud(J,))= rank(J,) (by an earlier remark, this last step is easily verified).

+w

To facilitate our handling of the hierarchy, we “stratify” the J,’s by defining an auxiliary hierarchy <S,\ a E OR) as follows:

Clearly, the J,’s are just the limit points of this sequence. In fact:

Lemma 24. (i) Each S, is transitive (ii) a LO S, C Sp (iii) J, = u v < w , S, = sw,. +

Proof. (i) By Lemma 21. (ii) Immediate. (iii) By induction: Jol+l= rud(J,) = “nEw

Lemma 25. (S,I

SW,+H

= +,sw

u,,

Sn(J,) =

u,,,

Sn(S,,)=

= Sw(a+lY

u < m a ) is uniformly

X k for all a.

Proof. Set c p ( f ) ~ ‘ ~ i s a f u n c t i o n ” dAo m ( f ) € O R

Af(O)=OA

(VuEdom(f))[(succ(u)+f(u)=S(f(u-l)))

b ( v ) + f ( v ) = u,,.f(.>lI

A

.

Clearly, is uniformly X k .And by definition, y = S, +-+ 3f(@(f)A y =f(v)). Thus it suffices to show that for any a, u < wa, the existential quantifier here


143

can be restricted to J,. In other words, we must show that whenever T < oa, then (S,I v < T) E J,. This is proved by induction on a. For a = 0 it is trivial. For limit a the induction step is immediate. So assume a = p + 1 and that T < wp -+ (S,I v < 7)E Jp. Then, by our above remarks, it is clear that (S,I v < 00) is Zip. So by Lemma 20, (S,I v < u p ) E J,. Thus for all n < o, (S,I v < o p + n ) = (S,I v < U P ) U {(Sm(Jp),UP + m)l m < n } E J,, as J, is rud closed.

Lemma 26. (J,I v < a ) is uniformly E k for all‘a. Proof. By an easy induction, (ovlv < a ) is uniformly E p for all a. Since J, = S,,, The result follows by Lemma 25. Lemma 27. There are well-orderings map of a bounded subset of wa cofinally into war. The above two results illustrate our earlier remark concerning the difference between a function being partial and being total, and the corresponding difference between a predicate being En and being A n . The next two results, which strengthen the last two, and are also due to Kripke and Platek, also highlight this distinction. Theorem 42. The following are equivalent: (i) wci is admissible. (ii) (J,,A)isamenable forallA E A1(J,). (iii) There is no (J,) function mapping a y < o a onto J, (Of course, any such function would in fact be A I(J,).)

-

~

Proof. (i) + (ii). By Lemma 15 ( A , -Comprehension). onto (ii) -+ (iii). Assume (ii) h l(iii). Let y < wa, and let f : y J, be C1(J,).

-

Thenfis A,(J,),sod= { v l v $ f(v))is A,(J,).Thus b y ( i i ) , d = d n y E J , . So, d = f ( v ) for some v < y, so v E f ( v ) ++ v E d v 4f ( v ) , a contradiction. (iii) (i). Assume (iii) A l ( i ) . If a = 0 + 1, we can easily construct a Zl(J,) map of wp onto wa, so Theorem 39 yields the required contradiction. Assume lim (a). By Theorem 40, there must be a T < wa! and a Z1(Ja) m a p f o f T cofinally into ma. Let f be Ek((p}). Pick y < a with T,p E J,. Let h = h, be the canonical El Skolem function for J,. Set X = h"(w X J,). As J, is closed -+


153

under ordered pairs, Lemma 35 tells us that X < z1J,. Let n : X Jp. Thus IT r J, = id F J,. By an argument as in Theorem 39, n X = id X,so X = Jp. Now, f isXF({p}) a n d p € X - ( q J,, soXisclosedunderf. B u t T C X a n d so f ’’7C X , which means, since f ”7 is cofinal in wa and X = Jp is transitive, that wa C J p . Thus@= a, a n d X = J,. Define a Zl(J,) map i: w X T X J , + J, as follows. Let H be a X k relation such t h a t y = k ( i , x ) -(3t€J,)H(t,i,x,y). Set y , if ( 3 t E S f , v ) ) H ( t , i , x , y ) K(i,v,x) = 0, otherwise.

i

Then h is total on w X T X J,, and i ” ( w X 7 X {x}) = kf’(w X {x})for any x , as f “ T is cofinal in we. Hence h”(oX T X J,) = X = J,. By Theorem 39 there isg E J,, g : o y w X 7 X J,. Then K e g is a Z1(J,) map of wy onto J,, contrary to (iii). Theorem 43. The following are equivalent: (i) wa is strongly admissible. (ii) (J,, A ) is amenable for all A E C,(J,). (iii) Tkere is no X l(J,) function mapping a bounded subset of wa onto J,.

Proof. (i) + (ii) + (iii). Similar to the above. (iii) + (i). Assume (iii) n l ( i ) and proceed much as before. So, we assume lim (a),f i s (by Theorem 41) a E l(J,) map of some a C r < wa cofinally into ma, f E Zp({p}), and T < oy, p E J,, y < a.As before, if k = k , and X = k ” ( o X J,), then X = J,. Now, since we do not need to bother about functions being total, we can easily contradict (iii). By Theorem 39, let g E J,, g : wyw X J,. Setf(v) =h(g(v)).ThenSis a Z1(Ja) map of a subset of w y onto J,. Note that an immediate corollary of Theorem 42 is: Theorem 44. If K is a cardinal, then K is an admissible ordinal. Using admissibility theory, we can give a quick proof that L = UaEORJ,. Theorem 45. I f

MYis admissible, then

J, = L,

.,

154

K . J . DEVLIN

Proof. If a = 1, then J , = L, = the hereditarily finite sets. Assume a > 1. Thus o E J,. Since J, is admissible, the recursion theorem tells us that rud(x) = Un_o)f'(T,y) = 6 .

For each limit ordinal /3 5 a we define A 2 - cf(0) to be the least h such that there is a A2 function with domain h and range unbounded in 0.

I .8 Lemma. A2

-

cf (a) = A2

-

cf (a*).

Proof. Let f : a -+ a* be a one-one a-recursive function. Note that y n rng(f) is a-finite for each y < a* since rng(f) is a-r.e. Hence {o If(a) < y } is bounded for each y < a * . Hence f "X is unbounded in a* whenever X is unbounded in a. Suppose h 5 a and g : A + a is A, and rng(g) is unbounded in a . Then fg : h -+ a* is A, and by the previous paragraph rng(f) is unbounded in a*. So A, - cf(a) 2 A, - cf(a*). For the converse suppose h : h +a* is A, and rng(h) unbounded in a*. Define k : h a by -+

Using 1.7 it is easy to see that k is A2. Furthermore rng(k) is obviously unbounded in a. So A, - cf(a) 5 A, - cf(a*). rn If (Ip[p < v ) is an a-finite sequence of (canonical indices for) a-finite sets, then of course {Zpl p < v) is a-finite. The following trivial lemma provides another useful sufficient condition for the union of a sequence of a-finite sets to be a-finite.

u

1.9 Lemma. Suppose v < A, - cf (a)and let (I,, 1 p < v ) be a simultaneously a-r.e. sequence of a-finite sets. Then U {I,, 1 p < v } is a-finite.

172

S.G. SIMPSON

Corollary. Put X = A -- cf (a) and suppose f : A + a is A 2 . Then (i) ( f ( p ) I p < v ) is a-finite for each v < A; (ii) the sequence (Cf(p)I p < u)l v < A) is A,. Proof. Let f : a X X -+ a be an a-recursive function such that f ( v ) = limo f ( u , v) for all u < A. Such anfexists by 1.7. Let us say that f (v) changesvalueatstage u i f ( V u ‘ < u ) ( ~ ~ ) ( u ’ ~ ~ < ~ & f ( ~ , ~ ) # f ( uLet ,v)). I , be the set of all u such that f(v) changes value at stage u. Then each I,, is afinite, and the sequence U,Iv < A) is simultaneously a-recursive. Conclusions (i) and (ii) are now immediate from 1.9.

2. The a-jump operator

’

Friedberg determined the range of the jump operator in ordinary degree theory by proving the following theorem: for every a-degree b 2 0’ there is an w-degree a such that a’ = a U 0’ = 6 . (For a proof see Rogers [ 10:p. 2651 .) We now generalize Friedberg’s theorem and its proof to a-degree theory as foliows:

2.1 Theorem. Let b be an a-degree 2 0’. Then b is regular if and only if there is a regular, hyperregular a-degree a such that

a ’ = a u O ’ =b . Proof. Theorem 1.5 implies that the a-degree 0’ is regular. (In fact, it can be shown that a‘ is regular whenever a is regular and hyperregular.) The “if’ part of 2.1 follows immediately. For the “only if” part we shall employ a forcing construction. This in itself is not surprising since Friedberg’s original proof may be viewed as a forcing argument. However, unlike Friedberg or Cohen, we shall d o our forcing over a possibly uncountable ground model, L,. A condition is an a-finite sequence of 0’s and 1’s. We use p , q , ... as variables ranging over conditions. Thus p is an a-finite function from an ordinal lh ( p ) , The main result of Section 2 was first proved in August 1971. It was presented in an invited address to this Symposium.

DEGREE THEORY ON ADMISSIBLE ORDINALS

173

the length of p , into (0,l). We say q extends p if p C q . Let Cond be the set of conditions. A set D C Cond is dense if every p E Cond is extended by some q ED.

For X C a we denote by cx the characteristic function of X . Conditions are thought of as a-finite approximations to cx. If D is a set of conditions, we say X meets D if cx y E D for some y < a. Our notion of forcing is defined as follows :

(9 P I t [ E l (r)= 6 iff ( 3 ~ ) ( 3 q ) [ ( y~, , E , Q ) E R ' , ~ & ( Pp) ' ' ~ ~(1) & p " ~ , (011; (ii) p decides [el (y) iff p H- [ E ] (y) = 6 for some 6 ; (iii) p H- ( [E](y) is undefined) iff no q 2 p decides [ E ] (7); (iv) p determines ( [ E ] (y) I y < 0) iff either p decides [ E ] (y) for all y < or p ( [E](yo) is undefined) where yo is the least y such that p does not decide [el(7).

c

c

Note that "p decides [ E ] (y)" is a-recursive as a relation of p , E , y. We now define certain important sets of conditions. The definition splits into cases depending on the nature of a.

Definition.

Case I. a* is a regular &-cardinal. Then for each 0 < a* put

Case I f . Otherwise, i.e. a* = a or a* is a singular a-cardinal. Then for each p _ H ( U , U ) & A ~ Z A < ~ ,

u+ 1

if

g(< u , E ' )

otherwise.

E'

2 H ( u , p ) for some p such that

H ( p ) changes value at stage u ,

Define

Bu = BconstructZfrom ES)). Aand LetH(S)= ( ~ ~ ( ~ U ) ( ( U + ~ , U ) ~ S & ( O ,We r as follows.

-

v

Then Z is an i.d. and E E C. We can easily prove by induction on A. fo(x) E 2) i) (VA) (VX) (x E A, ii) (vA)(A 2, Prewellordering (EL). By using this fact we obtain Corollary 3 from Theorem 5 immediately. Finally, we have that PD and DC imply Prewellordering(Hii-l) and Prewellordering ( X i i ) . (See Moschovakis 1970, p. 3 3 , or see Martin 1968.) Hence Corollary 4 follows.

INDUCTIVE DEFlNITIONS AND THEIR CLOSURE ORDINALS

219

5 . Further results We shall now state some results without proofs. A full treatment will be published elsewhere.

Remark 10. Let w , be the first non-recursive ordinal. Spector showed that l n l1- m o n I = I I I ~ - m o n I = o l < I A ~ l . A c c o r d i n g t o T . G r i l l i oIC:-monl= t, IE!l.

S. AANDERAA

Acknowledgement I wish to thank Jens Erik Fenstad for having encouraged me to work on the problem of deciding the order relation between I Ili I and I E; I. I am also indebted to Peter Aczel and Wayne Richter for helpful comments on the preliminary draft of this paper. I am also grateful to Leo Harrington for helpful discussions.

References [ 11 Aczel, P. and W. Richter, Inductive definitions and analogues of large cardinals, in: Conference in Mathematical Logic, London, 1970. Lecture Notes in Mathematics Nr. 255 (Springer, Berlin, 1971).1-9. [2] Addison, T.W., Some consequences of the axiom of constructibility, Fund. Math. 46 (1959) 123-135. [ 31 Addison, T.W. and Y .N. Moschovakis, Some consequences of the axiom of definable determinateness, Proc. Nat. Acad. Sci., U.S.A., 59 (1968) 708-712. [4] Martin, D.A., The axiom of determinateness and reduction principles in the analytical hierarchy, Bull. Amer. Soc. 74 (1968) 687-689. 151 Moschovakis, Y.N., Determinacy and Prewellordering of the continuum, in: Math. Logic and Foundations of Set Theory, Y. Bar Hillel (ed.) (North-Holland, Amsterdam, 1970) 24-62. [6] Moschovakis, Y.N., Uniformization in a playful universe, Bull. Amer. Math. SOC.77 (1971) 731-736. [7] Rogers, Hartley, Jr., Theory of Recursive Functions and Effective Calculability (McCraw-Hill, New York, 1967). (81 Spector, C . , inductively defined sets of natural numbers, in: Infinistic Methods (Pergamon Press, Oxford and PWN, Warsaw, 1961) 97-102.

J.E.Fenstad, P. G. Hinman (eds.), Generalized Recursion Theory @ North-Holland Publ. Comp., I974

ORDINAL RECURSION AND INDUCTIVE DEFINITIONS Douglas CENZER University of Michigan and University of Florida

1. Introduction An inductive operator r over a set X is a map r fromP(X) to P ( X ) such that for all A S X , A C r ( A ) . r determines a transfinite sequence {P:a € ORD (ordinals)}, where F a = U{Yp : fl < a } for a = 0 or (Y a limit and Fa+' = r(r7.r is monotone if, for all A , B inP(X), A E B implies

r(A)C r ( B ) .

The closure ordinal I I of r is the least ordinal a such that Fa+' = F a ; clearly I r I always has cardinality less than or equal to F. The closure of r is rIr',the set inductively defined by r. Inductive definitions are basic to the development of recursion theory. Following the methods of Kleene [ 1 1,121, we will define ordinal recursion and recursion in a partial functional by means of inductive definitions. Given a class C of inductive operators, one would like to characterize the closure ordinal ICI = sup{lrl : r € C} and the closure algebra {A : A is 1-1 reducible to T for some r E C}. We write C-mon for the class of monotone operators in C. The following is a brief summary of results on inductive definitions over the natural numbers. The first significant results on inductive definitions were obtained by 0 who showed that Spector [24],~ _ _(IIl-monI _ = wl, the first non-recursive ordinal, and that n y - m o E l l i -mon = nt. Candy (unpublished) later showed that In: I = w 1 and l l y = n!. We make use of slightly generalized versions of Spector's results in section 3. I t is easily seen that In: 1 > w1 and lli # II!. Richter [19] demonstrated that even In!l is a rather large admissible ordinal. Anderaa [ 11 recently proved that In{1 < 1 1. On the other hand it follows from the work of Aczel [ 21 on

r

c=

221

D. CENZER

222

Ei operators that I Ei 1 = I Ei-monI. Aczel and Richter [3,4] have characterized I II; /,1 Ei /,and, for all n , lI:1 in terms of reflection principles in the constructible hierarchy. Pu tnam [ 181 showed that I A; 1 = Si, the first non-A; ordinal; it is also known that A; = A;. More generally, for all n > 1, A: = A:, IIA-mon = II,,1 and CA-mon = EA.(However, E -mon # E .) Using the techniques of Lemma 9.1 2, we can now show that I A: I = 6; for all n > 2. (See Cenzer [8] .)

i

2. Summary of results I n this paper we explore further the relation between non-monotone inductive definitions and ordinal recursion. As pointed out above, ordinal recursion can be defined by an inductive operator, and in return the theory of inductive definitions can be developed within the framework of ordinal recursion. For any ordinal a , let a+ be the least recursively regular (admissible) ordinal greater than a and let a* be the least stable ordinal greater than a. Let S be the class of stable ordinals. (See section 3 for a brief development of ordinal recursion theory.)

Theorem A. (a). III; I is the least ordinal a which is not @+-recursive; (b). in: I is the least ordinal a which is *+-stable; (c). 1 1 is the least ordinal a such that L , is a E1-elementaiysubmodel of La, '

Part (b) was proven independently by Aczel and Richter [ 3 , 4 ] .

Theorem B. (a). I Eil is the least ordinal a which is not a*-recursive in S ; (b). I Zi I is the least ordinal a which is a*-stable in S ; (c). I E;l is the least ordinal a such that L , is a E2-elementary submodel of La*. -

Theorem C. (a). II; is the class ofsets of numbers which are 1 II: I-semirecursive (the - domain of a 1 KIi !-partialrecursive function); (b). Zi is the class of sets of numbers which are I EiI-semirecursive in S. We derive results similar to Theorem C regarding E; and IIi inductive operators.

ORDINAL RECURSION AND INDUCTIVE DEFINITIONS

223

We extend the above results to combinations of operators in which the components need be inductive. For example, if denotes the class of inductive operators which are the composition of two ll: operators, then l ( l l ~ j 2 is 1 the least ordinal a which is not a*-recursive. Similar extensions of Theorems A, B, and C obtain for the classes (ll: j" and (C!)" for all n. It is well known that for A E o and a admissible, A EL, iff A is a-recursive. We generalize this result to the ordinal recursive arithmetic hierarchy and thus obtain results concerning constructibly analytic inductive operators. (A relation is constructibly analytic if definablt by means of quantifiers restricted to the constructible reals.) Analogously to the results of Aczel [2] on Ef and inductive operators, we define a functional GT and prove the following theorem.

Theorem - D. (a). Ill: I = w?', the least ordinal not recursive in GT; (b). is the class of sets of numbers which are semirecursive in G t .

3. Ordinal recursion This section is intended to provide the necessary background of ordinal recursion theory. Proofs omitted here can be found in Cenzer [7]. Let ( ) be a natural sequencing function from U,,,ORDn to ORD. ( ( a ,,...,a , ) ) j = a j ; l n ( ( a O ,...,a,))=n+ l ; ( a o,..., a,)*(f10 ,..., (a,, ..., an,fl,,-.,&). We give the inductive operator (actually a class of operators) which defines ordinal recursion in full detail here since we will later want to discuss two related operators with reference to this definition.

on)=

Definition 3.1. For any ordinal y, any I < w , and any f = (fo,..., fi- 1), n,[f] is the monotone operator such that for all k and n < w , all i < k and j a > n ) E Q, [f I ( A ); (1) ( ( l , k , I , i ) , aa, j ) € n,[f](Aj; (2) ( ( 2 ,k , l , i ) , a ,a j +1)E Q,[f](A); ( 3 ) g < { implies((3,k+4,I),g,{,u,7,a,o)EQ,[f](A); E > { i m p l i e s ( ( 3 , k + 4 , i ) , ~ , { , ~ , 7 , a ,E7 )n , [ f ) ( A ) ; (4) for all m, b,cO,..., and cmPl < a ,and all T ~ ..., , 7m-.1< y, if for all i < m , 3

D.CENZER

224

( c i , a, 7 )€ A , and ( b ,f , p ) E A , then ((4, k , f, b, c), a,B) € R,[f] ( A ) ; (5) i f V ~ < o 3 5 ) < ~ [ ( ( 5 , k + l , I , b ) , 4 . , a , 5 ) E € A ] , a n d VC < 7 3 , $ < o 3 p < ~ [ >p { A ( ( 5 , k + l , Z , b ) , 4 . , a , p ) E A ]and , ( b , T , u , a , p ) E A , then((5,k+l,Z,b),o,a,P)€R,[f](A); (6) i f V o O [ ( b , o , a , 7 ) E A ] , a n d ( b , p , a , O ) E A , then ( ( 6 ,k , t , b),a,P)E Q,lfl ( A ) ; (7) i f ( b , a , P ) E A , t h e n ( ( 7 , k + l , I ) , b , a , p ) € a , [ f ] ( A ) ; (8) iffi(Lyj)=p, then ( ( 8 , k , f , i , j ) , a , p ) E R y L f ] ( A ) ; (9) p E R,[f] ( A ) iff p is put in by one of clauses (0) to (8).

We write

u{a;[f]:

f 3 for (a, E ORD).

and

5,[f] for a,[f ] ;

[f]is

c,[f);

Definition 3.2. (a). {a},(a, f ) -0 iff (a, a,p) E (b). F is y-recursive iff 3a < w.F = {a},; (c). F is weakly y-recursive iff 3a < w 3a < y . F = ha, f . fa},(a, a, f). (F may be partial in the above definition.) We point out that O U T definition of a-recursion yields the usual w-recursive functionals and that every o-recursive functional is y-recursive for all y > w . Notice that for any a and any (Y < 0,{a}, C {a}p. The following lemma is easily verified.

Definition 3.4. y is recursively regular (RR(y)) iff for all y-recursive functions F, all a and 0< y , if Vo < 0.F ( o , a ) 4,then sup, = {aI,(Ca,f). In order to prove that various functionals are y-recursive, we want to show that the set of y-recursive functionals is closed under the following two schema: (1) Strong Composition (G,Fo, ..., Fm-l, Go, ..., Gn-l) = H iff for all a and all f: H ( a , f ) =FF(FO(a,f),...,Fm- l ( a , f ) ,hB .Go(P,a,f), ..., W - G n - 1 @ , a , f ) , f ) ; ( 2 ) Strong Primitive Recursion (C) = F iff for all /3, a , and f: F(P,a , f)= G(P, a , ( h a . F(o, a , f N do), i f u P. The following two propositions, which demonstrate the desired closure, can be proven by means of the recursion theorem. See Cenzer [7], pp. 14-18, for details.

r B,n,

[

Proposition 3.6. For all m, n < w, there is a primitive recursive function Cmpmrn such that for all a, b,, ..., b,-,, co, ..., c ~ - all ~ y,, all a< y, and all f:{Cmpmrn(a, ( b ) (c))},(a,f> , = {aI,({bO}Ja,f), ... ..., {bm-II,(a>f)j .{CO>,(P,a,f), ..., X P {c,_l},(P,a,f),f). Proposition 3.7. There is a primitive recursivefunction Spr such that for all a < w , all 7,all a,P < y, and all f : {Spr(a)Iy(P, a , f >1: { ~ I ~ (aP, (hu{Spr(a)Iy(u, , a,f))I

8,f).

Definition 3.8. Sup ( a , f )1: ifffis total on a and = sup {f ( u ) : u < a}. It is easily seen that the functional Sup is y-recursive for any y. For y > w, w = l e a s t a < y [ a # O ~ S u p ( a , h ~ . ~ + 1 ) =sow a ] , isy-recursive. Let {c,} = Sup and {c,} = w . We now distinguish an important class of functionals over ORD. Definition 3.9. (a). POR is the smallest set of numbers containing cs, c,, (0, k , l , n ) ,(1, k , l , i ) ,( 2 , k , l , i ) , (3,k+4,1),and ( 8 , k, l, i,j ) for all k,l,n,i I , and closed under CmpmJnfor all m , n <w and Spr.

226

D. CENZER

(b). F is primitive ordinal recursive (p.0.r.) iff 3a E POR. E = {a}_ The usefulness of p.0.r. functionals lies in the following proposition, which can be proven by induction on POR.

Proposition 3.10. For all a E POR, if F = { a } _ , then (a) F is total on total functions; (b) for any recursively regular y > w , any a< y, and any y-recursive f : F ( a , f ) = {a>,(a,f). We say that a relation or predicate is p.0.r. or a-recilr+~eiff its characteristic function is. We list some properties of the p.0.r. relations and functions.

Proposition 3.1 1. (a). The followingfinctions and relations are p.0.r.: (1) for all i, ( )i,( ), In, and * ; ( 2 ) , >,and = ; ( 3 ) lim (a) i f f a is a limit ordinal; (4) the operations +, * , and exp of ordinal arithmetic; ( 5 ) T, defined by a TP = u iffP+u = a o r ( a 2 0 A u = 0); ( 6 ) all arithmetic relations over wk X (“a)’, for all k , 1 < w ; (b). the p.o.r. functions andlor relations are closed under the following: ( 1) union, intersection, and complementation; ( 2 ) bounded quantification; ( 3 ) definition by cases; (4) bounded search operator (least a,~A > A N ( u , n , A ) ; -


2351

Proof. (a) This follows from Proposition 5.10. (b) Let uf = a ; by Proposition 5.6c, laA[xAJ I 5 a. Since i l a [xA] parallels aA[~Al,wehav~ e E [ x A =I a , t x A j , so {~>,(IC~>~I,XA) = I { U > ~ Iiff

I{uPI),x~).

TYa,a,(a,I ~ ~ I ~ I , (c) This follows from (b) above and Proposition 3.14b. Proposition 5.2 is a corollary to 5.11

Proposition 5.12. (a) For any functions f , any a recursively regular in f,and any function qf~ weakly a-recursive in A if W ( @ ) ,then I@I < a; (b) the relation W , , defined by Wo(a,@)i f f W ( @ ) A I@I = a, is p.0.r.

.

Proof. (a) Define H by recursion on u so that H ( u , @) = the unique m Iml, = (5 least u,per-Mahloordinal, arid similarly f o r I [I$,

..., ny]1.


243

Using the techniques of 6, it is not difficult to prove

Proposition 8.3. For all n, I

[m] < n

I

y2.

We obtain a result analoguous to Theorem 8.2; the proof is omitted since i t is similar to that in Richter [ 191.

Proposition 8.4. (a) 1 [lli,ll:] I is the least recursively regular ordinal y such that y is not y+-recursiveand y is a limit of ordinals a which are recursively regular and not a+-recursive; (b) 1 [II:,@] 1 is the least recursively regular ordinal 7 such that y is not y+recursive and such that every normal weakly y-recursive funcfion has a fixed point QI which is recursively regular and not a'-recursice. While the ordinals of Theorem 8.2 present a nice hierarchy of recursively large ordinals, those of Proposition 8.4 and its obvious extensions seem of less interest. The natural ordinals to consider above y1 would appear to be y2, y 3 , and so forth. It turns out that there are natural classes of inductive operators with these as closure ordinals.

For r = Po * ... * r,, we do not require each operator Ti to be inclusive, but C since for any operator only the composition r. For example, r, if ro(A)= w - r(A) and F1(A) = w - A , then r = ro* r l . We now state the primary result of this section.

Theorem 8.6.For any n > 0, (a) I ( m " I = 7,; ___ (b) for all A E w , A f(ll!)" iff A is y,-semirecursive. Theorem 8.6 is proven in a manner similar to the proof of Theorem 6.1.

Proposition 8.7. For all n > 0, I(I'Ii)" 1 2 y,.

D. CENZER

244

Proof. We sketch the proof for (lli)2.Given r =ro-rl E (IIi)2,we have by Proposition 5.l(a), p.0.r. functionals Fo and F , with rg(Fi) S {0,1} such t h a t f o r a l l m andA: m E F i ( A ) i f f 3u.Fi(5,m,xA)= 1 iff 3u<wA l .Fi(u,m,xA)= 1. We definep.0.r. functionsHandI by: I ( r , m , p ) - - S u p ( ~ , X ~ . I ( ~ , m , iflim(7) P)), or r = 0; H(T+I,m,b) = Sup(P,Xa.Fo(a,m,hP.J(r,P,P))); I ( T + l , m , b ) SUP ( b , XQ.Fl(@,m,X P . l ( T , P , P ) ) ) . N_

The following lemma is proven in the same manner as Lemma 6.3.

Now for recursively inaccessible ordinals a , m E r" iff I ( a , m ,a ) = 1, and m E r a + ' iff 3 u < a + + F l ( a , m , h p. H ( a + l , p , a + ) ) = 1. Asin Proposition 6.4, it follows that if m E P z + ' , then for some a < y2, m E so m E rrz; thus I l - l S y z . Next we construct operators A, E (llf)"such that lA,I = 7,. Definition 8.9. A(A) = {(x,y): L ( { x } ~b}A)}; , A, = (A),-'.

A,.

The following lemma is easily verified. Lemma 8.10. For any A C w, let wf = a ; then (a) A(A) is a pre-well-ordering of length a ; (b) $ A is p.0.r. in a, then w t H ( A =) a(,); (c) i f A isp.0.r. in a, then A,+'(A) = { ( a , r n , t ) : {a},(,,(rn)

N

t}.

Applying this lemma and proceeding as in the proof of Lemma 6.6, we have


245

Proposition 8.12. For all n > 0, = { ( a , m ,t ) : {aITn(m)= t ) ; (a) (b) IAnI = 7".

I\n

It follows from Proposition 8.7 that = A?, which equals { ( a , m , t ) : {a},(m)-tby Lemma8.11. (b) T h i s follows from (a). This completes the proof of Theorem 8.6(a); 8.6(b) ((n;)"= yn -semirecursive) is easily obtained. This follows from Lemma 8.8(a) and Proposition 8.7. (2)This follows directly from Proposition 8.12(a). Finally, we can extend the results of 97 to (n;)" and 7,.

(c)

Following the development of $7, we can prove Proposition 8.14. For all n > 0 and all k < w , # (a> = yn ; (b) for all f : wk -+ w , f is G,#-recursivei f f f is 7,-recursive; ( c ) for all A C w , A is G,#-semirecursive iff A is y,-semirecursive.

9. Z! inductive definitions The main result of this section is a characterization of 1 E! 1 which is the dual of the characterization of I lIi I = y1 given by Proposition 4.2 1 (in combination with Theorem 6.1).

Theorem 9.1. 1 Zi I is the least ordinal u such that for all a < w, < 0 {a}T+(T>J) {a}o+((J)i.

(VT

--f

We also prove an analogue to Theoren 6.1 (b).

Theorem 9.2. For all A

-

C w ,A E Zi i f f A is 1 Z; 1-semirecursive.

Anderaa [ I ] has recently obtained a significant result regarding dual

246

D. CENZER

classes of inductive operators which implies that I IIfI < I Z! I and ICi1< In:\.Combining the former inequality with the techniques involved in Theorems 9.1 and 9.2, we are able to characterize the spectra of the two classes. Definition 9.3. For any class C of inductive operators, Spectrum(C)= ( 1'1: rEC). Theorem 9.4.(a) Spectnim(;F.i)= {a:+a_ w,, x,,(m) =I(T, m, p ) ; (bjfor all 7 , wyr 5 o,+, . Lenima 9.6 is proven after the pattern of Lemma 6.3. Proposition 9.7. For any Xi inductive operator accessible, then y is y+-recursive.

r, i f y = 1 r I is recursively in-

Proof. For all inaccessible y, m E P iff /(y, m , y) = 1 and m E rY+' iff Vo


25 1

there is a B which is A; in A . I t follows that {a),(a,XA) < S i - A . (2) For any u < 6; - A , we show that u is not stable in A . Given u < 6: - A , we have a A: - A well-ordering @ of type u.I t follows from Proposition 10.1 that @ is 00-recursive in A , so if u is stable in A , then @ is u-recursive in A . But then by Proposition 5.12, I @ / < u, a contradiction. We remark that Propositions 10.1 and 10.2 could be combined and restated as follows: Proposition 10.6. A relation over natural numbers and sets of natural numbers is iff it is -semirecursive iff it is H -semirecursive.

11. Stability and the ordinal arithmetic hierarchy A crucial point in the study of ni inductive definitions in 9 6 is the fact that the relations RR and RI are p.0.r. A comparison of $ 5 with 5 10 makes it clear that in the study of Z inductive definitions the relation “stable” will have a large part.

i

Definition 11. I . (a) Sl(01)iff 01 is stable; (b) for all n > 0, Sn+’(a) iff 01 is stable in Sn. We say that 01 is n-stable if Sn(a). By Proposition 4.12, n-stables exist for all n > 0. It is interesting to note, however, that for n > 1 there need not be any countable n-stables and uncountable cardinals need not always be n-stable. Let So be all of the ordinals for the sake of simplicity. Recursion in the S n is closely related to the ordinal arithmetic hierarchy, defined similarly to the usual arithmetic hierarchy. Definition 11.2.R & ORDk X (OmoRD) is y - X n iff there is a p.0.r. relation P and an alternating sequence 30, < y ... QnPn < y of ordinal quantifiers such that for all a and f:R(a ,f ) iff 3p1 < y ... Q, 6, < y .P(fl, a , f);y -n, and 7 - En are defined analogously. These y-arithmetic classes are comparable to the usual arithmetic hierarchy for sufficiently regular y. Proposition 11.3. (a) For any f,any y recursively regular in f,and any partial function F, F is y-recursive in f if graph (fl is 7 - C in f ;

D. CENZER

252

(b) For any regular cardinal K (or K = -) and any partial functional F, F is K recursive iff graph ( F ) is K - X 1 . We need a notion of relative n-stability.

Definition 11.4. a is n ---stable (RS"(a,P))iff RR(@ and for all relationsR and all t < a , 3 y < @ . R ( t , y ) + 3 y < a . R ( t , y ) . Lemma 11.5. For all n

-

En

> 0 , RS" is p.0.r.

Proof. This is an easy application of Propositions 3.14 and 3.15. In contrast to Lemma 11.5 is the following result.

Lemma 11.6. For all n, Sn+l is not -recursive in S". Proof. If S"" were 00-recursive in S", then least a! .Sn+l(a)would also be 03-recursive in S n ,contradicting its n +I-stability. We can now prove an ordinal arithmetic "Hierarchy Theorem".

Theorem 1 1.7. For all n > 0, (a) for all a, S"(a) i f f for any 00 - X,, relation R and any t < a, 3 0 . R ( t , P) 30 < a tR(t,P); (b) for any n-stable ordinal P and any a! < (3, Sn(a)iff RSn(a,p); ( c ) S n i s m - n , b u t n o t m - X n'. in S n . (d) for any R E ORDk, R is 00 - Zn+l i f f R is 00 ++

Proof. Let n = 1; for n > 1 the proof is similar but more involved. (a) This follows from Proposition 11.3(a). (b) This is immediate from (a). ( c ) sl(a) iff V ~ V U V r< atla < w [ ~ ' ( u , o,(a, r,y)) 3 y < a ] ;ifS1 were also m - E l , it would be.-recursive, contradicting Lemma 11.6. (d)(+) 3PVy.P(P,y,a)iff 3P30[S1(a) A ( a , P ) < o A V y < u , P ( P , y , a ) ] . (+) 3 ~ {a)_(@, . a,S1) N _ 1 iff 303to[~l(o)A ( a ,P ) < u A Tl(o,o,(a,P,a,l),X~. RS1(7,u))], whichism- E2 since by(c)S' i s m - l l , (We identify S" and RS" with their characteristic functions for the sake of simplicity.)

253


We can relativize Theorem 1 1.7 (d) to large ordinals.

Proposition 1 1.8. For all n > 0, all ordinals a such that a is a limit of n-stables, and all R 5 ORDk, R is a - &+, i f f R is a - XI in Sn. In particular, R is H, - Z2 iff R is H, - El in S. Although the S n are not w-nncomplete (for example, since for any 00recursive f and any m , f (m)< 6 and is therefore not stable, the only A w reducible to S by an m-recursive function is the empty set), they play the role in Theorem 11.7 of the n’th “jump” of 4. We can think of stability as a jump operator in the following sense (proven as I1.7(c)).

s

Proposition 11.9.ForanyA C O R D , { a :cuisstableinA}ism-llI not 00- ZIin A.

i n A but

For the remainder of the section we discuss S’ or S for short.

Definition 11.10. (a) E ( a , f )iff a is recursively regular in f ; (b) a is inaccessibly stable (IS (a))iff E ( a , S ) and a is a limit of stable ordinals. Proposition I I .I I . (a) IS (a) iff= (a,S ) A a = 6,; (b) is p.0.r. and I S is p.0.r. in S; (c) for all a, 1,6 is recursively regular in S; (d) a stable in S implies a is recursively regular in S.

-

Proof. (a), (b), and (d) are similar to results on RR, RI, and stability. To prove (c), notice that by Theorem 11.7 (b), S(p) p = 6, v RS’( p, 6,) for so that S 16, is weakly 6,+1 -recursive; (c) now follows by the < regularity of a+,., It is clear that Sf must be inaccessible stable, but as 6 s need not be countable (see 6 13), we need something else.to construct a countable inaccessible stable ordinal.

Lemma 11-12.For all ordinals 0, S(p) i f f for all a < F ( a ) # 13.

and all -recursive F,

D. CENZER

254

Proof. By Proposition 4.10,Dp = { F ( a ) : a< A F is w-recursive} is a countable initial segment of ORD such that Sup (Dp)is the least stable ordinal greater than or equal to 0. The lemma follows directly from this fact. Proposition 11.13. There are countable inaccessibly stables. Proof. The least ordinal which is not w-recursive in S is clearly regular in S and will be stable by Lemma 11.12. We are interested in ordinals much larger than the first inaccessibly stable because of the following result, which is parallel to Proposition 4.16.

Proposition 11.14. If a! is any of the following: 6 the least inaccessibly stable ordinal, the least hyper-inaccessibly stable ordinal, then a is a* -recursive in S. Proof. For example, 6, = least a < 6,. S(a). n

For any a, let a*" = cr*...*_ Definition 11.15.0, = least 0.0is not p*n-recursive in S. It is clear that the 0, exist and are less than the least ordinal not wrecursive in S, and therefore countable. The 0, are large with respect to stability as the yn of 94 are with respect to regularity.

Proposition 11.16. For all n > O,p, is inaccessibly stable. Proof. For any a < Pn, each aiis a;'-recursive in S and therefore 0;"recursive in S ; any w-recursive function F is equivalent on pn to a pi"recursive function Fo by the stability ofpi". By the definition of &, F ( a ) = F,,(a) 0., Hence by Lemma 11,12,0, is stable. The proof that 0, is regular in S and is a limit of stables is parallel to the proof of Proposition 4.18 (a).

+

We can characterize the 0, with a proposition similar to Proposition 4.21. We state our result for n = 1.

255


Proposition 11.17, pl is the least ordinal such that for all a < w , {a}p*(P,S)J. 3a.l.

12. Zi inductive definitions In this section we present the following theorems.

Theorem 12.1. For all n > 0, (a) I(Z12>" I = fl, ; (b) for all A C_ o, A E (Ei)"iffA is &,-semirecursive. Theorem 12.2. I Ili I is the least ordinal 0such that for all a < w , if V a < p . {a),*(a,s)J, then {aIp,(P,S)J.. Theorem 12.3. For all A C a,A E

-

"i iff'A is III; I-semirecursive in S.

Theorem 1 2 . 4 . ( a ) ~ p e c t r u m ( ~ i ) ={ a : a < ~ ~ } ; (b) Spectrum (ni)= { a < IIIi 1 : a is a*-recursive in S } .

i

implies I I' I Iol. For any E operator I', We begin by showing that I' E we have by Proposition 10.1 a p.0.r. F so that m E F(A) iff 3a .F(a,m, xA) = 1 iff 3a < Sf F(a,m, xA) N 1. Let I be defined from F as in Definition 6.2. Parallel to Lemma 6.3, we have

.

Lemma 12.5.(a)Forall manda,xr,(m)Yf(a,m,SLY); (b) for all a, I Now for any inaccessibly stable ordinal fl and any m: m E iff 3a 1 is straightforward except for the construction of an operator T, with IT, I = 0,. We need the following lemma, a fairly difficult corollary to the Uniformization Theorem ( 1 0.4). (See Cenzer 171 for a proof.)

(c)

r=

Lemma 12.10. There is a

relation L such that for all A

5w ,


257

{(u,u) : L2(u,u,A)}isa well-ordenngof type :6 Let ri+l(A) = { 0, there is an index s, such that for all p and all limit ordinals r 2 , any Xi - L relation Q, any m and any constructibleA C w , 3 @ € L.Q(m,+,A) i f f 3$(6is Ah-Lin A ) .Q(m,4,A). Parallel to Proposition 10.3, we have

Proposition 14.6. For aN n 2 1 and all constructible A

E o,&,-A = (I~,,+,-A)~

Theorem 11.7 can be relativized directly to n - Hf -stability and Hfi - L, relations. We leave the details to the reader.

Definition 14.7. For any n 2 1 , un is the least ordinal u which is not %"(a)recursive in Sr. Notice that u1 is the ordinal p1 defined in 5 1 1. The following results are proven as in 8 1 1.

D. CENZER

262

Proposition 14.8.For all n 2 1, (a) S:(o,) and un is afixed point of the n - tsf stables; (b) un is the least ordinal u such that u is ntl-scn(u)-stable; (c) u,, is the least ordinal u such that for all a < w , {a)sc n ( L l ) ( u , s ? ) J

+

-

37.< 1s {a)scn(T)(7,s;)J.

We can now prove the main result of the section. Theorem 14.9.Foralln>_l,1X~+11=u, Sketch of proof: Let n = 2 for simplicity.

(9For any Xi - L operator F, we have by Proposition 14.4 a p.0.r.

functional F w i t h rg(F) G {0,1} such that for all m , A : m E r ( A ) iff 3 ~ < 6 , - A .Vp_ 1, (a,scn(a)) is X n ( l l n ) reflecting iff for any q n , ) f0rmula a,Lscn(,)t= @[a1 30< L S C " ( @t= )@[PI.

+.

.


263

Parallel to Proposition 13.6 we have

Proposition 14.14. For all n 2 1, 1 (a) I En+l - L 1 is the least ordinal a such that (a, scn(a))is Xn+l-reflecting; (b) - L I is the least ordinal a such that (a, scn(a))is IIn+l-reflecting. The results of this section can be extended in an obvious fashion to obtain characterizations for l ( X A -L)kl and -L)kl

References [ 11 S. Aanderaa, thisvolume. [ 21 P. Aczel, Representability in some systems of secon xder arithmetic, Israel J . Math. 8 (1970) 309-328. [ 31 P. Aczel and W. Richter, Inductive definitions and analogues of large cardinals, Proc. Conf. Math. Logic London 70, Springer Lecture Notes #255. [ 4 ] P. Aczel and W. Richter, this volume. [ 5 ] J.W.Addison, Some consequences of the axiom of constructibility, Fund. Math. 46 (1959) 337-357. [6] J . Barwise, R.O. Candy and Y.N. Moschovakis, The next admissible set, J . Symbolic Logic 36 (1971) 108-120. [ 7 ] D. Cenzer, Ordinal recursion and inductive definitions, Ph.D. Thesis, University of Michigan, 1972. [ 81 D. Cenzer, Analytic inductive definitions, to appear. [ 9 ] K. Godel, The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory (Princeton Univ. Press, Princeton, 1958). [ 101 R.B. Jensen and R.M. Solovay, Some applications of almost disjoint sets, in: Y. Bar-Hillel (ed.) Mathematical Logic and Foundations of Set Theory (NorthHolland, Amsterdam, 1970) pp. 88-104. [ 111 S.C. Kleene, Recursive functionals and quantifiers of finite types, I, Trans. Amer. Math. SOC.91 (1959) 1-52. [ 121 S.C. Kleene, Recursive functionals and quantifiers of finite types, 11, Trans. Amer. Math. SOC.108 (1963) 106-142. [ 131 M. Kondo, Sur I'uniformization des complementaires analytiques et les ensembles projectifs dela seconde classe, Japanese J . Math. 15 (1938) 197-230. [ 141 G. Kreisel and G . Sacks, Metarecursive sets, J. Symbolic Logic 3 0 (1965) 318-338. [ 151 S. Kripke, Transfinite recursion, constructible sets, and analogues of large cardinals, in: Lecture notes prepared in connection with the Summer Institute on Axiomatic Set Theory held at UCLA, July-August, 1967. [ 161 A . Levy, A hierarchy of formulas in set theory, Mem. Amer. Math. SOC.No. 57, 1965. [ 171 R.A. Platek, Foundations of recursion theory, Ph.D. Thesis, Stanford University, 1966.

D. CENZER

264

[ 181 H. Putnam, On hierarchies and systems of notations, Roc. Amer. Math. SOC.15

(1964) 44-50.

[ 191 W. Richter, Recursively Mahlo ordinals and inductive definitions, in: R.O. Gandy

[ 201

[21] [ 221

[23] [24]

and C.E.M. Yates (eds.) Logic Colloquium '69 (North-Holland, Amsterdam, 197 1) pp. 273-288. H. Rogers, Theory of Recursive Functions and Effective Computability (McCrawHill, New York, 1967). J.R. Shoenfield, The problem of predicativity, in: Y. Bar-Hillel (ed.) Essays on the Foundations of Mathematics (The Magnes Press, Jerusalem, 1961 and NorthHolland, Amsterdam, 1962). J.R. Shoenfield, Mathematical Logic (Addison-Wesley, Reading, Mass., 1967). C. Spector, Recursive well-orderings, J. Symbolic Logic 20 (1955) 151 -163. C. Spector, Inductively defined sets of natural numbers, in: Infinitistic Methods (Pergamon, Oxford, 1961) pp. 97-102.

J.E.Fenstad, P.G. Hinman (eds.). Generalized Recursion Theory @ North-Holland Pu bl. Comp., I 9 74

INDUCTIVE DEFINITIONS Robin 0. CANDY Mathematical Institute, Oxford University

$ 0 . Introduction Mathematical Logic is certainly permeated with inductive definitions. Here are some examples of concepts which are usually or readily defined in this way. In syntax; the notions of expression, well-formedformula, proox theorem. In semantics and model theory; the satisfaction relation, validity, Morley’s notion of rank. In set theory; well-founded set, ordinal, constructible set, the forcing relation, Bore1 set. In recursion theory the question is rather: are there any fundamental notions which are not inductively defined? All this suggests that a study of inductive definitions in general should produce interesting and applicable results. Of course it could be that it is always particular features of the definitions which are significant, so that a general study will only yield trivial results. But in fact this is not the case. One example is Barwise’s completeness and compactness theorems: the theorems are consequences of the form of the inductive definition of derivation, not of its particular details. (This is discussed in 5 2 below.) Another example is Moschovakis’,notion of hyper-projective;the original definition was by rather elaborate schemata. Under the new title hyper-elementaryMoschovakis (in 141) has reworked the material as a study of first-order positive inductive definitions; the proofs are more general, shorter, and more transparent. A final example is the study of extended systems of notations for ordinals which flourished in the 1950’s. The authors (no names, no pack-drill!) were often at pains to verify, case by weary case, that their systems of notation had certain simple properties (e.g. of belonging to A;). But this verification was quite unnecessary; all that was needed was to observe that the definitions were built up using arithmetical (not necessarily monotonic) clauses, and then to apply a trivial theorem about such definitions. 265

266

R.O. GANDY

General studies, then, are worth pursuing. And once this has been accepted, it would be unreasonably Draconian to de-ny them autonomy. The view taken here is that inductive definitions are interesting in their own right. Of course we are also interested in applications; but we do not have to back up each line of enquiry with a promise of applicability. My original intention was to give in this paper a fairly systematic account of first-order inductive definitions on an admissible set. But the recent work of Moschovakis [4], Aczel [ 11 and Barwise (this volume) made my account obsolete. So what is presented here consists, in effect, of remarks and reflections. In $ 1, I discuss the various methods which have been used to investigate certain particular classes of inductive definitions. Section $2 represents the residue of the first draft; it describes the method of semantic tableau which may have further uses. The results of 2.3. are certainly, the result of 2.4. possibly, more expeditiously proved by other methods. The results of 2.5. about I l l positive inductive definitions are new; but it is not clear if that class is significant. In $3 an effort is made to present Kleene's theory of recursion in a type 2 object as a branch of inductive theory. In so far as the effort is successful (when 'E is recursive in F') it gives a clear indication (already discussed by Aczel in [ 11) of how to set up the theory for structures other than the natural numbers. There is some inconclusive discussion of the contrary case. In $4 I draw attention to some of the problems which were ignored in $ 1-3, and make propaganda for the investigation of the forms of inductive definition which occur in proof theory. (This propaganda is directed as much at recursion-theorists as it is at proof-theorists.)

5 1. Preliminaries and a discussion of methods 1.1. Let 'u be an arbitrary first-order structure with domain A . An n-place inductive operator CP is a map P(nA) + PC.4) (P for power-set, n A for A X ... XA). For simplicity we shall always suppose that @ isprogressive, i.e., R C CPR;if not, replace @ by a', where CP'R = R U CPR. The a-th iterate, CPQ(Ro),of @ applied to R , is defined by

INDUCTIVE DEFINITIONS

267

If R, = @ we write simply CPa. The closure ordinal 1 @(Ro) I of @ from R, is defined by

The closure, @-(I?,) of CP fromR, is simply CP"(Ro) where a = ICP(Ro)(.If a E aa(R0),the stage I a I a(Ro) of a is defined by

It is often convenient to set ( a on we supposeR, = 6.

= I@(R,) I for a 4 (Pm(Ro).From now

1.2. We shall be interested in these things as CP ranges over some given collection C. So we define

The class Cm of C-fixed points is defined by Cm = interst is the class C(") of C-inductive relations:

{ap" : CP E C}. Of greater

(where R is m-place and CP is n + m-place). A relation is C-co-inductive ('E C(-w)') iff its complement is C-inductive. It is C-bi-inductive ('E @('=)') iff it is both C-inductive and eco-inductive. 1.3. One is normally interested in inductive operators which can be defined in some language L. Naturally L must contain variables ranging over A , and a relation symbol R . We do not automatically assume that L contains '=', nor that identity is a relation of%. Then CP is defined by a formula cp(x,d) of L (which shall not contain any free variables other than x (= xl,...,x,)) iff (1)

CPR =

{a E "A : (%,

...,R) k cp(ii,k)};

here ._. indicates whatever enlargement of % is necessary to give a realisation of L. We adopt the convention that cp, @; $, 9;..., are always related as in (1).

268

R.O. GANDY

Note that this useful convention may sometimes conceal the true nature of @. For example, if

where R is one-place and $ is quantifier-free and T is a term then the corresponding cp is not quantifier-free. In what follows, however, we shall mostly be concerned with rather broad classes C for which this difficulty does not arise, and we shall use syntactic classes C of formulae to characterise the corresponding classes of inductive operators. If 9is a class of formulae we shall be particularly concerned with the class 9+ of inductive operators defined by formulae of 7 in which R occurs only positively, and the class T m of inductive operators which are defined by formulae of 9and which are also monotonic; i.e. which satisfy R C S + @RC W. There are two classic cases; much of the recent work on inductive definitions stems from trying to understand them and to generalise them. In both, the underlying structure is % = IN, O,S,=).

(R) (Post-Smullyan). If (? lies in the range from Rud + to Zym then (?("I = Xy and (?('m) = Recursive. (Here Rud is Smullyan's 'rudimentary' (= constructive arithmetic)). (H) (Kleene-Spector). If lies in the range from ny+to ll:m then e(-)= Iii and = Hyperarithmetic. 1.4. The standard problems of inductive theory for a given a, C are to determine I C? ),to characterise (?("), to determine the closure properties of (?(-) and (?(""), and to uncover any additional structure which these sets may have. Further problems arise from relativisation - that is by considering @ which depend on parameters. The methods which have been most used may be summarised as follows. 1.4.1. Direct methods. One proceeds by constructing particular inductive definitions. A good example is the definition and use of 0 in hyperarithmetic theory. This way of proceeding appeals naturally to the purist. For many recursion theorists it also has a psychological attraction: there is a pleasure in working out the details of an intricate recursion which is akin to the pleasure of constructing a tangible object. And for investigating the fine structure of

INDUCTIVE DEFINITIONS

269

C(”)i t seems to be the only method available. In classifying the r.e. sets, for example, one has actually to construct simple sets, maximal sets, and so on, in order to prove their existence and discover their properties. 1.4.2. Use of higher lype recursion. In case (H) it is possible to consider

as the class of relations which are recursive in the jump operator. This was shown by Kleene in [ I ] . The first study of the generalisation of (H) (by Moschovakis in [ 1-31) was based on this method. But it has turned out that the results are more easily obtained by other methods (in particular 1.4.1. and 1.4.4.).

1.4.3. 7’he use of normal forms. In case (H), for example, many of the closure properties of C(”) and d’”) and a certain amount of the structure of these classes can be most easily derived from the fact that C(”) = ll:. Recently (in [4]) Moschovakis has obtained a normal form for the generalisation of case (H) to arbitrary structures by using a ‘game quantifier’ (the idea underlying this is discussed in $2.3 below). But I think it would be a mistake to place too much reliance on this method. For I believe that the future development of the subject will be concerned with finer classes C. These will not have the sort of broad and simple syntactic characterisation of the classes so far considered; i t is not to be expected then that C(-) will have a simple syntactic form. 1.4.4. The method of embedding. We can enlarge a first order structure 91 to a structure (91 ,S,E), where S is a subset of the cumulative hierarchy of types

VA formed with the elements o f A as individuals (urelements).Le., VA = u(V; : a E On} where V; = u { P( V p UA ) : 1.3 0 u p ( p * n )= u,(p)

* vsn ;

(Id) if u,(p) is a point of T at whichy E X stands Up(P

* 1) = U , ( P ) * P(Y)

;

( l e ) in all other cases

We consider a tree I'whose infinite branches correspond to different choices of p . A point p ( p ) of this tree is determined by the values of p up to and including the value at p :

We say the branch p is secured at p if u,(p) is a tip of T. Let

I" = { p ( p ) : (Vq O: aEF"(X)}

1.4. Theorem (LCvy). f f n > 0 then Hn+l(Rg) CH,f(Rg), (Hf)A(Rg), etc.

ADMISSIBLE ORDINALS

309

Let us now turn to the strongly indescribable cardinals. These are defined using reflecting properties of the cumulative hierarchy of sets. Let R(a) = UB 0 an ordinal is strongly n; (Ck)-indescribable if and only if it is strongly inaccessible and isn; (EL)-indescribable. So, assuming the GCH, the two notions coincide when n > 0. Let L, be the set.of constructible sets of order < a, (i.e. La= UB,.Def(Lp) where Def(x) is the set of subsets of x definable in (x, E 1x, I I ) ~ ~ ~ ) .

1.6. Definition. L, reflects cpon X if

La reflects cp if L, reflects cp on On. If this definition is used as in Definition 1.5 the resulting indescribability notions may easily be seen to coincide with those of Definition 1.1. In order to obtain the classes of ordinals that we are interested in we restrict the language L. Let -C, be the sublanguage of 6: obtained by only allowing E as a relation symbol. 1.7. Definition. a i s n ; @“,-reflecting sentence of L,.

n; (c;)

[on X ] if L, reflects [on X ] every

Some properties of this definition are summarised in the following theorems, which should be compared with Theorems 1.2 and 1.3.

31 0

W. RICHTER and P. ACZEL

1.8. 'I'heorem. a is Il!-reflecting iff a is an admissible ordinal > a.

This result and Theorem 1.9 below w d be proved in 5 2. Let Ad = {a > w : a is admissible}. a E Ad is recursively Mahlo if for every a-recursive function f : a -+ a there is an ordinal 0 > 0 closed underf such that (3 E X n a. 1.9. Theorem. (i) The following are equivalent a) a is n:-repecting on x b) a' is Z!-regecting on X c)(~=sup(~na). a is recursively Mahlo on X. (ii) N is n$reflecting on X

--

(iii) is n:-rejlecting on x a is I:,O+l-reflectingon X. (iv) If n > 0 or m > 2 ( n > Oor m > 3 ) then a isn; (Z",-reflecting X -a isnk (Zk)-reflectingonX 17 Ad.

on

As i t is often easier t o work with ordinals rather than the constructible hierarchy the following characterisations will be useful. Let L, be the sublanguage of C that has relation symbols only for the primitive recursive relations on sets (see [8] fo1 the properties of this notion).

1 .lo. Theorem. a isn; (z:k)-reflecting [on X I i f and only i f a reflects [onX ] every n:, (c;)sentence of I,. The pIiniitive recursive relations in the language L, are needed for reflecting propeities on ordinals in order to compensate for the richness of the € relation for reflecting properties on the constructible hierarchy. Theorem 1.10 will be proved in 33. Lh(Ad) is the class of )\-recursively inaccessible ordinals, while if RM(X) = (a E X : a is recursively Mahlo on X} then RMX(Ad) is the class of )\-recurswely Mahlo ordinals. Let M,(X) = { a E X : a is @-reflecting on X } . Then M , = M , = L and M 2 = KM. The next result indicates the relative magnitndes of rhe vrdinals inM,l(Ad) and should be compared with Theorem 1.4. I.ll.Theorem.Zfn> Othen

31 1

ADMISSIBLE ORDINALS

T h s will be proved in $4. 1.12. Definition. Let

.”,(0;)

be the least lI; (Z”,-reflecting

ordinal.

By 1.9 n$ = IT: = w and $ = w1 are the recursive analogues of the first two regular cardinals. What can we say about T!? By 1.9 and 1.11 7r! is greater than the least recursively Mahlo ordinal, the least recursively hyperMahlo ordinal etc. In fact n! appears to be greater than any “reasonable” iteration into the transfinite of this diagonalisation process. When one thinks of a corresponding cardinal in set theory (with “recursively Mahlo” now replaced by “Mahlo”) the cardinal which comes to mind is the least niindescribable cardinal. We shall now try and justify the view that n:-reflection is the recursive analogue of H:-indescribability. The same ideas with some additional notational complexity provide an analogy between n:+2-reflection and nA-indescribability for all n > 0, but we shall concentrate on the case n = 1. The analogy is obtained as follows. A class of cardinals, called the 2-regular cardinals, is defined, as well as a recursive analogue of this class whose members are called 2-admissible. We then show that a cardinal is 2-regular if and only if it is strongly lI -indescribable, and an ordinal is 2-admissible if and only if it is @-reflecting. Certain properties of infinity can be stated in terms of f x e d points of operations. For example K > w and K is regular if and only if: (1) for every f : K + K there is some 0 < a < K such that f ’faC a. (We say a is a witness for f.) If we modify (1) by requiring that the witness be regular, we obtain the Mahlo cardinals, etc. An alternative way of modifying (1) is by using higher type operations on K . Let F : K~ .+ K ~ F . is K-bounded if for every f : K + K and 8 < K , the value F ( f ) ( l )is determined by less than K values off. More precisely, F is K-bounded if

i

V f 3 Y < Kvgk 1Y = fE Y * F ( f ) ( U

0 < a < K is a witness f o r F if for every f : K

f “aC a * FCf)“a C a.

+K,

= F(g)(t)l .

312

1.13. Definition. K witness.


> 0 is 2-regular if every K-bounded F :

1.14. Theorem. K is 2-regular iff

K

K~

-+ K~

has a

is strongly IIi-indescribable.

We now look at a recursive analogue of 2-regularity. Roughly speaking the following definition of 2-admissible is obtained by replacing in the definition of 2-regular, “bounded” by “recursive” and the functions by their Godel numbers. In the following definition we write {t}K: K -+ K to mean that {,$}K is total on K .

1.15. Definition. (i) Let K E Ad and [ < K . to K -recursive functions if

{ t } Kmaps K-recursivefunctions

(ii) Suppose {E}, maps K-recursive functions to K-recursive functions. a € K n Ad is a witness for .$ if t < a and {t}, maps a-recursive functions to a-recursive functions. (iii) K E Ad is 2-admissible if every t; < K such that {t}Kmaps K-recursive functions to K-recursive functions has a witness.

1.16. Theorem. K is 2-admissible iff

K

is n $reflecting.

Theorems I . 14 and 1.16 will be proved in 3 5 . Certain classes of ordinals, defined in terms of reflecting properties, also have characterisations in terms of stability properties. LetA <x; B ifA and cp for every Zy sentence B are transitive sets such that A S B and B cp * A cp of XE that only has constants for elements o f A . Kripke has defined the notion of an ordinal Q beingo-stable (see [lo]). His definition used his systems of equations for defining recursion on ordinals. For admissible fi he gave the following characterisation, which we shall take as a definition:

+

1.17. Definition. Q is fl-stable if a < /3 and L,

icy

Lo.

When 0 is not admissible, this notion may well diverge from Kripke’s original one.

ADMISSIBLE ORDINALS

31 3

1.18. Theorem. a is JI:-reflecting ifand only i f a is at1-stable. 1.19. Theorem.For countable a, a is ni-reflecting ifand only ifa is a+-stable, where a+ is the first admissible ordinal > a.

These results will be proved in $6. Given A C nON all of our definitions and results wdl relativise to A . As we shall need the relativisations in Part I1 we spell out exactly what this means. Definition 1.6 is relativised by using (La [ A ]: a E ON) instead of (L, : a € ON). Here L,[A] = Up.,,DefA(Lp[A]) where DefA(x) is the set of subsets of x definable in (x, E bx,A f'x,a)aEx.The language L, must be replaced by the language &(A) which is L, with an added n-ary relation symbol to denote A . Definition 1.7 becomes: a is n",A)-reflecting [on X ] if L,[A] reflects [on XI every nk sentence of &(A). Similarly for X k ( A ) reflecting. Theorems 1.8 and 1.9 relativise in the obvious way. Ad must be replaced by Ad ( A ) = {a> w I a is admissible relative to A b a}. The language L p ( A ) is defined by allowing relation symbols for all relations primitive recursive in A . Most of the proofs relativise in a routine way.

5 2. Elementary facts In order to prove our theorems we shall need to assume some familiarity with the notions of primitive recursive set function; admissible class, admissible ordinal and ordinal recursion on an admissible ordinal. We shall use [8] as our basic reference and will usually follow the terminology they use. We shall also need to refer to [6] when we use Jensen's notion of a rudimentary set function. The notion of a primitive recursive function with domain M has been mON and mV. As shown in [8] all formulated for various classes M e.g. these notions turn out to be special cases of the following: F : M + V is primitive recursive if M is a primitive recursive function with domainM has been to M of a primitive recursive set function. In [8] a transitive prim closed class M is defined to be admissible i f M satisfies the X :-collection principle (there called Ly-reflection principle) which we shall formulate as follows: For every prenex C formula 0 of L, if M I= Vx E a0 then

:


3 14

+

Vx EaOb for some b E M , where i f 0 is 3 y 1 ... 3yk\k, with 9 E :, then O b i s 3 y 1 E b ... 3 y k E b q . We shall find it more useful to use the characterisation in [ 6 ] .

M

2.1. Definition. The transitive class M is admissible if M is rud closed and satisfies c:-collection. This definition is relativized by replacing Cy-collection by Cy(A)-collection, obtained by using L,(A) instead of L, ,and adding the condition that a€M*A naEM. A relation R on a transitive set M is Zy on M if R is defined on M by a C y formula of L,. A partial function with arguments and values in M is Ey on M if its graph is. We shall assume some familiarity with the closure properties of these relations and functions on an admissible M , as presented for example in [ 81. In particular we shall need the following: 2.2. Proposition (Definition by Ey-recursion). Let M be an admissible set. Let C b e a function such that G E M : M X M - + M a n d G PMis 72: onM. Let

r

Then F M : M -+ M and F 1 M is Zy on M. Moreover the Zy definition of F r M depends only on the .E definition o f G 1M (and not on M ) .

y

Usually we will only be interested in F r M n ON. For the notion of an admissible ordinal a and a-recursion we shall follow [ 81. An ordinal a is admissible if L, is admissible. f : na + a is a-recursive if it is on L,. The following lemma will be useful and the proof will illustrate some of the techniques of a-recursion.

~7

2.3. Lemma. If a > o is an admissible ordinal and f : a + a is a-recursive then there are arbitrarily large limit ordinals < a that are closed under fProof. Let a > w be admissible and let f : a + a be a recursive. Define g : a -+ a by g ( x ) = Max (xt 1, Supylxf(y)). Then g is a-recursive, x < g ( x ) and f ( x ) < g ( y ) for x < y < a. Given yo < a let yn = gn(yo). Then

ADMISSIBLE ORDINALS

yo < y1 < ... < a and x 27, * f ( x ) 5 Y,+~. Let y = Sup,, is a limit ordinal such that yo < y and y is closed underf as

315

7,.Then y Ia

So it only remains to show that y < a. For this we need 2.2. Let F ( x ) = G(x, F 1x) where G(x,y) = g(z) if x is a successor ordinal,y is a function such thaty(x-1) is defined with value z < a, and G(x,y) = yo otherwise. Then it is not hard to see that y, = F(n) for each n € w , and that as G L, : L, X L, + L, and is Xy on L, it follows that F I' a is a-recursive and hence y = Sup,, y, = Sup,< F(n) < a.

r

Proof of Theorem 1.8. Let a be KI:-reflecting. If a < a then La k l ( a €a). Hence there is a P < a such that Lp k l ( a €a); i.e. a w. Using Lemma 6 of [6] it is not hard to show that L, is rud closed for any limit ordinal a. Hence it remains only to show that L, satisfies Ey-collection. So let L, i=Vx E a 0 where 0 is a Xy formula of L,. Then by II$reflection there is a B < a such that Lp k V x E a 0 . Now if b = Lp € L, then La t= Vx € a @ as required. Conversely, let a > w be admissible, and let cp be a @ sentence of L, such that L, k cp. We may assume that cp has the form Vxl ... x, 3 y l ...y , ?Ir where ?Ir is Zoo. Hence L, Vx, ... x, 3y0 where 0 is the E! formula 3y1 E y _..gym € y + . For simplicity we shall just consider the case when n = l . I f p < a a n d a = L p thenL, k V x l E a 3 y 0 . H e n c e byzy-collection Vxl E a 3y E b0. But b C L, for some there is a b E L, such that L, 7 < a so that Vx, E a 3y E L,0. Let f(0) be the least such 7 < a. Then f. : a + a is a-recursive. Let Po < a such that every constant of 0 occurs in Lp,. Then by the Lemma 2.3 choose a limit ordinal (3 such that flo < 0 < a and (3 is closed under f.Then we must have Lp Vx, 3yB so that a reflects the @ sentence cp. In order to prove (iv) of Theorem 1.9 we shall need

+

2.4. Theorem. There is a KIg- sentence uo of L, such that the transitive class


316

M is admissible ifand only i f M

0,.

Proof. By Lemma 6 of [6] there are binary rud functions F,, ..., F, such that the classM is rud closed if and only if it is closed under F,, ..., F,. By Lemma 2 of [6] there are Xt- formulae cpi(x,y,z) of L, that define the graphs of F, for i 5 8. So M is rud closed if and only i f M k 0, where O o is the Il!- sentence /Ii w . (ii) a, b < a *a+b < a. (iii) b < a =$ 2b 0 (i) For each n; sentence 8 of Lp there is a Ilk sentence OE of that for admissible a

L, such

(ii) For each l3; sentence 8 of LE there is a n ; sentence O p of Lp such that for admissible a

Using this lemma let us conclude the proof of Theorem 1.10. Let n > 0 or m > 2 and let a benh-reflecting on X . Let 8 be a n ; sentence of Lp such that a F 8. Then 0, is a n; sentence of L, such that La F OE as a is admissible. Hence there is a 0E X n a such that Lo OE. As X C Ad, /3 is admissible so that 0 F 8 . Hence L, reflects 8 . Similarly if ( n > 0 or m > 3) and a is E; reflecting on X and 8 is a E; sentence of Lp then 1 8 is a n ; sentence of X p so that l(1O), is a X; sentence of LE and the argument is as above. The proof of the converse implications is exactly similar using (ii) of the lemma instead of (i).

+

hoof of Lemma 3.6. (i) By thestability Theorem 2.5 of [8] we may easily associate with each primitive recursive relationR a Xy- formula pR(xl,...,x,) of L, such that for admissible a and a l , ..., a, < a

Now let 8 be a sentence of L p . If 8 contains individual constants for sets that are not ordinals, then a F 8 can never hold, so let Og be ( 1 E 0). Otherwise define 8, as follows. First replace each constant for an ordinal 0 by a constant for N ( 0 ) . Then replace each occurrence of a relation symbol R(sl, ..., s,) in O by pR(sl, ..., s,). Then for admissible a it is clear that

ADMISSIBLE ORDINALS

32 1

Now if 8 isn; and n > 0 then 8, is alsoII2 and so we can let OE be 8 * . If 8 is n; ( m > 0) then we have to be more careful. We may assume that O is in prenex form. So it has the form of an alternating sequence of m blocks of universal and existential type 0 quantifiers followed by a II! formula \k(xl, ...,x k ) . Now \k(xl, ..., x k ) is built up from primitive recursive relations and ordinals using the boolean operations and restricted quantifiers. Hence there is a primitive recursive relation R and ordinals P1, ...,P1 such that for all (Y

" k W"1,

..., " k )

-"

i=R(P1, ..., P I , "1, ..', " k ) .

Now define OE as follows: If m is even, replace 9 ( x 1 , ..., x k ) in 8 by qR(N(P1), ..., N(P1),xl, ...,x k ) and if m is odd, replace \k(xl, -..,x k ) in 8 by 7 q ~ , ~ ( N ( ..., @ N(P1), ~ ) , xl, ..., x k ) . Then OE is IIk and has the desired properties. (ii) Let 8 be a sentence of L,. If 8 contains constants for non-constructible sets, then L, k 8 never holds so we can let 8, be (0 = 1). Otherwise define O0 as follows. First replace each individual constant for the set a by the constant for ordinal ar such that N(ol) = a. Then replace each occurrence of s E t in 8 by R,(s, t ) , where RE(a,P)e N ( a ) EN(@. (When proving the relativised version of 3.6 there may be occurrences of an atomic formulaA(sl, ..., sn). These must be replaced by R A ( s , , ..., sn) where RA is the relation primitive recursive inA such that RA(al,..., an)e A ( N A ( a l ) ,...,NA(a,2)).) Clearly for admissible ordinals a

Now if 8 i s n k with n > 0 then O 0 is also II; and hence we can let 8, be O O . If 8 isn; with m > 0 then we must again be more careful. We can assume that 8 is in prenex form with a sequence of quantifiers followed by a II! formula \k(xl, ..., x k ) . Now % determines a primitive recursive relation R and ordinals PI, ..., such that for all a

Now define 8, by replacing \ k ( x l , ..., x k ) in 8 by R(P1,..., P1, xl, ..., x k ) . Then O p is a rlk sentence of L, satisfying the lemma.

322


We conclude this section with a characterisation of admissible ordinals that will be useful in the appendix. We state it in relativised form.

3.7. Theorem. Let A be a relation on ordinals. The ordinal (3 is admissible relative to A (3 if and only if for all a < (3 and all R C ON that is primitive recursive in A if

then there is a < A < /3 such that V x < h 3 y < AR(a,x,y) Proof. Note that this characterisation uses a restricted form of KIs *-reflection. Hence it is only necessary to observe that this special form is sufficient for the proofs of 3.2 and 3.3.

54. The relative sizes of the first order reflecting ordinals In this section we shall need some m r e results about ordinal recursion on an admissible ordinal. Iff is a partial function on the admissible ordinal a then f is a-partial recursive if the graph off is definable on L, by a Xy formula of fE. As in Theorem 4.4 of [8] we may prove: 4.1. Normal Form Theorem. For each n 2 0 there is a primitive recursive relation T, and there is a primitive recursive function U such that if a is admissible and f is an n-aly a-partial recursive function then there is an e < a such that for a l , ..., a, < a

Moreover e depends only on a Zy formula of L, that defines the graph o f f on L,. If this formula contains no constants then e < a.e is called an a index o f f .

A D M I S S I B L E ORDINALS

323

Note the uniformity in this theorem. For example i t follows that if F : ON -+ ON is primitive recursive then there is an e < w such that F I‘ a is a-recursive with a-index e for all admissible ordinals a. Let us write {e},(al, ..., a,) for U(p,yT,(e, a l , ..., a,,y)). It will be useful to allow n = 0. The next result is a uniform generalisation of Kleene’s S - m - n theorem.

4.2. Theorem. For each m > 0 there is a primitive recursive function S , suck that for all admissible ordinals a if e, a l , ..., a,, al,..., a, < a then {el,(al, ---,a,, ~ 1--, , a,)= {S,(e,al, .-.,am)l,Ja1, --,a,>This theorem may be proved roughly as follows: Iff is an m t n - a r y apartial recursive function whose graph is defined by the x: formula O(x1, .*.,xrn,xm+l, ...>xm+n) on L, then for al ... a,< a Xa, ...or, f ( a l , ...,a,, al,...,a,) is also a-partial recursive, with graph defined by the C: formula O(al, ..., a,, x l , ...,x,) on La. Now , S is chosen so that if e is the index off determined by O(x,, ..., x,+,) then Sm(e,al,..., a,) is the index of ha,, ... ...,a,f(al ... a,, al _..a,*)determined by O(al, ..., a,, x l , ..., x,). We leave a detailed definition of S , as a primitive recursive function independent of a to the imagination of the reader. We now use Theorem 4.1 to define universal KI:+l

and E:+l

formulae of

1,. For each n 2 Olet Cl(xo,...,x,) be 3yT,(xo, ...,x , , y ) and let C,+l(xO, ..., x,) be 3yIl,(x0, ..., x, , y ) f o r m > 0, where KIm(xo, ..., x k ) is 1Xm(xO,..., xk). Clearly C,(xo, ..., x,) is a G,0 formula of L, and Il,(xo, ..., x,) is a n ,0 formula of L, for each m > 0, n 2 0. Let us call two formulae of 1, O,(xl, ..., x,), 02(x1,...,x,) equivalent on a i f f o r a l l e l , ...,a,.

4.3.Lemma.Let m > 0. Z f q ( x l ,...,x,) is a Ek- (n:-)-)ormula ofL, then there is an e < w such that q ( x l , ..., x,) and E,(e,xl, ...,x,)@l,(e,xl, ...,x,)) are equivalent on evely admissible ordinal. Proof. This is by induction on m. Note that then:

case follows from the


3 24

EL case by taking negations. I f m = 1 and e(xl, ...,x,) is a Ly- formula of L,, then, using the stability Theorem 2.5 of [8], we may find a Ey- formula cp(xl, ..., x,, x,+]) of LE such that for admissible a and a l , ...,a,, p < a

But cp(xl, ..., x , , ~ ) defines the graph of an a-partial recursive function on each admissible ordinal with index e < o independent of a. Hence if a is admissible and a l , ..., a, < a then

Hence U(xl, ...,xn) is equivalent to Xl(e,xl, ...,x,) on admissibles. Now suppose that the result has been proved form > 0 and let cp(x,, ..., xn) be Z:+]. Then we may assume that it has the form 0 3y ... 3yk U(y I , ...,y k ,x l , ...,x,) for some n, formula B(y ...,y k , XI,

‘..,X,).

Now let G be the graph of a primitive recursive function mapping k-tuples of ordinals one-one onto the ordinals. Then cp(xl, ..., x,) is equivalent on every admissible to 3y@’(xl,...,x,, y ) where U’(xl,..., x,, y ) is the rIL formula VYl

..’ VYk(G011, ...,YkJ)+eB(Y1,

.-.,.Yk > X I >...>X,))’

By induction hypothesis there is an e < w such that O‘(xl, ..., x,,y) is equivalent to ll,(e, x l , ..., x,, y ) on every admissible. Hence q(xl, ...,x,) is equivalent to Em+l(e,xl,...,xn) on every admissible. 4.4. Corollary. I f X

5 Ad then for n > 0

Proof. By Theorem 1.10 a E M n ( X ) if and only if a E X and for every n: sentence cp of L,, a cp * (3P E X n a)P cp. By Lemma 4.3 and Theorem

+

+

ADMISSIBLE ORDINALS

325

4.2 if cp is an: sentence of L, then there is an ordinal a such that cp is equivalent to n,(a)on every admissible. The corollary now follows when X C Ad. Below we shd be concerned with operators F on classes of ordinals that have the following properties. (i) F ( X ) C L(X) (ii) X C Y =$ F ( X ) E F(Y) (iii) h < a E F ( X ) 3 a € F ( X f l @ , a ] ) where(h,a] = { p : h < f l l a } . I t follows from (iii) that for all A 4.5.

F ( X ) E F ( X n( h , ~ U ] )( h + l ) where ( h , ~=](0 : h < p}. Examples of such F are L, M , H,, RM,M,. Moreover, if F has these properties, then so does F A for h > 0 and also F A . 4.6. Definition. If F satisfies (i)-(iii) above and n > 0, then F is JIE-preserving if there is a primitive recursive function f : ON + ON such that if X = {a E Ad : a k then

n,(a)}

a) F ( X ) = { a E Ad : a I= n,(f(a))} and b) M,(Ad) E X U p *M,(Ad) C F ( X ) U p for all p E ON.

0 4.7. Lemma. For n > 0,M, is II,+l-preserving.

+

Proof. If X = { a E Ad : a n,+,(a)} then by 4.4 a E M , ( X ) if and only if a E Ad & a I= [n,+,(a) & Vx 3y(n,(x) - R ( a , x , y ) ) ] where R is the primitive recursive relation such that R(a,b, 0)-0 k n,(b) & 0E Ad & /3 n,+,(a). So by 4.3 M J X ) = {CY E Ad : a l=n,+l(e,a)} for some e < w . Now i f f = h x S l ( e , x ) thenfis primitive recursive and

Now let M,+l(Ad) C X U p and let a €M,+,(Ad), We must show that a EM,(X) U p. If a < p, then we are done. Otherwise CY E X so that a E Ad and Q k n,+,(a). Now suppose that a i= n,(b).Then CY t= n,(b) A n,,,(a).

326


As a is flf+l-reflecting on Ad there is a 0E Ad n a such that 0 k n,(b) A KI,+](a). Hence P E X f~a and 0 I= Il,(b). Thus we have shown that a EM,(X). 4.8. Lemma. If F is HE-preserving, then so is F A

Proof. Let F be @-preserving and letf be a primitive recursive function such that F ( {a E Ad : a n,(a)}) = { a E Ad : a k n,(f(a))}. Our first aim is to find a primitive recursive function g such that for admissible a and a, c E ON

So let 8 ( x 1 , x 2 , x 3 )be the formula

where R = { ( u ,u) : f ( U ( u ) ) = u } is primitive recursive. Clearly this is n:- so that f?(x,, ~ 2 , ~ is3 equivalent ) on admissibles to KIn(eo,~ 1 , ~ 2 , for ~ 3some ) eo < w . By a uniform version of the second recursion theorem on admissible ordinals there is an e < w such that {e},(a,x) * S3(eo,e,e,x)for a,x < a and admissible a. Now l e t g = ha,xS3(e,,,e,a,x). Then on admissibles n,(g(a,c)) is equivalent to n,(eo,e,a,c) which is equivalent to O(e,a,c). Hence for admissible a

so that (1) is proved. Let F@’(X) = {a > p : a E FO(X)}

Our next aim is to show that for all 0 E ON

ADMISSIBLE ORDINALS

327

This will be proved by induction on /3. Let X = {aE Ad : a /=II,(a)). By induction hypothesis, for b < /3 < a

So that (2) is proved. Now we shall find a primitive recursive functionf' such that (3)

F A ( { &E Ad : a k n,(a)}) = { a E Ad : a kn,(f'(a))} .

L e t X = { a E A d : a kn,(a)}.The formulaVxVy[g(z,x)=y+nn(y)] i s a n:- formula so that there is an e l < w such that for admissible a and a E ON

But F A ( X ) C X C Ad so that F A ( X ) = { a E Ad : a k n,(f'(a))}where f' = h x S l ( e l , x ) . So (3) is proved. It now remains to show that if X = {a E Ad : a k n,(a)}and


328

M,(Ad) C X U p thenM,(Ad) E F A ( X ) U p. So let X,p satisfy the above assumptions. We first show that for all /3 E ON:

This will be proved by induction on /3. By induction hypothesis, if b < /3, then M,(Ad)

CF b ( X ) U Max ( p , b + 1) C F ( b ) ( X )U Max (p,b + 1) .

But as F isn:-preserving, M,(Ad)

by (2), if b < 0, then

C F(F(b)(X))U Max (p,b + 1) C F(Fb(X)) U Max ( p , B + 1)

by 4.5 (iii)

.

Hence

Hence (4) is proved and now if a €M,(Ad) then if a < p we are done. Otherso that a EFA(X)Up. ThusM,(Ad) C wise, by (4) Q € n,,.Fp(X) FA(X)u P. We can now prove Theorem 1.11. 0 -preserving, thenM,+l(Ad) C F(Ad) as Ad = {a E Ad : I f F is n,+, a n,(e,)} for some eo < o.Hence Theorem 1.1 1 follows from the previous two lemmas. 4.9. Remark. If Y is a primitive recursive class of ordinals such that Y C Ad

and Ad is replaced by Y in Definition 4.6, then the proofs of the previous two lemmas still hold so that we get that for n > 0:

ADMISSIBLE ORDINALS

329

55. Reflecting ordinals and indescribable cardinals In this section we will prove Theorems 1.14 and 1.16. 5.1. Lemma. If K is 2-regular then

K

> w and K is regular.

Proof. Let K be 2-regular. It suffices to show that every g : K + K has a witness. For a given g, let F : K~ + K~ be defined by F ( f ) (f)= g(f(0)) for all f : K + K and all t < K . F is clearly K-bounded. Let a be a witness for F. We show a is a w i t n e s s f o r g . L e t p < ~ a n d f : K + K such t h a t f ( f ) = p f o r a l l ( < ~ . T h e n f " a C a and hence F(f)"a C a. Thus

Hence g"a C a. 5.2. Proof of Theorem 1.14. K is 2-regular iff

We show

(a)

K

2-regular*

i

(b) & (c)

3

(d)

=+

(a).

K

is strongly n:-indescribable.

K

is strongly inaccessible

K

is Hf-indescribable

K

is strongly IIi-indescribable

We first show (a) =+ (b). Let K be 2-regular. Since K is regular it remains to s h o w h < ~* 2 ' < ~ . S u p p o s e n o t . LetX w 2 .

c

7.3. Remark. If R is L then we still have (1) and the first equivalence in (2) so that 7.2 still holds if Xy replaces n! everywhere. Note thatn!S [llt,ll!] C [II!,ll:,II:] C...I'Iy.Thusw<w2< w 3 < ... < I@l. We have

ADMISSIBLE ORDINALS

7.4. Theorem(Gandy).

I@I=

339

IZiI = ol.

As I Z! I 2 IXIy I 2 I ny-rnonl 2 w1 we have one half of the theorem. For the other half we will use the next two lemmas. These will also be used for getting upper bounds for other classes of first order inductive definitions.

7.5. Lemma. Let r En:. Then (I'E : r; < h ) is unifarmly E on L, for h E Ad. Hence for h E Ad ?l is Ey on L,.

fit

Proof. If h E Ad and x S o such that x E L, then r ( x ) is on L, as it is defined by a formula with quantifiers restricted to w < A. Hence r ( x ) E L, as r ( x ) C w. So if G ( x , y )= U { r ( y ' z n a): z E x ) then G L, : LAXL,+ L, Moreover G r L, is uniformly Ey on L, for h E Ad. Let F ( x ) = G ( x , F 1x ) . Then by 2.2 F r LA: L, + L, and is uniformly Ey on L,. By an easy induction we see that I'E = F ( [ ) for all t E ON, so that (I'E : t < h ) = F h is uniformly Xy on L, for h E Ad. Hence I" is Ey on L, as x E r CJ

r

( 3< ~ X ) X E rs.

7.6. Lemma. Let (I'E : r; < h ) be Ey on L, where h E Ad. Let R be recursive. Then < rE) . V x R ( n , x ,FA)* 3 ~ hVxR(n,x,

Proof. Suppose V x R ( n , x ,r') where h E Ad. Then for eachx, V z < x R ( n , z , FA).Since h is a limit, by ( 2 ) , V z < x R ( n , z , I")for some $, < A. Let f ( x ) = & < hVz < x R ( n , z , r").Then f : w + h is A-recursive. As w < h a = Sup,,, f ( n ) _ y f, ( x ) = a. But by definition off this clearly implies V x R ( n , x ,Fa). We can now complete the proof of Theorem 7.4. We must show:

W.RICHTER and P. ACZEL

340-

-

Proof. Let r be E!. Then iz E r(X)

3yVxR(n,y,x,X)

for some recursive R . Then

n E r(rwl) v x ~ ( n , y , xrwl) ,

=. V X R ( ~ , ~ ,r5) X,

for some y I rl and I n d ( C ) 2 { X : X 5 , rm}. If A E (2 then A r and hence by 8.2 I A I 5 I rl and Am 5, r". Hence I C I 5lri and Ind (C) E { X : X 5, rm}.

we may write the formula as e ( X , o - X , x ) where O(X, Y , x ) contains only positive occurrences of X and Y . Then

Now if m is odd let

A2(X) = { ( e , x ): e({e)-'X, { e } - ' ( W - X ) , x ) ) and if m is even let

0 Then in each case A 2 isn,,,.

8.8. Definition. is closed if (a) There is a C-complete operator; (b) r1,r2ee * r l u r 2 , r l n r 2 ~ e ; (c) Every recursive operator is in C! . The following result is now trivial.

8.9. Theorem. n&+l and X & + l are closed.

,344


In order to obtain characterisations of I C I and Ind(C) for closed classes we will need a method for constructing notation systemsW = ( M , 11) which is more general than that mentioned in the introduction. We shall first give an example which bears some resemblance to Kleene’s systems of notations for the constructive ordinals. We define a transfinite sequence of sets (M6 : [ E ON). In the definition la1 =&(a hx[b](x,X) is the b’th function primitive recursive in X in a recursive enumeration, uniform in X , of the functions primitive recursive in X .

Mo = 0 Ma+l = M a u { 0) u {C 1,a, b ) : a E M , & Vx[b] ( X d y U l ) EM,} MA= US 1 [ny,n:] =H; and by 1.9(i~)M,+~(Ad)=M,+I(ON). 9.3. Corollary.

0

Hence InfI= lZ,,+ll = 7 ~ : + ~ by 9.1 and 9.2. By 1.9fiii) ?r: = (i) J

9.4. Corollary.

00,

for alln.

[ny,Fl! ] I is the least recursively inaccessible ordinal;

I [ny,n Il: ] 1 is the least recursively inaccessible limit of recursively inaccessible ordinals, etc. (ii) 1 [ny, n 1 is the least recursively Mahlo ordinal; I I l l y , f l y , ny]I is the least recursively hyper-Mahlo ordinal, etc.

y]

We now turn to the proof of the theorem. By 7.8 it only remains to prove:

9.5. Lemma. (i) Q E Ad for some Q I In:/; (ii) ~ E M , , + ~ ( A d ) f o r s o m el [~I '_ wl. To give a direct proof let 311 = 31' where 0 = Z, and let a = IMI. T h e n W is a good notation system so that (YEAd, by the Coding 5 IKIyl. Lemma. As 0 E I I y , so is 0, so that Q = I@,[ (ii) First assume that n isodd. Let% = W ' where a E 0 ( X ) a E Z(X) v [z(x> c T(X) &a E Q ; ( x >Q ;~( X, ) = ~ ( 3 , e: )e E +,(T(X))) a n d a E @ , , ( X ) ~ ( V x , E X ) ( 3 x 2 E X,..(Vx,, ) E X ) [ a ] ( x l ..., , x,,)EX. Here A x l , ...,x,, [ a ] ( x l..., , x,) is the a'th n-ary primitive recursive function in a recursive enumeration of them. An easy argument shows that 0, = so that 0, E [@,IT,"]

-

-

[&,(@A) 0 somewhat more difficult. In the following lemma the class A is some relation on ordinals.

E~A,

10.1.Lemma. Suppose m, n > Oand r En;. Let X be a class of limit ordinals greater than 0, and K E X be nr(A)-rejlecting on X . if (I“: < X) is uniformly Z: on L,[A] for X E X , then I rls K . Similarly with n F ( A ) and n r replaced by Z T ( A ) and Z respectively.

A

T,

Proof. Let I’E nr. Then for some JIr formula q(y, Y ) of L, , for all Y C o n € r ( Y ) w w l=q(n,Y). Let $o(z) be the formulaz E w , and $k+l(Zk+l)be V Y k [ Z k + l ( Y k ) IC/k(Yk)l Then each $k+l is a n : formula (in the constant a).Let q*(n,X) be obtained from cp by restricting each quantifier of type k to G k . Then q* E n; andforh>wand Y L u ,

Let q * ( y , Y )be Q Z . $ ( Z , y , Y )where Q Z is a sequence of quantified variables of appropriate type for a prenex n: formula and $ is X:. Then for AEX, (1) s E r ( r h ) ~ L h [ Ak1 Q Z . $ ( Z , s , r x ) La [ A ] I= Q z V$ E On 3 6 E On [t; _< 6 A $ ( Z ,s, r6)] - L a @ ] I= Q Z V $ E On 36 E O n 3 y [ ( < 6 ~ R ( h , y ) ~ $ ( Z , s , y ) j LAMI I= PdS) >

-


352

where R is a X Y EL,[AI

A formula of L,(A)

(independent of h E X ) such that for 6 < A,

and cpI(s) is the sentence appearing immediately above it in ( I ) . Clearly cpl is a nr formula of &(A). Now let K E X be nr(A)-reflecting and s E r ( r K )We . show s E r K L, . [ A ] k q l ( s ) by (1). Since cpl is a nr formula of Lc,(A) there is some h E X n K such that L,[A] k cpl(s). Hence by (l), s E r(rh)5 P. 10.2. Definition. For a given i.d. r let A , ( x , y ) -y E On & x E n,"(r) be the least n?(A,)-reflecting ordinal; similarly for u,"(r).

ry.Let

10.3. Theorem.Letm, n > Oand r be complete 11;. Similarly 1 z; I = r) i f r is complete ,".

r

OF(

z:

Then

Inrl=n,"(r).

Proof. We prove this for rIr. The proof for E r is similar. Let r be complete 11;. For A € Ad@,) a n d t < A , I+= {x E o : A , ( x , ~ ) }E L,[A,], by IIg-separation. And since f o r y E L, [A,],

(r": t < X) is uniformly Ey on L,[A,] for h E Ad (A r). LettingX= Ad(A,) and K = nF(r)in Lemma 1, we see that IrIrlI n,"(r). To show Inrl2nr(r) it suffices to show Inrl isn;(r)-reflecting. Let O b e a s i n theproof0f8.19andlet31i'=~M,ll)=~~~.By8.14IMI=I~~l. If T, and h are as in the proof of 8.19 then i t follows that

Since h is recursive, the predicate h ( x ) = y is Ey on L, and hence X! on L,[T,] for X > w . Hence A , is X: on LJT,] for h > w. So if cp is a 11," sentence of &(A,) there is a n ; sentence cp* ofX,(T,,) such that for h > w Lh[Ar] 1cpL,[Tlm] l= cp*. Hence it suffices to show that IMI isrI,"(T,). reflecting. This will follow from the next lemma. Let ~ ( u , ..., , up) be an;formula of Lp(T,) with the indicated free variables.

ADMISSIBLE ORDINALS

10.4. Lemma. There is a IIF i.d. \k such that for%-good hand cl,

35 3

..., cQE M X

Roof. This will be in five parts. Assume throughout that A ranges over%-

good ordinals. (1) If R is primitive recursive in T , then by the Coding Lemma there is a primitive recursive function h, , independent of A, such that for a l , ..., a, E M

In particular

( 2 ) Let 9 ( Y , X ) if and only if X E w and Y E w X w is the graph of a bijection f : w Q C X such that (i) x ,y E Q & h=(x,y ) E X * x = y , and ( i i ) y E X * 3x E Q h , ( x , y ) E X . It should be clear that Q is arithmetical. Moreover Q ( Y , M , ) holds if and only if Y is the graph of a bijection f : w Q E M A such that Y * = Ax If(x) I is a bijection; w S A. Hence 3 Y Q ( Y , M X ) . (3) IfR is primitive recursive in TclKlet B R ( Y , X , x l ,...,x k ) be the Zyformula of Lp

Then if S ( Y , M X )and ~ 1..., , ak E o

(4) Let p * ( Y , X , u l , ...,ua) be obtained from q ( u l ,...,u a ) by replacing every atomic formula R ( x l , ...,x k ) by 6,( Y , x , x l ,...,x k ) . Then p* is a formula of Lp, and if S ( Y , M h ) and a l , ..., up E w then

l37-


354

( 5 ) q ( X ) may KIOWbe defined to be the set of ( c l , ..., c,) such that c , , ..., < :f 9tX') arid tor all Y such that S( Y , 9( X ) ) and all a ..., ap, b , , .,b,, 11 A , 5 f ~ l ( Y ( a f , b~l k) ( h ~ , ~ ~ ) E S ( X ) ) t h e n

,,

~

Then 9 is a Kl? i.d. that satisfies the lemma.

We L a n now ccmplete the proof of the theorem. Let /MI k cp where cp is a fl; sentence of fr,( T m). We must find A < /MIsuch that h cp. cp must have the f o r m p ( ( a lI, .., l a y ] )foi s o m e I l r - formula of Lp(Tm)a n d a l , ..., au EM. Lxet 111be the 1.d. given by 1,ernmu 10 4. Then as 9 is11," and I' is complete 11: there 19 a g q ( X ) :

Generalized Recursion Theory

Fundamentals of generalized recursion theory

Fundamentals of generalized recursion theory