Generalized Recursion Theory

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

-

Amsterdam

North-Holland Publishing Company, Ltd.

-

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

’

P. ACZEL and P.G. HINMAN

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.

RECURSION IN THE SUPERJUMP

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

P. ACZEL and P.G. HINMAN

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

RECURSION IN THE SUPERJUMP

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

P. ACZEL and P.G. HINMAN

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<,, this can be determined andq computed fromJ$. Then by 1.5(c) we may compute an index of K r l p from S(K,,,, a ) and y and thus, using the induction hypothesis and 1.4(b), from S(J&,) and y. Since y is recursive inJ: and Id1 5 l@(q)I,it suffices to take @ ( p ) = 2@(4)and $ ( p ) an appropriate index. Finally, suppose 1Iy rpll is limit ordinal. Let 0 be defined by: e(0) = the unique q E Fld (y) such that II y l' 411 = 0;

.Ti,

Ji

Ji

O(m +1) = least q * e(m) < ,q < , p .

RECURSION IN THE SUPERJUMP

11

It is clear that 0 is recursive in y and oJ, hence i n J i , and that f3 defines a strictly increasing cofinal sequence in < r r p . Let q ( m ) = @(O(m))and let a be an index of r ) from Then 3d 5a E Os and we claim that Krr? is recursive It suffices to show that for all r , S(K(,,.p),.r) is recursive in J f d S 5 . (uniformly in r , of course!). If r 4:, p , then K ( , r p ) r r= oJ. If r < p , we compute the least m such that r < ,O(m). Then I@(r)l< I@(€J(m))I = Iv(m) I SO by 1.4(b) and the uniqueness lemma, S ( K ( , l p ) p r )is recursive in Jq(ml S

Ji.

,

-

which in turn is recursive inJfd.5Q. Thus we take $ ( p ) = 3d 5“. o

Theorem 1.10. There exist primitive recursive f and g such that f o r any a, m,

and n :

{a} (m, K ) = n +f(a, (m)) E0 and Ig(a)I(m,Jfia,(,,,) -n

.

Proof. We define f and g by recursion over computations in K . Most cases are straightforward - we consider only the case where K is applied. Suppose

By the induction hypothesis, V p f (b, ( p ,m))E 0‘ so by the boundedness lemma we can computed E 6’ such that V p I f ( b , ( p , m ) ) l < Idl. Hence y = Xp * {b}( p , m ,K ) is recursive in J i with index (say) e , computable from g(b). Then appropriate values for f(a,(m)) andg(a) can be computed using the functions of Lemma 1.9. 0

-

Corollary 1.11. For any A C w , i f A is recursive in K , then for some a E 6 ’, A is recursive in J:. Proof. Immediate from 1.10 and the boundedness lemma.

0

9 2.The ordinals 5 I6 I In this section we shall characterize the ordinals ,:$(for

a E 6’) and 161 ’.

P. ACZEL and P.G. HINMAN

12

Definition 2.1. For any ordinals u and r: (a) u is 0-recursively inaccessible iff u > w and u is admissible; (b) u is r t 1-recursively inaccessible iff u is .r-recursively inaccessible and a limit of r-recursively inaccessibles. (c) if r is a limit ordinal, u is r-recursively inaccessible iff u is p-recursively inaccessible for all p < r. Our aim is to prove: JS Theorem 2.2. (a) For all a E 6’,wla is the least u which is lal-recursively inaccessible ; (b) I 01 ’ is the least u such that u is u-recursively inaccessible.

This follows easily from the following two results: (1) For any jump J and any a E 6’, if Jf is recursive in J , then is la I -recursively inaccessible; (2) for any ordinal u and any a E 0’ if u is la I-recursively inaccessible, S 5 u. then

wi

wp

S

Proof of Theorem 2.2. By ( I ) is la]-recursivelyinaccessible and by (2) it must be the least such. By (2) of 5 I , J f is recursive in K for alla E 6’ SO by (l), of = 10’1 is lal-recursively inaccessible for all a E 0’.As 10’1 is a limit ordinal, 101 ’ is thus 10’1-recursively inaccessible. I f u < 1 6 ’ 1 ,u = la1 for JS some a E 8’. Since la1 < wla, it follows from ( 2 ) that u is not u-recursively inaccessible. n We now turn to the proof of (1). For any jump J , let

R(J)

= {my : F

is type 2 and J is recursive in F } .

Lemma 2.3. For any jump J and ordinal u, i f u E R(S(J)), then u E n(f) and u is a limit of ordinals in n(J). Proof. Let u = w p with S ( J ) recursive in F. By 1.4(a), J is recursive in F so u E R ( 4 . For any r < u there exists a set A C w such thatA is recursive in F and .r < wf (for example, A codes a well-ordering of w in type 7). Then

RECURSION IN THE SUPERJUMP

But clearly ufA€ Q(J).Hence u is a limit of ordinals in

13

n(J).0

Theorem 2.4. For any a E 6‘ and any ordinal u, if u E R(J:), recursively inaccessible.

then u is la I-

Proof. We proceed by induction on lal. If la1 = 0, thenJf = OJ and it is wellknown that any u € Ci(oJ) is admissible and greater than w, hence O-recursively inaccessible. If a = 2 b , n((Jf) = n(S(J,”)). By the induction hypothesis, elements of are Ib I -recursively inaccessible so any u E C l ( J f ) is la I recursively inaccessible by Lemma 2.3. If a = 3b * S e and @(n)= { e }( n ,J,”), then any u E n(Jf)is clearly also in Ci(J&H$ for each n and thus by the induction hypothesis is I@(n)I-recursively inaccessible. Since la I = sup { I@(n)I : n E w} and is a limit ordinal, u is la ]-recursively inaccessible. 0 This establishes result (1) and we now turn to (2). Our proof will require some detailed information about recursion on ordinals. As there is at present no good reference for this material, we digress here to state the facts that we need. First we note that the class of primitive ordinal recursive functions has closure properties similar t o the class of ordinary primitive recursive functions; in particular, if g is primitive ordinal recursive and

then so is f . With each admissible ordinal K and each a < K is associated a k-ary partial function {a}, on K for a natural number k “decodable” from a. A function f on K is K-partial recursive just in case f = { a } , for some a < K . By allowing indices from K (not just a),we include all constant functions with values < K among the K -recursive functions.

(A) (Uniform Normal Form Theorem) For each k > 0 there exists a primitive ordinal recursive relation Tk such that for any K (i) if K is admissible, then fur any a, p = ( p o , ..., p k - 1), and v < K ,

(ii)

K

is admissible iff for any a and p


14

P. ACZEL and P.G. HINMAN

(B) (Uniform Iteration Theorem). For'each i E w there is a primitive ordinal recursive function Sbi such that for any admissible K and any a, p < K

( C ) (Recursion Theorem). For any admissible K and any K-partial recursive

function A there exists an esuch that

As corollaries of (A) we have:

(D)

{K

: K is admissible} is primitive ordinal recursive.

(E) For any admissible K and X and any a , p,v < A, h
A

{a},@) = v

-+

{a},(p)

=v .

-

(F) For any 1-recursively inaccessible K and any a, p < K , (i) {a},(p)$ c-, 3,X,, [A is admissible A { a } , ( ~ ) & ] ; (ii) Vpp<w{a},(p)$ 3h,,, [A is admissible AVpP<,{a},(p)$].

Lemma 2.5. For any admissible K has order-type < K .

> w, any K -recursive well-ordering of o

Proof. Let R be a K-recursive well-ordering of w of type A. By the Recursion Theorem (C) there exists a K-partial recursive function f such that for all m E w ,

The next definition is the key to our proof of (2):

Definition 2.6. For any jump J , any

K

> w, and any e < K , J is K-effective

RECURSION IN THE SUPERJUMP

with index e iff K is admissible and for any a < K such that {a},

I5

r wE

w,

J is K -effective iff J is K -effective with some index. E f , , is the set of all K such that J is K -effective with index e. Now (2) follows immediately from:

Theorem 2.7. For any jump J and any K , (a) if J is K -effective, then w{ i K ; (b) If a E 0’ and K is la J-recursivelyinaccessible, then J; is effective. This, in turn, will follow from some lemmas.

Lemma 2.8. There exists a primitive recursive g such that for any K and e such that J is K-effective with index e, any a, any c such that {c}, 1 o E 0 and any m = m o , ...,mk-l < w

Proof. The proof is very similar to that of Lemma 1.4(b). g is defined as there except for two cases: (i) if {a}(m,a,J) = a(mj),then g(a, e) is chosen so that

(ii) if {a)(m,a,J)= J(hp.{b}(p,m,a,J))(q),wechooseg(a,e)to be the “natural” index such that

The proof that g is as required is essentially identical to that of Lemma 1.4(b). 0

16

P. ACZEL and P.G. HINMAN

Part (a) of Theorem 2.7 follows immediately from Lemmas 2.5 and 2.8. Part (b) will follow from the next three lemmas, which are the induction steps in the definition of a recursive function f such that if a E 6' and K if la Irecursively inaccessible then J: is K -effective with index f ( a ) .

Lemma 2.9. There exists a co < w such that OJ is K -effective with index co for every admissible K > w. Proof. Let R be a primitive recursive relation such that

For any admissible K , let

Since the definition off, is uniform with respect to K , there exists an index

co such that f , = { c ~ for } ~every admissible K . Then OJ is K -effective with index co. 0

Lemma 2.10. There exists an primitive ordinal recursive f such that for any > w, any jump J , and any e, if K E Ef J,e and K is a limit of ordinals h E E f , , , then K E Efs(J),fl(e).If e < 0, then also f l ( e )< w.

K

We claim first that for K as in the hypothesis of the lemma, if {a}, I' w E w w , then for all m E w,

RECURSION IN THE SUPERJUMP

17

Since K is 1-recursively inaccessible, it follows from (F)(ii) that h,(e, a ) 4 so that f,(e,a,m) is also defined with value 0 or 1. Then

The second equivalence uses (E) and the third Lemma 2.8. To complete the proof, it suffices to show that f , is uniformly K-partial recursive, that is, that for some fixed index c l , f, = { c ] } ~for all K satisfying the hypothesis. Given this, f i ( e ) = S b l ( cl , e ) is the required function. Let

It follows from (A), (D), and closure under bounded quantification that

Ef, is primitive ordinal recursive. We claim that for K E E f , , ,

The implication (+) is immediate from the definition of E f , , . Suppose h E Ef, and {a},, w E w w . Then by (E), {a},, r w = {a}, w so that

r

J({a),, w ) = J ( { a ) , r w ) = { S b l ( e,a)), r w = {Sbl(e,a)), r w . Now in the definition of h , we may replace E f , , by Ef, without altering the value when K E Ef,,,. Thus h , and f , are uniformly K-partial recursive for K satisfying the hypothesis of the lemma. 0

Lemma 2.1 I . There exists a primitive ordinal recursive f 2 such that for any > 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

P. ACZEL and P.G. HINMAN

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.

RECURSION IN THE SUPERJUMP

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 < I , if {a}(m, a,S ) = n then M a , e>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.

P. ACZEL and P.G. HINMAN

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


and set

Clearly for any admissible A,

h is closed under f

++

a,ho < h and h E E f , , .

Since K is recursively Mahlo and f is K -recursive, there is an admissible h < K such that h is closed under f and hence a h E EfJ,, with A, < h < K . 0 Theorem 3.6. For any

K,

if

K

i s recursively Mahlo, then S is K -effective.

Proof. Suppose K is recursively Mahlo and let e be an index such that

where fl is the function of Lemma 2.10. We claim that S is K -effective with index e. Let J be any jump which is K-effective with index d : K E Ef,,,. By the preceding lemma, K is also a limit of ordinals A E EfJ,,, so by Lemma 2.10 S(J) is K-effective with index f l ( d ) . But then S ( J ) is also K-effective with index S b l ( e , d ) as required. 0 Corollary 3.7. of< p 0 .

RECURSION IN THE SUPERJUMP

21

For ordinals K which are projectible to w , the converse of Theorem 3.6 holds also:

Theorem 3.8. For any K which is projectible to o,K is recursively Mahlo i f f S is K -effective.

Proof. Suppose S is K-effective, say with index e, and let h be a 1- 1 K-recursive function which projects K into w . We define a hierarchy of jumps J , for u < K as follows:

and for limit u

We define, by use of the recursion theorem, a K-partial recursive functiong such that for all u < K , J , is K-effective with indexg(u):

and for limit u , g(u) is an index b such that

Now let f be any K-recursive function. We aim to find an admissible h < K such that h is closed under f . Let T be the jump defined by

22

P. ACZEL and P.G. HINMAN

Using the function g and the fact that W i s recursive in S , it is easy to see that T is K-effective. Hence S ( T ) is also K-effective so

Then X is admissible and it remains to show that X is closed under f . If u < X, then u = 11 y I1 for some y €9 and recursive in T . Hence Jf(.)+, = h a T ( y ,a ) is recursive in T and
up

<

Corollary 3.9. p g is the least K such that S is K -effective.

34. A countable hierarchy for 1-SC ( S ) We next define an extension of Platek’s hierarchy and show that a set of natural numbers is recursive in S iff it appears in our hierarchy. To guide the reader through the inevitable forest of indices and recursions we sketch first the main points of argument. We first define a set of notations 6, and for each u E 6 a jump J,, by adjoining the method of [Ri] to Platek‘s hierarchy that is, whenever Platek’s inductive definition grinds to a halt but the set of notations is closed under some new primitive recursive function { d } , we add 7 d to tlic set ofnotations, collect a11 previous J,’S to form J,,, and go on. With relatively minor altcrations to the proofs of [Sh] and [PI] we establish uniqueness and boundedness lemmas. We aim to show that computations relative to S can be done relative to the various J1,( u E 0)as in Theorem 1.10. The difficult point, which failed for Platek’s hierarchy, is to show that from such a method for computing F we can find one for S ( F ) . Here we accomplish this roughly as follows (the details differ somewhat from this sketch). From the instructions for F we can compute a primitive recursive .$ : 0 + 6 such that for any u E 6 and any a recursive in J,, we can compute the value F(a) from Jt(,). If d is a primitive recursive index for g, then 7 d € 0.A computation similar to that of the proof of 1.1 0 shows that computations from F can be done relative to J , d and hence that the “diagonal set” { a : { a } ( F ) $ }is recursive in S ( J 7 d ) .Then to compute S ( F ) ( a )for any a computable from the J,’s we apply this procedure to a recursive join of F and 01.

RECURSION IN THE SUPERJUMP

23

Definition 4.1. We define by ordinal recursion a sequence of sets 6" and, for u E 0,an ordinal Iu I and a jump J, as follows: let [a,u] denote the function (a) Let 6(")= u{0' : 7 u} and for u E 0(") (b) Let (i) (ii) (iii) (iv)

<

- {aHm,J,).

0;be the smallest set with the following properties: O(0) S O ; ; 1 E 67;

~ v [Ev ~ ( o-+) 2" E 071; for any v E S(") and any 6, if [ b , v ]is a total function y such that y ( O ) = v a n d f o r a l l m , y ( m ) E 0'") andIy(m)I
c!")

7dE

0;.

07,

Then 6" =

if

S(U)#

60 1 ,.

05,otherwise.

- d"),

For u E 0"

Iu I = u and we define

J, as follows:

u = l :J,=oJ; u = 2" : J, = S(J,);

u = 3 " * S 6 : Ju(LY)((m,n))=J~b,"l(,)(LY)(n),

( ~ , ( a ) ( n ) , if m E O("); u = 7 d : J,(cu)((m,n))=

10,

otherwise.

Lemma 4.2. For any a, b, d , u , and v : (a) ( u I < 13".Sbl -+ 12'1 < 13".Sbl; (b) l ~ l < 1 7 d l - + 1 2 U I < 1 7 d / ; (c) I u ( < 17dland 3".Sa E 6 a n d V m ( l [ a , u ] ( m ) l <17dl) 13U.5al < 17dl; -+

24

P. ACZEL and P.G. HINMAN

(d) d is a primitive recursive index and Vv [ v E 6 + { d } (v) E 61 and VvVv”lvl I IV’I < I01 I {d)(v)l I ICdI (v‘) I < I6 I I 7 d E 6 ; (e) 7 d E 6 and I v I < I 7 d ~ I i d ) (v) I < I 7 d I ; ( f) d is a primitive recursive index and I 7 e1 < 17d l + 3v( I v I < I 7 1 5 I { d l ( v ) I). +

+

-+

Proof. Most are immediate from the definition and are included to facilitate understanding the nature of 6.(c) depends on the fact that if 17dl = u, then a(”)= 6;. For (0, if there were no such v , then for u = 17el, 7 d E 65 = 6 ” so I 7dl 5 I 7el. 0

Lemma 4.3 (Uniqueness). There exist partial recursive f and g such that for all v E 6 : (a) [ f ( v ) , v ] is the characteristic functions of { u : IuI < Ivl}; (b) for any u, i f Iu I 5 I v 1, then J , is recursive in J , with index g(u, v). Proof. The functionsfandg are defined by the recursion theorem over 6 much as in [Sh, p. 104-106]. Our equivalent of [Sh, (1)-(3) p. 1041 is Lemma 1.4 and the fact that for all v E 6 , o J is recursive in J , with index computable from v. Thus all cases not involving a notation of the form 7 d may be treated as they are there and we consider here only the cases that are new. (1) u = 7 d and v = 2,: (a) I u I < I v I ++ lu I < I w I & 3e(w = 7 and 1 u I 5 I w I). The first clause can be checked from J , by the induction hypothesis. By 4.2(f) the second holds just in case

which again can be checked from J,. l withindexg(u,w), (b) l u l ~ l ~ l + l u lS~O Jl ,~isrecursiveinJ, hence inJ, with index computable from 1.4(a). ( 2 ) u = 7 d and v = 3 W * 5 a : exactly as in [Sh]. (3) v = 7d: (a) lul
RECURSION IN THE SUPERJUMP

25

Corollary 4.4. For any u E 0, w JI u > Iu I. We shall now prove that for all u E 6 ,J , is recursive in S. In earlier instances of this type of hierarchy (e.g. [Sh]) the analogous result was nearly trivial. This is not quite so here and in fact the proof eluded the authors until Leo Harrington provided the key idea of Theorem 4.6 below. Lemma 4.5. There exists a functional F recursive in S such that for all u, m, a, and T,

J,(a)(m)t F(u,m,a,y) =

1, i f y ~ ~ ~ u ~ ~ l l y l l ;

0, otherwise.

Proof. The definition of F is similar to that of the corresponding functional F in the proof of Theorem 1.3. We shall describe intuitively how F is computed and leave it to the reader to define the appropriate auxiliary functional G and apply the recursion theorem. The proof that F is as required will be by induction on llyll for y €9, so to compute F(u,m,cr,y) we may assume y EW and that for all p E Fld (y), F(u, m, a,y r p ) is the correct value. Hence we may decide recursively in S, whether or not a given v belongs to O(llrll) by computing whether or not 3 p [ F ( v , m , a , y r p ) > 01, and if the answer is yes we may compute the unique p,, such that I v I = II y t' p,,ll. Now we proceed by cases on u . u = 1:

F ( u , m , a , y ) = oJ(cr)(m) ;

26

P. ACZEL and P.G. HINMAN

otherwise, F(u,(m,n),a,y) = 0 .

u = 7 d : ifp, exists,F(u,(m,n),cw,y)= F(u,(m,n),a,yEp,);otherwise, if the following three conditions are satisfied: (i) d is a primitive recursive index; (ii) V v b , exists - + p ~ ( ~ exists]; )(~) (iii) VvVv‘[p, and p,,’ exists A pv 5 pv’ p(d)(”)5 P { ~ } ( ~ ~ ) I ; then +

F(m,n,a,yrp,), F(u, (m,n ) ,a,y) =

0,

if pm exists;

otherwise;

otherwise, F(u,(m,n),a,n) = 0. u is not of one of these forms: F(u, m , a,7)= 0. 0 Theorem 4.6. There exists a functional G partial recursive in S such that for any u, m, and a,

u E 6 -+ G(u,m,a) = J,(a)(rn) . Proof. We define G by recursion over 6 . The clauses for u = 1, 2”, or 3 ” . S b are routine (in the sense that they differ little from the corresponding clauses in [Sh] or [PI]) and we omit them. If u = 7 d ,l e t H be defined by:

By the preceding lemma H is partial recursive in S. Furthermore, the conditions of the first clause imply IvI < IuI = 17dl and hence I{d)(v)l< IuI, so that by the induction hypothesis for this stage, G({d}(v),m,a) is defined for all m and a. Thus H is a total functional. Let I be a jump recursive in S such that both F and H are recursive in I (F being as in Lemma 4.5). We claim that w i 2 IuI. If not, then as w i is a limit ordinal one of the following must hold:

RECURSION IN THE SUPERJUMP

21

(i) u { = 1 3 ” . 5 b ( < l u l :thenIvI<w{sothatthereexistsyEW,yrecursive in I , such that Iv I I I1 yll. Then

So J,, is recursive in I. Similarly, for all m , J F b , v j ( mis) recursive in I with index computable from m . HenceJg,.5b is recursive in I so by Corollary 4.4, 13’- 5bl < w { , contrary to assumption. (ii) a{= 17el < Iu I: then by Lemma 4.2(f) there exists a v such that Iv I < 0:5 I{d} ( v )I. Choose y E 94,y recursive in I , such that Iv 1 < IIyll. Then J { d } ( ” ) ( c w 4= H ( v , m , f f , 7 7)

so J { d ) ( , , l is recursive inI, which again leads to a contradiction by use of Corollary 4.4. Hence, w i 2 lu I. There exists a fixed index a such that for any jump J , hp {i}( p , S ( J ) ) codes a well-ordering of type 0:.Hence it suffices to set 0

Corollary 4.7. (a) For all u E 6 , J , is recursive in S ; (b) l O l < w f . We now turn to proving that every set of natural numbers recursive in S is recursive in some J, (u E 6).To this end we need a boundedness lemma for 6. As in [Sh] we shall derive this through the use of a partial recursive “addition” function @ on 6 .Apparently it is not possible to define such a function with the property Iu @ v I = Iu I + IvI. Fortunately, some weaker properties suffice.

Definition 4.8. An ordinal u is superadmissible iff u = I 61, or for some. 7dE0,u=17dl. Lemma 4.9. There exists a partial recursive function @ such that f o r any u,vEO:

28

P. ACZEL and P.G. HINMAN

(a) u@visdefinedandu@vEO; (b) for any superadmissible u, if Iu I , I P I < u, then I u v I < u; (c) Iu @ vI 2 max {lu I, Iv 11; (d) v f 1 -+ Iu I < Iu @ v I; (e) IU’IIIUI’IU’@VI~IU@VI. @

Proof. We define @ by recursion on v E 6 (much as in [Sh]) as follows: (1) u @ 1 =u; (2) u @ 2 v = 2 u e v ; @ Suchac (3) @ 3 ” . 5 b = [email protected],where [ c , u @ v ] ( m ) = u[b,v](m). exists and can be easily computed by the uniqueness lemma and the induction hypothesis that I v I 5 I u @ v I ; (4) u k 7 d = 37d*5C,where,ifO= 1 and% = 2m, 7 d @ r n , if lu1<17dl ;

7d,

if

u @ E , if

and m = O ; and m>O.

By (a) of the uniqueness lemma, the cases are recursive in J7. so such a c exists. We prove (a)-(e) by induction on v E 6 . Cases (1)-(3) pose no unusual problems - for (b) we use 4.2(b),(c). For case (4), let y = [ c ,7 d ] .Then clearly y(0) = 7 d and for all m , y ( m + l )= 27(m) so Iy(m)l< l-y(m+l)l. Hence u @ 7 dE 0. Properties (b), (c) and (d) are obvious. Suppose l u ’ l < l ~ l . I f l u 1 < 1 7 ~ lt,h e n a l ~ o l u ’ l < 1 7 ~sou’@v=u@vand(e)issatisl fied. Otherwise, by the induction hypothesis, for all m , Iu’@iiilI Iu @EI so a g a i n l u ’ @ v ) I l u @ v l0. Lemma 4.10 (Boundedness). There exists a primitive recursive 5 suck that for any superadmissible u and any a, u suck that I u I < u, [a,u ] is total, and vm( I [a,ul(m)I < u): ( a ) SupII [a,ul(m)l:m E o ) < It(a,u)l< 0; (b) If IvI 5 Iu I and [b,v]is a total function suck that for every m, [b,v](m)is equal either to 1 or to [a,u](m),then {(b,v)5 {(a,u). Proof.Let{(a,u)=3U.5C,where [c,u](O)-u and [c,u](mtl)-[c,u](m)@ 2[n7u1(m).Then (a) is obvious from 4.9(a)-(d) and 4.2(a).

RECURSION IN THE SUPERJUMP

29

Suppose b and v are as in the hypothesis of (b), and < ( b , v )= 3 Y . 5 d .It suffices to show

For m = 0 this is just the hypothesis IvI [b,v ] ( m )= 1, then by 4.9(d)

5 lu I.

Assume it is true form. If

If [ b , v ] ( m )= [ a , u ] ( m ) ,the result follows by 4.9(e).

0

To provide a convenient setting for discussing computations relative to various initial segments of the Ju's, we next define a sequence of computation systems similar to d of [Ri, 283-2841. The intuition is that for a given u, a function is u - computable iff it can be computed effectively except for references to "indexed oracles" for all JU( Iu I < 0). Definition 4.1 1. For every u 5 I0I : (a) the relation [ x ] " ( m )= n is defined inductively as in [IU] for type-2 with a clause S8 for functional application of the form:

(b) 31 7 is the class of partial functions @ such that for some x , @(m)2 [x]'(m). Such a @ is called u-computable with a-index x . (c) 32 5 is the class of partial functionals F which are u computable on Wp - that is, F is defined on all (total) a E % and there exists a u-computable @ such that for all x , if [ x ] " is a total unary function, then @ ( x )% F ( [ x ] " ) .A u-index for @ is also called a u-index for F . When u = 10I, we shall usually omit the superscript u.

Lemma 4.12. There exist primitive recursive f and p such that for any x , m, and n, and any v €6,i f O = l v l : (a) uxnu(m) N- [ f ( x 7v), Vi(m); (b) i f u is superadmissible, [[x]"(m)J. p(x, (m))E a("), and ifso T = l p ( x , ( m ) ) l ,and [ x ] " ( m ) N- [ x ] T ( m ) .

-

30

P. ACZEL and P.G. HINMAN

Proof. f and p are defined by straightforward recursions and the properties established by corresponding inductions. For (b) we use the boundedness properties of superadmissibles. 0 Corollary 4.13. There exists a primitive recursive '17 such that for any superadmissible u and any x, if [x]lais total, then ~ ( xE)6") and [ x i = [ x ] ' q ( x ) ' . particular, h [ x j Uis recursive in J q ( x ) .Hence the total ucomputable finctions are just those recursive in some J,(lu I < u). Proof. If [[xi"is total, take q(x) to be an element of O(O) such that for all rn, Ip(x,(rn))l
p,

p.

Lemma 4.14. There exists a primitive recursive O such that for any F E 7? 2 with I 6 I -index x, O(x)E 6 , u = I O(x)I is superadmissible, and F E 311 with a-index x. Proof. Since F has I6 I -index x , we have from 4.12(b) that for total [ [ y1, F ( [ I y ] )= o[x](y)= [Ix]'(y)for T = Ip(x,y)l. For each v E 6,let A, = { y :

ny]lv' is total} .

It is obvious that lvl 5 Iv'I -+A, 2 A"! and it follows from 4.12(a) thatA, is semi-recursivein J , and hence recursive in J,v. Hence there exists an index c computable from x such that for each v E 0 and ally:

Let t(v) = { ( c ,2 " ) ({ from 4.10) and set O(x)= 7 d where d is a primitive recursive index f o r t .

RECURSION IN THE SUPERJUMP

31

It is clear that t maps 6 into 6, so to show 7 d E 0 we need only check the monotonicity condition of 4.2(d). But if Iv I I I w I, then clearly the functions [c, 2'1 and [c, 2"'] satisfy the hypothesis of 4.10(b) so I ,$(v)l= INC,2")I I 1s(c,2w)l= It(w)l. The ordinal u = lO(x)I = I 7dl is superadmissible by definition and it remains to show that x is a u-index for F. For anyy such that [ y ] " is total, F(uyn0) =F(uyn)= u x n w = u x n w for7 = iP(X,y)i. BY 3.9, uyn" = [ y p ( j + so thaty € A , @ ) and T = Ip(x,y)I < { ( c , 2q(-'") = g(q(y)).Since Iq(y)l< u, by 4.2(e) so is [ ( q ( y ) )< u, hence T < u. Thus F ( [ y ] " )= [ x ] " ( y ) as required. 0

Lemma 4.15. There exists a primitive recursive f such that for any superadmissible u and any F E 925 with 0-index x, for all a and m

hoof. The definition off is by a standard application of the recursion theorem similar to those in the proofs of Lemmas 1.4(b) and 2.8 and we consider only the case

By the induction hypothesis

for some primitive recursive (substitution) function g. Then we pick f(a,x) to be an index such that

The first term is, of course, to ensure that Xp {b}( p , m,F ) is a total function.

32

P. ACZEL and P.G. HINMAN

Corollary 4.16. For any superadmissible u 5 I 6 I and any F E 92 5, 1-sc ( F )

CW?.

-

In the introduction we defined S as a function on jumps. I t is obviously equivalent to consider it as a function on functionals defined by 0,

F ) .1 ; if (a(0))(a+,

1,

otherwise.

This form is more convenient for treating computations in S .

Lemma 4.17. There exists a primitive recursive 71 such that for any total F and any x , i f F E 3 '2 with I 6 1 -index x , then S ( F ) E 9? with 1 6 1 -index .(X>.

Proof. Suppose F E W 2 with 161-index x . Let f be a primitive recursive function such that for any total [ y ] E q 2f,( x , y ) is an 161-index for a recursive join Fy o f F and [ y ] .By Lemma 4.14,0(f(x,y))€%, u ( y ) = l e ( f ( x , y ) ) l is superadmissible, and Fy E %?;Q). By 4.15 and 4.12(a) we can define a primitive recursive g such that

and thus a primitive recursive h such that

Thus the relation {a} (1y 1,F ) .1 is uniformly semicompu table from J,cf(x,v)) and hence computable from S(J,cfcx,y)))which isJ2eu(x,y)). From this it is easy to compute a I 61-index for S(F). 0

Theorem 4.18. There exists a .primitive recursive f such that for any a < o, m = m o , ..., m k p l
nmn(m, = n.

RECURSION IN THE SUPERJUMP

33

Proof. This is similar to the proof of Lemma 3.3, and we again proceed immediately to the main case where

with F = hp {b}(rn,a,0,S ) and a = A4 * { c } ( 4 ,m , a , S). The assumption that ( a } ( m , a , S)S guarantees that F and a are total. Furthermore, from the induction hypothesis it is clear that there exists primitive recursive g and h such that F E 92 with I 0 I -index g(b, m , x ) and a E 02 with I 6 I -index h(c,m,x).Hence by 4.17S(F)(a) = [[n(g(b,m,x))n(h(c,m,x)). Then it suffices to let f ( a ) be an 161-index for this as a function of m and X. 0

Corollary 4.19. For any @ and F : (a) @ partial recursive in S -+ $ E 92 ; (b) F partial recursive in S and defined on all a E ‘2

+

F E% ‘ 2.

Corollary 4.10. (a) For any LY,a is recursive in S iff for some u E 6, a is recursive in J,, ; (b) 101=.us. Proof. (a) is immediate from 4.19(a) and 4.13; (b) follows from (a) and 4.7.

0

$5. Functionals equivalent to S Along with the superjump S , Gandy introduced in [Gal two other functionals and sketched proofs that all three generate the same class of type-1 functions. In [Pl] , Platek made use of some slightly altered versions of Gandy’s functionals. As most details of the proofs of equivalence were not included in either article, we thought it worthwhile to present here (reasonably) complete proofs. T o avoid repetition we shall treat in detail only Platek’s versions of the functionals and mention at the end how the proofs may be altered to handle also Gandy’s original versions. For completeness’ sake we also include the equivalence of a fourth functional. The main ideas of this section are due to Gandy. In order to formulate our results we shall need to extend the interpretation of Kleene’s schemes SO - S 9 in [Kl] .He only considers computations for partial functionals hn cp( a’) where a is a sequence of total objects. Also he

P. ACZEL and P.G. HINMAN

34

only defines relative recursion relative to such functionals. We wish to consider these notions for partial functionals where some of the arguments may range over partial type 2 objects. We shall continue to use F , G , H , ... to denote total type 2 objects while we shall use F,G, ... to denote partial type 2 objects. Kleene's scheme S8.2 must be supplemented by a new scheme SS.2 introducing the new type of variable:

with appropriate modifications in the indexing to take care of the new scheme and the new type of variable. By using the scheme SO as in [Kl] we may relativise to a sequence ..., ql of functionals, some of whose arguments may be partial. But it will be necessary to insist that q l ,..., 'kl are consistent in the following sense: if \ k j ( n ) $ and n ' results from n by extending some of the partial type 2 objects occuring in n then 'Pi(.') = \ki(a). Without this restriction the inductive definition for the graph of the enumerating functional (see 3.8 of [KI]) would no longer be monotone, so that a crucial ingredient of [Kl] would be missing. I t is not hard to see that if cp is partial recursive in q l ,..., ql where q l ,..., are consistent then cp is also consistent. It is important to distinguish between variables F and F even in some contexts where this might not appear necessary. For example let 3 O = XF.0 and 30 = hF.0. Clearly these are distinct and both are partial recursive, using essentially the same schemes. But what about 30=hfi.30(ha*(a))? ' 5 is extensionally identical with 30, but it is not partial recursive, and hence must be distinguished from 3 O on the grounds that 30and 30 have arguments of different type. In the next two definitions we introduce the three functionals we shall compare with S. Definition 5.1. For any total F ,

0, if

& ( F )=

recursive in ' E , F and F(a) = 01;

~(Y[(Y

1, otherwise.

We recall from [PI] that a partial functional F is called acceptable iff the domain of F includes all (Y recursive in 2E, F.If F is acceptable and F C G,

RECURSION IN THE SUPERJUMP

35

then G has the same 1-section as F Definition 5.2.For any acceptable F, S+(LZ,F ) =

&+(F)=

0, if

{ a ) ( k ) ~; .

1, otherwise. 0, if 3a [arecursive in 2E, F and k(a)= 01 ; 1, otherwise.

Both St and E+ are undefined for non-acceptable F. We note that S+ does not seem to be naturally interpreted as a jump since the jump of an acceptable functional would not in general be itself acceptable. Computations relative to &+ are the same as what are called “weak computations from 8’’ in [Pl].Of course the proof in [Pl, Corollary 111 that the same a are weakly computable from 8 as are strongly computable from 8 depends on the incorrect hlerarchy result. Note also that both S + and 8’ are consistent. This follows from the fact that if fi is acceptable and F E G, then G is also acceptable, for any a and m,{ a } (m, 2E, k)= ( a } (m, 2E,G) (because only values k(a) for a recursive in 2E, F are used in the computation), and 1-sc( 2 E ,k)= 1-sc( 2 E , G). Lemma 5.3. (a) 8 is recursive in &+; (b) S is recursive in S + ; (c) & is recursive in S ; Proof. (a) is trivial as &(F)= E+(XaF(a)).For (b) note that S(F)(a)(a)= S + ( p ( a ) , ( F ,a)) where p is a primitive recursive function such that { p ( a ) } ( ( F , a ) )= {a}(a,F).For (c) note that

E(F) = 0

-

3a{b}(a, 2E, F ) = 0 ,

where b E o such that { b } (a, 2E,F) = F(Am{a}( m ,2 E , F ) ) .Hence

P. ACZEL and P.G. HINMAN

36

w h e r e e E w such that { e ) ( a , ( 2 E , F ) ) =Oif 2E(ha{b}(a,2E,F))=0, and is undefined otherwise. Then 8 ( F ) = S ( ( 2 E ,F))(ht.O)(e). 0 Our aim now is to show that Z+ and S+ are both partial recursive in G . From this together with Lemma 5.3 and the appropriate transitivity results it follows that 8 , S , &+, S+ are all partial recursive in each other and hence have the same 1-sections. We shall use the notion of a computation tree relative to ’E, F. Our notion be the inductively differs somewhat from that introduced in [Kl]. Let defined set of all tuples ( a , m , n )such that (a}(m, 2E, F ) s n . L e t s F be the well-founded partial ordering of Q [PI corresponding to the relation ‘precedes or equals as subcomputation’ among computations. For example, if a is the index for the composition of
Q[a

r

Lemma 5.4.Forany ( a , m , n ) € a [ F ]TCm,n , isrecursivein 2 E , F . Proof. Construct in the usual way a primitive recursive f such that for ( a , m , n ) E Q [ P ] , f ( ( a , m , n ) )isanindexofTCm,, from2E,k. 0 Definition5.5. (a) dF(T)-for s o m e ( a , m , n ) € a [ F ]T=T&,,; , P (b) i f d (T),aT,mT,and nT are the unique a, m,and n such that

P

T = To,m,n. Lemma 5.6. There exists a function @ partial recursive in El such that (a) F i s acceptable c=$ AT@(T, F ) is total;

RECURSION IN THE SUPERJUMP

37

(b) If F is acceptable then XT@(T, F ) is the characteristic function of J F .

Proof. Let cl ‘(T) be the conjunction of the following clauses: (a) T is a well-founded partial order with largest element but no limit points; (b) all elements of the field Tare of the form (a,m,n); (c) for all u other than the T-largest element, J ‘(T r u ) ; (d) for all t = (a,m,n)in the field of T, one of the following holds: (1) a is an index for an initial function and T = {(t, t)}; ( 2 ) for some b and c, {a}(m,2E,p ) = {b}({c}(m,2E,P ) , m,2E,&’)and forsomep,sl=(c,m,p)ands2=(b,p,m,n)are theuniqueimmediate T-predecessors o f t ; (3) for some b , {a}(m,2E,P ) = 2E(OI>or k(a)where a = A ~ * { b } ( p , m , ~ E , FandVp3!4 ) suchthat(b,p,m,q)isan immediate T-predecessor o f t and all immediate T-predecessors o f t are of this form; further, if 2E was applied then q = 0 or 1 depending on whether or not one of the immediate T-predecessors o f t is of the form ( b , p , m , O ) ; (4) other cases similarly. J ‘is well-defined by recursion on 11 TII, and a proof that it is recursive in E , follows the lines of the proof of 1.3. Of course, El is needed for (a) and for the number quantifiers. For any T such that J ’ ( T )and any t = (a, m,n) in the field of T which describes application of F as in ( 3 ) , let & = hp * unique 4 [(b,p , m , 4 ) is an immediate T-predecessor o f t ] .Then set

-

@ ( T ,F)

-

0, if J ’ ( T )and for all t = (a,m,n ) E Fld ( T ) which describe application of F,&3,) = n; 1 , ifeither- d!(T)or d’(T)but f o r s o m e t = ( a , m , n ) E F l d ( T ) which describes application o f F , 3n’ (F(&)=n’ and n’#n).

Thus @ is partial recursive in and it suffices to prove for any acceptable F (i) ( a , m , n ) ~ a [ -F+]@ ( T [ ~ , ~ , P0,. ).(ii) V T [ @ ( T , F ) $and(@(T,F)---O+dJ(T))],andforanyF (iii) if AT@(T,F) is total then F is acceptable. The proof of (i) is a straightforward induction over Q[&’]. For (ii) if J ‘(T), @(T,F)N _ 1 so assume J ’(T).We now proceed by induction on II TI1

P. ACZEL and P.G. HINMAN

38

and by cases according to the T-largest element t = ( a ,m,n). We consider only the case where t describes an application of P. By the induction hypothesis Vp.@(Tr(b,p,m,p,(p)),F)$. If any of these hasvalue 1, thenclearly also @ ( T , F ) = 1.If they all havevalue 0, then V p . J F ( T r(b,p,m,P,(p))) which implies V p ( p ) ‘v { b }( p , m, 2E,g). Hence Pt is recursive in 2E,F, so, as F is acceptable, P(&)J and thus @(T,P ) $ . If @(T,#) = 0, then P(&) = n and thus d F ( T ) . For (iii) let AT@(T,F ) be total and let a = Aq{e}(4, 2E,P). We must show that ~ ( ( w ) $ . For each 4 E w Te,4,,(4)E J I . Let b describe the application of T b e theunique tree F t 0 a ; i . e . (b}(2E,&’)~~(hq(e}(q,2E,.~)).Nowlet in J’with T-largest element ( b ,O).and 7’&,(4), for each 4 E w , as immediate subtrees, i.e. T(s, t ) iff either 34T,&,(,)(s, t ) or (s= t = ( b ,0)) or (34T[4,,(4)(s,s)and t = ( b ,0)). If t = (a,m,n>EFld(T) describes application of F then either t = ( b ,O), when = a, or else t E Fld(T&,(4)) for some 4 , when F( &) N- n .Hence a p t

So if A T @ ( T , F )is total it follows that F(a)$ .

0

Definition 5.7. For any F and T: 0, if @(T,F ) = 0 and a , is an index describing application of Fand nT = 0;

k * ( T )N

1 , if @(T,F)= 0 but ( a , is not an index describing application o f F o r n, 0);

+

-

Lemma 5.8. For any F (a) F i s acceptable F* is total; and (b) ifk is acceptable then 1-sc(2E, F ) C 1-sc@*).

Proof. (a) follows immediately from the definition of F* and Lemma 5.6. For (b) we define a primitive recursive f such that

RECURSION IN THE SUPERJUMP { a } (m, 2E,

P ) =n

-+

39

hr { f ( a , h))) (r, P * ) is the

characteristic function of T&,n . The definition o f f for most cases is routine. We treat only the case when a describes an application of F,say

{ ~ } ( m , ~ E , @ ) = F ( f l ) where = n P=hp .{b}(p,m,2E,F). For any 4, let Tq be the tree with largest element ( a , m , q )and all T&,,,D(p) as immediate subtrees. Then dF(Tq)iff 4 = n. From the induction hypothesis we can define a primitive recursive g such that for all 4, hr {g(a, m , 4 ) } ( r ,P*) is the characteristic function of Tq. Then n = least q [$'*(Tq)I 1 J and f ( a , m)-g(a, m,n) is a proper value. Finally for any a, m such that { a } (m,2E, F)4, { a } (m, 2E,

F)= least n - { f ( a , m))((a,m , n ) ,F*)= 0 .

0

Theorem 5.9 (Candy). &+(F)= &(F*);hence &+ is partial recursive in 8.

-

Proof. Suppose first &'(j) = 0 so for some b , F(hp { b } ( p , 'E,.F)) = 0. Then if a is an index such that {a}(& N _ $(hp { b } ( p , 2E,F)),$'*(T$,o) = 0. By 5.4, T:4,0 is recursive in 2E, F and thus by 5.8 recursive in F*.Hence &(c;'*) = 0. Conversely, if &(r'*) = 0, say PI(?') = 0, then @ ( T ,F)= 0 and for some b, {aT}(mT,2E,P ) =P(hp * { b ) ( p , mT,2E, F)).But then d F ( T )and this is a correct computation from 2E, F.Hence h p { b } ( p , m T 2E, , P) is a function fl recursive in 2E, F and $(p) = n - 0. Thus &+@) = 0. +T : Lemma 5.8(a) shows that & ( F ) $ iff &(F*)$, so that the first part is proved. The second part follows from the fact that E l is clearly recursive in & and the passage from F to F* is uniform. 0

-

Definition 5.10. Let 3' be defined like d ' except that no provision is made for application of 2E. Then for any F,T , and a,

3 ' ( T ) and aT = a and mT = $; Q(T, F)= o but (- S'(T)or aT # a or mT + $1;

0, if @(T,F)= 0 and

1, if

2, if @ ( T , P ) = 1.

40

-

P. ACZEL and P.G. HINMAN

Lemma 5.1 1. For any a and F (a) F is acceptable F,** is total ;and (b) ifFisacceptable then l - s c ( 2 E , F ) G Proof. As for 5.8.

l-x(F,**).

0

Theorem 5.12. S+(a, F ) =

&(Pi*); hence, S+ is partial recursive in I%.

c)

Proof. Suppose first S+(a, = 0, so for some n , { a } ( F ) = n. Hence Fl*(T[o,n) = 0. By 5.4 T:Q,n is recursive in 2E,F and thus by 5.1 1 recursive

in F:*. Hence &(Pi*) = 0. Conversely, if &(&’:*) = 0, say F i * ( T ) = 0, then @(T,F)= 0 and 3’(T), so T is a correct computation tree involving only F. Hence {aT} ( F ) = nT so S+(a, F)= 0. Lemma 5.1 1(a) shows that S’(a, F)$ iff &(pi*)$, so that the first part is proved. The second part follows just as

in 5.9.

0

Thus from 5.3 and 5.12 we obtain the equivalence of all four functionals and, in particular, the identity of their 1-sections. Candy’s original functional was 0, if 3a [a recursive in F and F(a) = 01 ; 1, otherwise, and its natural extension gi defined for all recursively satisfactory F,that is, F such that the domain of F includes all a recursive in F. A minor modification of 5.3(c) shows g is recursive in S and the proof of 5.12 actually shows S+ partial recursive in is an extension o f & so is partial recursive in I%+. For the converse of this, let % be defined from 3’ as @ was - from 6’ and define F* from in such a way as to ensure - 2E is recursive in F * . Then for any recursively satisfactory F,g’(F) = z ( P * ) .Hence and E+ also have the same 1-sections as S. We leave it to the reader to check that $+,the extension of S+ to all recursively satisfactory F , also has the same 1-section.

E.Z+

5

E

RECURSION IN THE SUPERJUMP

41

Bibliography P. Aczel, The ordinals of the superjump and other related functionals, abstract, Conference in Mathematical Logic-London 1970, Lecture Notes In Mathematics 255, Ed. Wilfrid Hodges (Springer, Berlin, 1971) 336-337. R.O. Gandy, General recursive functionals of finite type and hierarchies of functions, paper given at the Symposium on Mathematical Logic, University of Clermont-Ferrand, June 1962 (mimeographed). P.G. Hinman, Hierarchies of effective descriptive set theory, Trans. Amer. Math. SOC.142 (August 1969) 111-140. S.C. Kleene, Recursive functionals and quantifiers of finite types I, Trans. Amer. Math. SOC.91 (1959) 1-52. Y.N. Moschovakis, Hyperanalytic predicates, Trans. Amer. Math. SOC.129 (1967) 249-282. R.A. Platek, A countable hierarchy for the superjump, Logic Colloquium ’69, edited by R.O. Gandy and C.E.M. Yates (North-Holland, Amsterdam, 1971). 25 7 -27 1. W. Richter, Recursively Mahlo ordinals and inductive definitions, Logic Colloquium ’69, edited by R.O. Gandy and C.E.M. Yates (North-Holland, Amsterdam, 1971) 273-288. J.R. Shoenfield, A hierarchy based on a type-two object, Trans. Amer. Math. SOC. 134 (October, 1968) 103-108.

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

THE SUPERJUMP AND THE FIRST RECURSIVELY MAHLO ORDINAL Leo HARRINGTON Massnchusetts Institute of Technology

The purpose of this paper is to continue the investigation of the superjump, which was first defined by Candy in [Gal, which was studied despite adversity in [Pl] and [Ac] ,and which has recently been examined with notable success in [A-HI .The main result of this paper is that of, the first ordinal not recursive in the superjump, is the first recursively Mahlo ordinal. This result was announced in [Ac], but its proof hinged on some corollaries of a fallacious theorem from [Pl] . These corollaries turn out to be correct. As prerequisites, a working knowledge of Kleene’s (or some equivalent) formulation of recursion on higher type functionals, [Kl] ,would be desirable. The reader would also be advised to acquaint himself with the discussions in [Pl] and [A-HI, and most particularly with $ 3 and $4 of [A-HI. I should acknowledge here that my debt to [PI], [Ri] and [A-HI is even greater than the derivative nature of this paper would suggest.

A few definitions seem to be in order. Definition. The superjump, S , is a type 3 functional defined by o if { e l F $ S(F, e ) = 1 otherwise (where F is any type 2 functional). 2E is a type 2 functional defined by 0 if X = @ 1 otherwise (where X is any real).

i

43

44

L. HARRINGTON

Let F be a fixed type 2 functional. F will stay more or less fixed till further notice. The rest of this paper will be developed relative to F. This will seemingly increase the generality of the results while in no way increasing the difficulty of their proofs. In fact F will in general be tacitly ignored. Definition. 1 - Sc (F,2E), [ 1 - Sc (F,S)], is the collection of reds recursive in F, 2~ [recursive in F, SI. Let u be an ordinal. Define L , ( F ) by: L J F ) = U6<(r{x C L 6 ( F ) ; x is first order definable with parameters over (L6(F),e,F 1(2w nL,(F)))). Let MJF) denote the structure (L,(F), E , F P (2w n Lu(F))). u is an F-admissible ordinal if M,(F) is an admissible structure. f~ is F-recursively Mahlo if u is F-admissible and if for any function f : u + a, A, over M J F ) , there is an F-admissible /3 < u such that f”/3 Go. wf(p:) is the first F-admissible (F-recursively Mahlo) ordinal. In [Sh] , Shoenfield constructed a hierarchy for 1 - Sc(F, 2 E ) which could be used to show that 1 - Sc (F, 2 E )= 2w f’ L F ( F ) . In 5 1 Shoenfield’s w1 hierarchy will be extended to a hierarchy for 2w n LpB(F).This is done in a way totally analogous to the extension in [A-HI of the hierarchy from [Pl]. The technique of [Ri] is the guiding influence in both cases. This new hierarchy will then be employed in $ 2 to demonstrate that 1 Sc(F, S) = 2- nL ( F ) . In $ 3 the possibility of finding a reasonable PF notion of partial recursiveness in S is discussed. ~

In this section the hierarchy for 2w n L (F)is defined, and the crucial 8 lemma concerning this hierarchy is proven. Definition. A set of notations, ‘ K F , a function \ I F : 7ZF+Ordinals, and for each 12 E q Fa real, H:, are defined by induction as follows (i) 1 E q F ,I I l F = 0, H: = w . F

( i i ) I f x E q F then: 2 ’ E n F , 12’IF=IxlF+ l , a n d H 2 F , = { ( e , a ) ; ( e } ~ x is total and F({e)H.F) = a}

THE SUPERJUMP AND THE FIRST RECURSIVELY MAHLO ORDINAL

45

(iii) If n E and if for all i E w {e}Hg(i) E % then: 3, * 5e E %“, 13” - g e l F = the 1st limit ordinal greater than InlF and I{e}HF(i)lF for all i E w , a n d H S n . = {(m,O);ImIFQ: 13n-5elF}U { ( m , a + l ) ; IrnIF< 13n-5elY a n d a E H i } . (iv) If there is an ordinal u such that (a) u is a limit ordinal; (b) u # I 3 a * 5b I F foralla,b;and(c)foralln,if I n l F < u then I{e}(n)IF
nF;

For an ordinal u , let %: = { n E In I F < a}. For n $92 let = sup { ~ I”; n n E % ”1. Let I Till further notice the superscript F will be systematically deleted.

nFl

Definition. For an ordinal u, u is %-admissible if u = 1 7el for some 7 e € % ; u is %-inaccessible if u is %-admissible and a limit of %-admissibles. Notice that for any n € % there is an %-inaccessible ordinal greater than

In I. Remarks. (1) For n , m €% ,if ( n1 5 Im I then %, and H, are both recursive in H,,, uniformly from n , m ; if In I < J mI then the Turing jump of H, is recursive in H, uniformly from n , m. (2) For n €92,if n is %-admissible (or even if In 1 = w -In I) then there is a real I,, recursive in H,, which codes M , , ,(F).The 1,’s are in fact recursive in the Hn’s uniformly. (3) For any ordinal u , { ( n , a ) ;n E% and a EH,} is C, overM,(F). By remarks(2) and(3) wehave: { X ; X I H , f o r s o m e n € % } = 2WnLI,I(19. It should be clear that p c cannot receive a notation, and so t 92 I Ip c . To demonstrate now that our hierarchy is in fact a hierarchy for 2w n L p c ( F ) , we need only show that 1% I is F-recursively Mahlo. The key lemma is the following.

L. HARRINGTON

46

Lemma. There is a recursivefunction p such that i f at least one of mo,m is in %, and $0 is the first %-inaccessible ordinal greater than min ( I mo I, I ml I), then A m o ,m 1 ) E 929, and Ip(mo, ml) I 2 min (I m0 I, Im I).

Proof. The proof is of course by effective transfinite induction on p = min( lmO1,Iml I), so assume that p(mb,m;) is defined and works whenever, min ( I mb 1, I) < p . The only novel cases are when one of m,, ,m is of the form 7e. (As may have been noticed, throughout this paper generalized recursive functions have been preferred to primitive recursive ones. When dealing with potential notations of the form 7 e or 3" * 5e, this could be a cause for worry as {e} or {e}Hnneed not be total. This difficulty is easily handled since an oracle for H , or H2,, can always be presumed, and so {e} or {e}Hn may be converted into a total function. For the remainder of this proof, this annoyance will be ignored.) As a notational convenience define, given a set of ordinalsA, the strict limit sup o f A , slsA, to be the 1st limit ordinal greater than every member o f A .

Case 1 : mo = 7eo,ml = 2x Find an e such that for all n E 72

(Such an e can be found: By remark (1) there is a recursive function k such H that for n E % {k(n)} 2" = % Let h be a recursive function such that for all integers n and all reds X if either mo or m l is in { k ( n ) } X

p ({ e o } ( n ) , x ) otherwise.

-

Get e so that {e}(n)= 32n 5h(n).)Notice that this respects the induction hypothesisas IXl>p*p= 17eol andso I n l < p * I{eo}(n)I< 1 7 e 0 1 = ~Let p ( m o , m l ) = 7e. Ip(mo,ml)l< p : By the induction hypothesisn €%, * {e}(n)€% ,. Let u = sls({ 1 {e}(n)1 ;n €92 p } U { p } ) . So u 5 0 . But u has a notation of the form 3m 5- (where Im 1 = p), and so u < 0 since 0 has no such notation. But f o r n E C ) l p- Y p , I{e}(n)I=sls{lnl}= I n l + o . T h u s 1 7 e l i t h e l s t % admissible ordinal greater than u, which is less than since 0 is 92-inaccessible. Therefore I 7e I < 0. Ip(mo,ml)12p: Supposenot.Then I n [ < 17eI*In1<pandso

-

THE SUPERJUMP AND,THE FIRST RECURSIVELY MAHLO ORDINAL

47

I{e}(n)l> Ip({eo}(n),x)l, and {eo)(n),x satisfies the I.H. (induction hypothesis).But ln1<17el*l{e)(n)I< 17eI<_IxIandtherefore I {eol(n) I'IIp({eo)(n),x) I < I {e}(n)I. But then n E 92 17el+ I { e o ) ( n )I < I {e)(n) I, and so I7eo I < - I 7e(
In a way similar to case 1, using the uniformities mentioned in remark (l), find an e such that for n €92

I {e)(n> 1 = jslsilnI'

M{InIlu {Ip({eOl(no),

if I n l 2 p

{el}(nl))l;no,nlE92,,,))

if In1

Notice that no,nl €92, *min(l{eo)(no)l, I~el)(nl)I)<min(17eol,l~e11)

= p and so the I.H. is maintained. Let p(mo,ml) = 7e.

By an argument entirely similar to that in case 1, we get lp(rno,ml)l < P . p(mo,ml)l 2 - p : Suppose not. Then 17el < 17e01 and so there is no E 92, such that I{eo)(no)l 2 17el. Similarly there isnl E 9217eLsuch that Iler(nl)I 2 17el. But pickn € 7 2 17e,so that In1 > lnOLInll. T en 17el > I {e)(n)l> Ip({eo)(no), {el)(nl))l ~ m ~ ( l ~ e ~ ~ ( ~ ~ ) I , I ~ e ~ ~ ( ~ ~ ) 1 ) Cuse 3 : mo = 7eo,m 1 -- 3n'

- se'

Claim. There is a recursive function f such that given 4 , if 14 I < p or if Imo I = p, then f(4) € no and If (4) I 2 min ( Imo I, I4 I )Proof. Find a recursive function e such that for all n E 9 '2

Notice that either 14 I < p , or l ~= p l= min( lmO1,I4 I ) in which case In1 < lmOl* I{eo)(n)I< ImOI= p . So if 1n1<min(IrnoI,Iq~)then one ofq, { e o ) ( n ) is in %, and thus the I.H. is preserved. Let f ( 4 ) = 7e(4). As usual If ( 4 )I lp({eo)(n),q)l.But Inl< lf(4)I+l{e(q))(n)I< 17e(4)l
L. HARRINGTON

48

{el l H n 1 ( i )If. n E % then the xi’s have substance, and the claim implies that f(xi)E%p andf(xi) 2 m i n ( i m o I , IxiI). Let m = 2f(n1).Find e such that

Notice that If(nl)l < lmOl =$ lnll I If(nl)l and thus thexi’s have substance, and therefore the I.H. is not violated. Let p(rno,ml)= 3M 5e. As always Ip(mo,m1)I < P . Ip(mo,ml)I 2 ~If 13m : ~ 5 ~ lmO1, 1 2 then lmO1> If(nl)l 2 Inll, and thus I{e}Hm(i)I > 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
Theorem 1 (Bounding). If cp is a total %,-recursive function, and i f the range of cp is a subset of %@,then sup {I cp(x)I ; x E a}< P.

Lemma 2 should be an immediate corollary of the Lemma in

Theorem 2 (Selection). There is a partial 92 -recursivefunction J. for some x , then [ e ] P( (e))4. for all e, if [el %@(x)

e, +

5 1. I)

such that

Theorem 2 is a soft consequence of Lemma 2. For an actual proof of a similar theorem in a different setting, see [Mo] or [Gr]. [Gal is the prototype.

Theorem 3. Let f be C, overMP(F),and assume that f maps a subset of P into 0. Then there is a partial %,-recursive function cp such that for all n , cp(n)$iff InIEdomf,inwhichcasef(InI)= Icp(n)l. Theorem 3 can be proven by combining remark ( 2 ) of 5 1 with a few applications of Theorem 2. Theorems 1 and 3 should now make it clear that /3 isF-admissible. But this, together with Theorem 3 and Lemma 1 for the case /3 = 1% 1, should make it equally clear that 1% I is F-recursively Mahlo. So to summarize.

Theorem 4 I % 1 = p c . The %-admissible ordinals are just the F-admissible ordinals which are less than The partial %-recursive functions are just the partial functions which are C , over M (F).

&.

P{

I t is much easier to show that our hierarchy is also a hierarchy for 1- Sc (F,S). This can actually be seen from its definition (the Lemma of is not needed). Arguments similar to those in 54 of [A-HI give:

51

L. HARRINGTON

50

Theorem 5.There are recursivefunctions f and g such that (a) [ e I W Z Y ' i f f Cf(e)}S%> 2 . Y (b) I f {e}'jF(x) r y then [g(e)ln(x) ' y . (a) can be proven fairly directly, as in 94 of [A-HI. (b) is best proven in the following form: there is a recursive function h such that, if [eoIn is total and if {el}S>F([eo]q , x ) z y , then [ h ( e o , e l ) ] y ( xz) y . h should be constructed by induction on the length, in the sense of [Gal or [Gr], of the [ eOln ,x). computation associated with {el}SJF( Theorems 5 and 4 immediately yield that 1- Sc ( F , S ) = 2w n L p ~ ( F ) . By letting F become unfixed, it is possible to catch a glimpse of t i e type 3 functionals recursive in S. They are just the total functionals of the form hF [ [ eIqnF(O)], or equivalently of the form hF [/.tx(M ( F ) k= ~ ( x ) )where ] &. ~ ( xis)a uniformized C, formula in the language appropnate to the structuresM ( F ) . Pg

§3 In 9 2 we were able to characterize the reds recursive in S. The next natural step would be to investigate the reds r.e. in S. Unfortunately the fact that S is of type 3 swamps all more delicate considerations. The reds r.e. in S are just the l 3: reds, which is definitely more than 1 - Sc (S) warrants - there are plenty of reds which are r.e. and co-r.e. in S and yet not recursive in S. This seems to arise as a consequence of the ability of a computation from S to diverge for totally gratuitous, rather than inherent or ineluctable, reasons. (See [Pl] pp. 262-263.) If these gratuitously diverging computations are systematically excluded, then S is seen to have a much more natural r.e. structure, as for example Platek's weak computability [Pl]. Before tlying to justify such systematic exclusions of divergent computations, it would be best to make this notion precise. Definition. Let G be a higher type functional of unspecified type. A recursive function f is a type n reindexing for G if for any type n a and any e, if {e}'(a) 2 x then {fe}'(a) z x. (Notice that {fe}'(a)$ is possible even if {e}'(a)t.) f is a minimal type n reindexing for G if, for any other type n

THE SUPEFUUMP AND THE FIRST RECURSIVELY MAHLO ORDINAL

51

reindexingg, there is a recursive function h such that for all type n a and all e, {feIG(a) k ( h e ) I G ( a ) . Iff is a type n reindexing for G then f gives rise to a natural enumeration of type n+l partial functionals which are recursive in G: the e* partial functional is just h a [ { f e j G ( a ) l .AIItotal type n+l functionals recursive in G appear in this enumeration, but some partial functionals recursive in G may be omitted. Any partial functional omitted in this way may be viewed as being gratuitously recursive in G. (For example: for many functionals G, it is possible to find H G such that there are more type 1 partial functionals recursive in H v G than there are recursive in G. These partial functionals have little right to be recursive in H v G, and this intuition is upheld by the fact that there is a reindexing for H v G which omits them.) Call a partial functional strongly recursive in G if no reindexing for G excludes it. This may seem a stringent requirement, but for the least pathological higher type functionals, the functionals of type n in which “E is recursive, this is no restriction since the identity type n reindexing is minimal. It is possible to have minimal reindexings different than the identity. The situation seems to be the following:

Definition.f is an acceptable type n reindexing for G if there is a type n + l partial functional p recursive in G such that for any type n ao,al and any eo el (i)ifp(ao,eo,al,el)=i t h e n i = O o r 1 and { f e j }G (aj).l,and 3

(ii) if {f(ei)IG(aj)4 for some i, i = o or 1, thenp(ao,eo,al,el)J. An acceptable reindexing supports a very reasonable recursion theory, which in some cases is more reasonable than the usual one. Type 0 selection, (the analogue of Theorem 2 of 52, see [Gal or [Gr]), though not necessarily true of the identity reindexing, is true for acceptable reindexings. Acceptable reindexings also possess some degree of naturalness:

Theorem.An acceptable type n reindexing for G is minimal. I f f and g are tun acceptable type n reindexings for G then there is a recursive permutation h such that { f e I G ( a )E {g(he)}G(a)for all type n a. Returning to S, Theorem 5 of 52 may now be summarized by saying that

52

L. HARRINGTON

S has an acceptable type 2 reindexing, which may now be viewed as giving S

a natural partial recursive structure. The type 3 partial functionals which are strongly recursive in S are the partial functionals mentioned at the end of $ 2 . The results of this paper can be generalized to other type levels. Define the type n +3 superjump, n+3S,by

"'3S(G,e) =

o

if

1

otherwise

{elG+

(where G is any type n + 2 functional). There is an acceptable type n +2 reindexing for n+3S, and this reindexing satisfies Grilliot's formulation of type n-1 selection (see [Gr] and [Ma]). There seems to be no model theoretic characterization of ( n +1) - Sc ("'3S), but it is possible to characterize = u, of type ( n + ~-)sc (a,n + 3 ~ dong ) lines similar to ( n + ~-) the characterization of (n+l) - S C " ( ~ + ~ in E )Chapter 3 of [Ma]. There is also a hierarchy for (n +1) - Sc ("'3S) which may be gotten by appropriately extending a hierarchy similar to the one for n+2E described in [Sa]. In all this n+3Sbehaves remarkably like a type n +2 object.

References [Ac] P. Aczel, The ordinal of the superjump and related functionak, in: Conference in Mathematical Logic London '70 (Springer, Berlin, 1972) pp. 336-337. [A-HIP. Aczel and P.G. Hinman, Recursion in the Superjump, this volume. [Gal R.O. Gandy, General recursive functionals of finite type and hierarchies of functions (A paper given at the Symposium o n Mathematical Logic held a t the University of Clermont Ferrand, June 1962). [Gr] T.J. Grilliot, Selection functions for recursive functionals, Notre Dame Journal of Formal Logic 10 (1969) 225-234. [Kl] S.C. Kleene, Recursive functionals and quantifiers of finite type, Trans. Am. Math. Soc. 91 (1959) 1-52; 108 (1963) 106-142. [PI] R.A. Platek, A countable hierarchy for the Superjump, in: R.O. Gandy, C.E.M. Yates (eds.) Logic Colloquium '69 (North-Holland, Amsterdam, 1971) pp. 257271. [Ma] D.B. MacQeen, Post's problem for recursion in higher types, Ph.D. dissertation, MIT (1 972). [ Mo] Y.N. Moschovakis, Hyperanalytic predicates, Trans. Am. Math. Soc. 129 (1967) 249-282. [Ri] W. Richter, Recursively Mahlo ordinals, in: R.O. Gandy, C.E.M. Yates (eds) Logic Colloquium '69 (North-Holland, Amsterdam, 1971) pp. 273-288. [Sa] G.E. Sacks, The 1-Section of a type n object, this volume. [Sh] J.R. Shoenfield, A hierarchy based on a type 2 object, Trans. Am. Math. Soc. 134 (1968) 103-108. -

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

STRUCTURAL CHARACTERIZATIONS OF CLASSES OF RELATIONS Yiannis N. MOSCHOVAKIS University of California, Los Angeles

For each object U of finite type over the integers and for each k

2 1 , put

ken(U) = the k-envelope of U = the set of all relations with arguments of type < k

which are semirecursive in U . We will give a fairly simple, indexing-free characterization of ,en(U) when U is of type m 2 2, m - 1 5 k 5 m + 1 and U is normal, i.e. the existential quantifier mE over the objects of type m - 2 is recursive in U . Our method will also give simple structural characterizations of some other interesting classes of relations, e.g. the E and the (positive) second order inductively definable relations with arguments of type 0 and 1. But our main aim is to make a start in the study of structural properties of envelopes. There is evidence that this study will increase our understanding of recursion in higher types. For example, we will also show that the 1-envelope of a normal object of type greater than 2 is not the 1-envelopeof any normal type-2 object. Also, the class of II,!, ( n 2 2) or E& ( m 2 1 ) relations on the integers is not the 1-envelopeof any normal type-2 object (this result is independently due to A.S. Kechris). Thus the I-envelope of a normal object u codes substantially more information about U than the 1-section of U which according to Sacks [ 19731 fails to determine whether type(U) = 2 or type(U)> 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 <_ 1, then these definitions agree with the classical definitions of primitive recursive functions and functionals. A partial function

f :x+w is a function with domain some D

E %which takes values in w . We write

to indicate that f is defined at x and has value n. We can think of a (total) function f : % -+ w as a partial function which happens to be totally defined on . The definition of recursive partial functions in Kleene [ 19591 adds only one more scheme to those needed for primitive recursion but is substantially more complicated in its interpretation. An inductive definition is given of a set of triples

Y.N.MOSCHOVAKIS

56

where e, n vary over w and a varies over tuples of arbitrary length of objects of arbitrary finite types. A partial function f : X + w with domain (a subset of) some fixed space X is then recursive, if there is some integer e (an index o f f ) such that for all x E X and all n E o, f(x)=n-(e,x,n)E%.

A relation R

C X is recursive if its characteristic

function

is recursive and semirecursive (or recursively enumerable) if there is a recursive partial f : X + w such that

R =Domain (f) = { x : f(x) is defined} These notions relativize to a futed object U of type m i.e. a partial function f : X w is rentrsive in U if

> 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

STRUCTURAL CHARACTERIZATIONS OF CLASSES OF RELATIONS

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-

STRUCTURAL CHARACTERIZATIONS OF CLASSES OF RELATIONS

59

tains all primitive recursive pointsets of type <_ k and if whenever X,y are of type < k R C ?J is in r and f : 5X + y is primitive recursive, then f-' [ R ]= {x : f(x) E R } is also in r. Using the basic definitions and the Stage Comparison Theorem, one easily proves that i f U is normal of type m 2 2 and 1 <_ k 5 m + 1, then ken ( U )is closed under primitive recursive substitution, &, V, 3' and V j for every

jlm-2. A k-pointclass r is o-parametrized if for every space X of type < k there is a relation G 5 o X 5X in r such that for every R E X, R isin

forsome e E o , R = G,

= {x E

X : G ( e , x ) }.

I t is immediate from the definitions thatfor each object Uand each k 2 1, ken ( U ) is o-parametrized. Finally we look at a more complicated structural property that pointclasses may posses. A norm on a pointset R is any function u :R

+ Ordinals

which assigns ordinals to the members o f R . (Sometimes it is convenient to insist that u is onto an initial segment of the ordinals, but we do not d o t h s here.) With each norm u on R there are naturally associated two relations,

I f R is in a pointclass r, we call u a r-norm i f both 5; and <: are in r. Notice that both 5: and <: are relations of the same type as R . A pointclass r is normed (or satisfies the Prewellordering Property) if every pointset in r admits a r-norm. The Stage Comparison Theorem implies immediately that if U is normal of type m and k 5 m f 1, then ken(U) is normed. This is the Prewellordering Theorem for semirecursion,relative to a normal object and implies directly most of the known structural properties of envelopes and sections, e.g. the existence or hierarchies on sections.

60

Y.N. MOSCHOVAKIS

2. The Main Lemma The key to our results is the closure of reasonably rich o-parametrized, normed classes under monotone inductive definability . Let : Power(%) +Power(%)

be an operator on subsets of % which is monotone, i.e.

Using the notation for induction of Moschovakis [ 19731, we define for each ordinal t the stage I$, by the recursion I$, = Q , ( U q < E I z ) . It follows that

v5t*I:Gz& and that the set

I,

=

u&

is the least (under inclusion) fixed point of Q,. A k-pointclass r is uniformly dosed under a monotone operator Q,

: Power(%)+Power(X),

with type (X) < k , if for every space y of type < k and every P C in r, the relation Q C % X y defined by

is also in I?. This says that if for eachy E

Py

=

(x’ : P ( x ’ , y ) }

y we define

X

y

STRUCTURAL CHARACTERIZATIONS OF CLASSES OF RELATIONS

61

by fixing a parametery in a I?-set P,then each set a@,,)is also obtained by f i i n g y in some r-set Q,

It may happen that @ :Power(XXw)+Power(XXo)

preserves single-valuedness, i.e. whenever R 2 X X w is (the graph of) a partial function, then so is @(R).Since the empty set 0 is a partial function, it follows in t h ~ case s that I , is a partial function too. For such operations @, we say that a k-pointclass r is uniformly closed for partial functions under a, if for each space y of type < k, whenever P E X X w X y is in r and for the set P,, = {(x’,n‘) : P(x’,n’,y)} is a partial function and every y E 9, Q C X X w X y is defined by

then Q is also in I‘.

Main Lemma. Let r be a k-pointclass which is w-parametrized, normed and closed under primitive recursive substitution, let X be a space of type < k and assume that r is uniformly closed under the monotone operator : Power ( X ) +Power

(X)

Then I , is in r. Similarly, if r is o-parametrized, nonned and closed under &, V , V’ and primitive recursive substitution, i f type (X) < k and @ : Power (XX w ) -+ Power ( X X w )

preserves single-valuednessand if r is uniformly closed for partial functions under @, then the partial function I , is in r. Proof. Choose G C w X w X X in

r which parametrizes the subsets of

‘62

oX

Y.N. MOSCHOVAKIS

X in r, let u : G + Ordinals

be a r-norm on G and put

is in r by closure under prinitive recursive substitution, so by the uniform closure under a, the relation

and define a norm r on R by

~ ( x=) u(e*,e*, x )

-

Immediately from the definitions, we have the equivalence

(*I

R(x)

x E a( {x’ : x ’ <; x } ) .

We now show that (*) characterizesZ@,i.e. any relationR wluch admits a norm r satisfying(*) must bel,. This will complete the proof, since we have found a relation in r which satisfies (*).

STRUCTURAL CHARACTERIZATIONS OF CLASSES OF RELATIONS

63

Step 1 . Zf R satisfies (*), then R ( x ) 3 x EI,.

Proof of Step 1 is by induction on ~ ( x )By . (*), from R ( x ) we get x E @( {x’ : x’ <: x } ) and by the induction hypothesis, { x ’ : x ’ <: x } , Z , so by the monotonicity of @, x E @(Z,). But @(Z@) = I , , since , Z is a fured point of @.

s

Step 2. Zf R satisfies (*), then x E I ,

*R ( x ) .

Proof of Step 2. We show by induction on l, that xEZ& * R ( x ) . For this, it will be enough to obtain a contradiction from the hypotheses

From the definition of <:, if l R ( x ) , then

From the definition ofZ$,, we have

and from l R ( x ) and (*), we have

x

4 @ ( { x ‘ : x’ <;

XI)

= @ ( R ).

Finally, from the induction hypothesis we have

which together with the two preceeding displayed formulas directly contradicts the monotonicity of @.

To prove the second part of the theorem, we first show that under the hypotheses on r, for each y of type < k there is a partial function on

UXY,

Y.N. MOSCHOVAKIS

64

GCwXYXw

which is in r and which parametrizes the partial functions in r, on y . (We could have substituted this in the hypotheses instead of the extra closure properties of r.) Choose H C o X Y X w

in r‘ which parametrizes the r-subsets of u H

+

yX

w and take

Ordinals

-

to be a r-norm on H . Put

G(m,Y > n )

f f ( m,Y, n )

& o(m,y,n)= infimurn{u(rn,y,n’): H(rn,y,n’)}

& n = infimum{n’ : H ( m , y , n ‘ ) & a(m,y,n‘)= u(rn,y,n)) It is immediate that G parametrizes the partial functions on r. That G itself is in r, follows from the equivalence

y which are in

Using this, choose G C w X w X ‘x X w in r, parametrizing the r-partial functions on w X ‘x, choose a r-norm u on G , put

and argue exactly as in the proof of the first part. This proof is a small variation on the proof of Theorem (9A.2) in Moschovakis [ 19731, which in turn was patterned on the proof of the First Recursion Theorem of Moschovakis [ 19711.

STRUCTURAL CHARACTERIZATIONS OF CLASSES OF RELATIONS

65

As a first and quite typical application of this Lemma, consider the following two simple but useful results. Proposition 1 . If a k-pointclass r is w-parametrized, normed and closed under vand primitive recursive substitution, then r is also closed under 3'. Proof. Suppose R

5w

X % is in

r, define @ by

Clearly r is uniformly closed under a, hence by the Main Lemma I , is in From the definition of @ it is immediate that

r

Conversely, a very simple induction on E shows that for all n ,

so that, in particular,

and

r is closed under 3'

Proposition 2. If a k-pointclass r is w-parametrized, normed and closed under primitive recursive substitution, v and &, then r is also clased under bounded universal number quantification, i.e. i f R 5 w X % is in r, so is P ( m , x ) defined by

P ( m , x )a ( V n < m ) R ( n , x )

Y.N. MOSCHOVAKIS

66

Proof. Given R

5o X

-

'x in F, define CP by

( m , x )E @ ( S )

[ m = 0 &R(O,x)]

v [ m > O & R ( m , x ) & ( m - 1,x)ESI It is easy to check that

( m,x) E I ,

-

(Vn 5 m)R(n,x)

3. Characterizations by closure under logical operations We start with a very simple characterization of the L (recursively enumerable) relations.

y

(I) For k = 1, 2 , the class of C k-pointsets is the smallest k-pointclass which is o-parametrized, normed and closed under primitive recursive substitution arid V.

Proof. It is well known that the class of Ey k-pointsets(k = 1,2) has all these properties, e.g. to prove that it is normed, write each E relation R in the form R ( x )- ( Y m ) P ( m , x ) with P primitive recursive and define u on R by u(x) = least m such that P ( m , x )

Conversely, if IT satisfies all these conditions, then I' is closed under 3' by Proposition 1, and since r contains all primitive recursive relations, it must contain all L? relations too. The next characterization is equally simple but more interesting. (11) Fork = 1, 2 , the class of Il k-pointsets is the smallest k-pointclass which is w-parametrized, normed and closed under primitive recursive substitution. vand v'.

STRUCTURAL CHARACTERIZATIONS OF CLASSES OF RELATIONS

67

Proof. It is again well known that the class of IT: k-pointsets ( k = 1,2) has all these properties. Here, the Prewellordering Property is a nontrivial classical result and it is proved using the fact that everyn; re1ation.k of the form

w i t h P primitive recursive. (We are using the standard notation,

In view of the Main Lemma, the converse is just the classical theorem of Kleene that every KI relation is inductively definable, see Spector [ 19611. Briefly, to prove that every k-pointclass r satisfying the conditions of the theorem must contain all II; relations, assume that R satisfies (*) with a primitive recursive P and put

( u , x )€ @ ( S ) . - P ( u , x )

V(Vt)(Uh(t),X)€S,

where u^v is the obvious primitive recursive concatenation function on sequence codes. Clearly I' is uniformly closed under.@, so by the Main Lemma I , is in r. Now verify

by showing directly from the definition of CP that

and then proving

by a reductio ad absurdum.

68.

Y.N. MOSCHOVAKIS

From this we get immediately the next result. (111) The class of X.$pointsets of type 2 is the smallest 2-pointclass which is w-parametrized, nonned and closed under primitive recursive substitution, v,Voand 3 ' .

Proof. That the class of E i pointsets has these properties is again well known - for the Prewellordering Property see Rogers [ 19671 or Moschovakis [ 19731, section 9B. Conversely, if I' has all these properties, it must contain all ni 2pointsets by (II), and hence it must contain all E $ 2-pointsets by closure under 3 ' . We do not know a structural characterization of this sort for the class of relations on w . (I)-(111) and the trivial closure properties of Z7,II and E determine completely a 2-pointclass r on the hypothesis that it is w-parametrized, normed, closed under primitive recursive substitution and v and also closed under the operations in a set

in fact r must be Ef,n or E i . To study closure under the full set of second order operations on w , including V ' , we now consider induction in analysis. Let d: be the customary two-sorted language of analysis (or second order number theory), with variables n , k , m , ... over w,a,P,7,... over T ( ' ) = with prime formulas

m = n , a = P , m + r ? = k ,m . n = k , a ( m ) = n , the logical connectives &, V, 1, + and the quantifiers 3 ', V o , El1, V 1 .We take it here with the standard interpretation. For each space % of type 1, we obtain the extension L(S) by adding a relation variable S to d: and the prime formulas S(x), where x is the proper sequence of variables so it denotes a point of % . An operator

:Power(%)-+Power(%)

STRUCTURAL CHARACTERIZATIONS OF CLASSES OF RELATIONS

69

is positive, second order definable if there is a formula (x,S) of L(S) with only positive occurrences of S such that for all S C X,in the standard interpre t ation,

@(S) = { x : Q ( x ,S ) } . Such operators are monotone. We call R 2 % absolutely second order inductive if there is some positive, second order definable operator

and integers n l , ..., n k , such that

These are the absolutely semihyperprojective relations of Moschovakis [ 19691. In Moschovakis [ 19731 we restricted ourselves for simplicity to induction on one-sorted structures and we allowed substitution of arbitrary constants from the domain of that structure in the language. One verifies trivially that a 2-pointset R is inductive on the structure of analysis in the sense of Moschovakis [ 19731 if and only if there is an absolutely second order inductive relation P in the present sense and a real ao, such that

(IV) The class of absolutely second order inductive pointsets of type 5 2 is the smallest 2-pointclass which is w-parametrized, normed and closed under primitive recursive substitution, V, 3 and V 1 . Proof. That the class of absolutely second order inductive pointsets of type 2 has these properties follows easily by the “lightface versions” of theorems (lD.l), (3A.3) and (5D.2) of Moschovakis [ 19731. To prove the converse, we show first that if ?7satisfies the hypotheses, then r is also closed under &, 3’, Vo. This is completely trivial for 3’ and Vo. For &, let

P(x)-G(eo,x), C?(x>-G(el,~),

Y.N.MOSCI-IOVAKIS

70

where eo, el are fixed integers and G S w X 5X in r a n d put

X parametrizes

the subsets of

( (eo,x> if a(o)= 0 ,

f(a,x)=

(el , x ) if a(0)> 0 .

Then

so that the conjunction of P and Q is in r. I t follows that a pointclass r with all these properties is uniformly closed under every positive, second order definable operator a, so by the Main Lemma r must contain every I , and hence every absolutely second order definable pointset. The results in (I)-(IV) can be summarized in the table below where each line is to be read as follows: the smallest 2-pointclass which is w-parametrized, normed and closed under primitive recursive substitution and the operations in the first column is the pointclass in the second column; it is also closed under the operations in the third column.

(9

Operations v

(11) v, vo (111) v, v0, 31 (IV) v, 31,v1

Smallest Class

x:

n: abs. second order inductive

Additional Closure v, &, 30,3l v, &, 3 0 , vo, v' v, &, 3 0 , v0, 3 l v, &, 3 0 , vo, 31, v'

Using this table and very trivial arguments we can easily compute the smallest w-parametrized, normed 2-pointclass which is closed under primitive recursive substitution, v and any set of operationsK C {&, 30 ,V 0 , 3 l , V'}. It is obvious from the proofs that the hypothesis of closure under primitive recursive substitution can be substantially relaxed in these characterizations. One can in fact put down characterizations that are almost algebraic in flavor, but which are naturally a bit more complicated.

STRUCTURAL CHARACTERIZATIONS OF CLASSES OF RELATIONS

7 1,

4. Characterization of envelopes We now turn to the characterization of envelopes promised in the introduction.

(V) Let U b e a normal object of type m 2 2and let m - 1 5 k I m + 1. m e n the k-envelope of U is the smallest k-pointclass r which is w-parametrized, normed, closed under primitive recursive substitution, v and & and which has the following two additional closure properties: (1) I f 2 <_ j < k and g ( d , x) is a partial function with graph in r, then the partial function f ( a ' , x ) = ai(Api-2g(ai, p i - 2 , x)) also has graph in r. x) is a partial function with graph in r, then the partial ( 2 ) If g(~w"-~, jhction f ( x ) = U(ham-2g(am-2, x)) also has graph in r. Proof. That ken ( U ) has all these properties under the hypotheses on U , k is immediate from the discussion in section 1. To prove the converse, assume that r satisfies all the conditions of the theorem. By Propositions 1 and 2 we also know that r is closed under g o and bounded universal number quantification. (a')i be the primitive recursive tuple-coding functions Let (ah, ..., put defined in Kleene [1959], and relativizing Kleene's { e } [ao,a', ..., &'I,

CIA),

where the number ni of variables in each type i < k is recoverable primitively recursively from the index e in the coding used by Kleene. The inductive definition of the relation { e } ( U ,a ) = n determines a monotone operator @ : Power(w X 5X X w ) *Power(w X

X Xu),

72

Y.N. MOSCHOVAKIS

The definition of ( e , x , n ) E @(S)is a disjunction of ten clauses, S1 -S9 and S8(U), the last corresponding to the clause for {e} ( U , a) = n which applies (I, S8(U)

f(b)

=

U(ham-2g(am-2,6 ) )

Certainly @ preserves single-valuedness. Now the closure properties of r imply that r is uniformly closed under @: we use 3" to express clause S4 for composition we use both 3' and bounded universal number quantification to express clause S5 for primitive recursion, we use (1) to express clause S8 and (2) to express clause S8(U) and of course we use closure under primitive recursive substitution all over. The explicit computation is a bit messy but direct and we omit it. By the second part of the Main Lemma then, I , is in r, hence the relation {e)'[a", ..., = n is in r, hence by closure under primitive recursive substitution, every pointset of type
The Main Lemma does not yield a characterization of this sort for the I-envelope of a normal type-3 object. The special case U = 3E makes precise the difference between the 2-envelope of 3E, the class of semihyperanalytical relations of Kleene and the class of absolutely second order inductive relations which we characterized in (IV). It is trivial that 2en (3E) is closed under V1 . Hence the only difference between

STRUCTURAL CHARACTERIZATIONS OF CLASSES OF RELATIONS

,73

these two classes is that the collection of absolutely second order inductive pointsets is closed under 3 ' , while 2en(3E) is only closed under a very restricted kind of existential quantification on T(');ifg(a,x) is a partial function (with graph) in 2en (3E), then

R(x)

-

(Va)[g(a,x) is defined] & ( 3 4 [g(a,x ) = 01

is also in 2en(3E). This is a kind of "restricted" 3 ' . It is known from Corollary 10.2 of Moschovakis [ 19671 that 2en ( 3 E ) is not closed under 3 . The hypothesis that U is normal is essential in (V), as otherwise ,en(U) need not be normed or closed under V. However, a similar characterization of the class of partial functions in ,en ( U ) for an arbitrary U of type m 2 2 (m - 1 5 k 5 m + 1) can be read off the results in Moschovakis [ 197 11. (To obtain such a characterization was one of the motivations for introducing axiomatic computation theories.) Briefly, the key property of these partial functions is that for each X of type < k there is an w-parametrization G : w X 5X -+ w of those with domain in X,which is in ken ( U ) and which admits a norm u : G -+ Ordinals having the natural properties of a measure of complexity (or length) of computations, as expressed in the axioms for a computation theory. This length function need not satisfy any definability conditions.

'

5. The type of a 1-envelope We show here that for sufficiently rich 2-pointclasses F, the class of 1pointsets in r cannot be the envelope of a type-2 normal object. As usual, we let 1F be the dual class of F,

ll?=( % - R : R R C , R i s i n F } and we put

We say that A isclosedunder 3' n V', if wheneverP(a,x), Q ( P , x ) are in A and for all x ,

(3a)P(a,x)

(VP) Q ( P , x )

Y .N. MOSCHOVAKIS

74

then the relation

is also in A . For each real (Y put

I , = {(n,rn) : a((n,m))= 0 ) and let WO be the set of reds which code wellorderings of w ,

a E WO

5, is a wellordering with field w

If a E WO, we let It , 1 be the rank o f t in the wellordering 5 a. The tool for the proof of the theorem is the following representation theorem for the 1-section of a type-2 object proved in section 2 of Moschovakis [ 19671 (see also Corollary 7.1 of [ 19671 and section 8). Let U be an object of type 2 and define a set N E o and for each z E N a function f , : w + w by the following simultaneous induction: (1) For each integer q , ( 1 , q ) E Nand for all t , f ( l , q , ( t=) q . (Introduction of constants.) (2) If w E N , then ( 2 , w ) E N and for all t , f ( 2 , w ) ( t ) = U(f,). (Introduction of U.) ( 3 ) If w E N and if the (absolutely) recursive partial function {e}(a,t ) is totally de.fined when (Y = f w and if for all t,

then ( 3 ,w, e ) € N a n d for all t , ft3,w,e)(t)= fecw,t)(t).(Diagonalization.) Then, a unary function is recursive in U if and only if it is f z for some z E N . Moreover, if U is normal, then N is semirecursive in U and the partial function u(z) = n

z E N & U( f,)

=n

is recursive in U. We understand the definition (1) - ( 3 ) as a transfinite recursion in the usual way, so that

STRUCTURAL CHARACTERIZATIONS OF CLASSES OF RELATIONS

75

where for each ordinal .$, N ( f ) consists of the integers put in N in 5 f steps. The induction clearly closes at some countable ordinal.

Theorem. Let r be a 2-pointclass which is closed under primitive recursive substitution, V, &, 3O, V o ,and assume further that the set WO of codes of wellorderings is in A = r fl 1r and that A is closed under 3 n V 1 .Then the class of 1-pointsetsin r is not the l-envelope of any normal type-2 object. Proof. Assume towards a contradiction that U is normal of type 2 and that len ( U ) consists precisely of the relations on w in r, let N , { f Z l z E N be the canonical hierarchy of the 1-section of U as described above. Put

P(a,B,y)

-

and then set

a E WO & ( 0 , ~ )code the definition of N , {f z } z E N along I,.

Y.N. MOSCHOVAKIS

76

Q(ac,P,r,x)- ( Y E W O & ( p , y) locally seem to code the definition o f N , Cf,),EN

lzt 1x1, .

It is trivial to check by transfinite induction along<, the two implications establishing

P(a,P,r)Q W X ) Q ( a A r , x ) . We now substitute for Q in this equivalence two approximations in Define Ql(ac,P,y,x)by replacing

r and 1r.

in the definition of Q by

where u ( w ) is the partial function, recursive in U , which is associated with the canonical hierarchy on sc (U). Clearly Q, is in r and again it is trivial to check by transfinite induction along 2,the two implications establishing

STRUCTURAL CHARACTERIZATIONS OF CLASSES OF RELATIONS

77

so t h a t P is in r. Now define &(a,P,y,x) by replacing

in the definition of Q by

Clearly Q2 is in 1r and the proof of

is exactly as before, completing the proof that P is in A . To complete the proof of the theorem, put

R(a,P,y,x) e P ( a , P , y ) &the recursion definingN, { f z ) z E N closes by 1x1,

Clearly R is in A and

so by the closure of A under 3' n V ' , N is in A . A similar trivial argument shows that the function

f z ( z ) + 1 if 0

2

EN,

otherwise

has graph in A , which is a contradiction since g is different from all thefz's, z E N . and hence not recursive in U.

78

Y.N. MOSCHOVAKIS

Corollary 1. I f U is normal of type m 2 3 , then len(U) is not the lenvelope of any normal type-2 object.

A

Corollary 2. The class of C ( n 2 1) or IIk ( m 2 2) relations on w is not the I-envelope of any normal type-2 object. Both corollaries are immediate, except for the case of in the second, where the result is well known, e.g. it follows from the fact that C is not normed, while the 1-envelope of every normal type-2 object is normed. The second Corollary was independently formulated and proved by A.S. Kechris. Several interesting problems suggest themselves. We state two of them as conjectures, for sharpness, although we do not have now any hard evidence that these statements rather than their negations hold.

Conjecture 1. If U is normal of type m 2 3 , then the I-envelope of U is not the 1-envelope of any normal object of type less than m. Conjecture 2. The class of X relations on w is not the I-envelope of any object U of finite type such that 2E is recursive in U. Of course it is quite probable that there are perverse, forcing-type counterexamples to these conjectures. We would consider positive answers (to these or similarly motivated questions) very interesting, as they would tend to show that there is a real notion of “type” in Kleene semirecursion, even if the types are lost when one looks only at the recursive cbjects.

6ibl iography Gandy, R.O. [ 19621 General recursive functionals of finite type and hierarchies of functions, Proceedings of Logic Colloquium CLermont Ferrand, 1962,5--24. Grilliot, T.J. [ 19671 Recursive functions of finite higher types, Ph.D.Tbesis, Duke University, 1967. KIeene, S.C. [I9591 Recursive functionals and quantifiers of finite types I, Trans. Arner. Math. SOC.91 (1959) 1-52.

STRUCTURAL CHARACTERIZATIONS OF CLASSES OF RELATIONS [ 19631

79

Recursive functionals and quantifiers of finite types 11, Trans. Amer. Math. SOC.108 (1963) 106-142.

Moschovakis, Y.N. [ 19671 Hyperanalytic predicates, Trans. Amer. Math. SOC.129 (1967) 249-282. [ 19691 Abstract fiist order computability 1,and 11, Trans. Amer. Math. SOC.138 (1969) 427-464.and 465-504. [ 19711 Axioms for computation theories - first draft, in: R.O. Candy, C.E.M. Yates (eds) Logic Colloquium '69 (North-Holland, Amsterdam, 1971) 199-255. [ 19731 Elementary Induction on Abstract Structures (North-Holland, Amsterdam, 1973). Platek, R. [ 19661

Foundations of recursion theory, Ph.D. Thesis, Stanford University, 1966.

Sacks, G.E. [ 19731

The I-section of a type-n object, this volume.

Spector, C. [ 19611

Inductively defined sets of natural numbers, Infinitistic Methods (Pergamon, New York, 1961) 97--102.

Added in proof. I thank D. Normann, A.S. Kechris and L. Harrington for noticing that Conjecture 2 was formulated in the first draft of this paper so that it was trivially false. L. Harrington and J. Moldestad recently disproved Conjecture 1 by showing (independently) that the 1-envelope of every normal object U of type 2 3 is also the 1-envelope of some normal type-3 object. (July 9, 1973.)

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

THE 1-SECTION OF A TYPE n OBJECT Gerald E. SACKS Harvard University, Massachusetts Institute of Technology

1. Introduction This paper is a puffing up of the proof of the plus-one theorem for the case k = 1. Let nE be the representing function of the equality predicate for all objects X , Y of type less than n : n E ( X , Y) = 0 if X = Y , and = 1 if X # Y .

Plus-One Theorem. Let U be o f type n. Suppose nE is recursive in U and k < n. Then there exists a V of type k + l such that the objects of type k recursive in Vare the same as those recursive in U. Furthermore kilE is recursive in V. Recursion in objects of finite type was discovered by Kleene [ 2 ] .An equivalent formulation, needed for the forcing argument of section 4, is given in section 2. The proof of the plus-one theorem for the case k > 1 will be given in [ 3 ] .It is largely a consequence of a stability lemma described in section 5 . kscU is the set of all objects of type k recursive in U , and is called the ksection of U . The plus-one theorem states that all k-sections generated by finite type objects (in which the appropriate equality predicates are recursive) are generated by type k + l objects. The first result on k-sections was Kleene’s [ 2 ]: I sc2E is the set of hyperarithmetic reals. He asked if the A: reals constituted the 1-section of some type 2 object. They do by virtue of the complete characterization of 1-sections of type 2 objects (in which 2E is recursive) developed in section 4. The preparation of this paper was partially supported by NSF contract GP-29079. Its principal results were announced in [ 11. The author is grateful to T. Grilliot for steering him towards k-section problems. 81

82

G.E. SACKS

Section 2 redoes some of the elements of recursion in objects of finite type. Section 3 introduces the notion of abstract I-section and shows that many familiar collections of reals, among then the I-section of nE ( n 2 2 ) and the set of lightface A; reals (min (i,j) 2 I), are countable abstract l-sections. Section 4 proves every countable abstract l-section is the 1-section of some type 2 object in which 2E is recursive by means of a forcing argument of the sort associated with generic classes rather than sets. Section 5 describes some further results based on the technique of section 4, and speculates on the nature of abstract k-sections when k > 1.

2. Recursion in higher types The objects of type 0 are the nonnegative integers. An object of type n > 0 is a total function whose arguments range over all objects of type < n and whose values are objects of type < n. Any object of type i can be inflated to an equivalent object of type j > i by adding dummy arguments. Any object of type n > 0 is equivalent to one of type n whose values are either 0 or 1; e.g. the function F(x) is equivalent to the representing function of the predicate F(x) = y . Any finite sequence of objects is equivalent to a single object; e.g. the pair F,(x), F l ( x ) is equivalent toH(n,x), whereH(n,x) = Fn(x) for n < 2 , and = 0 otherwise. The previous three sentences should make the ambiguities in what follows tolerable. Fix n 2 0, and let F , G, ... be objects of type n+2 called functiorzals. The definition of G 5 F , n+2E(read G is recursive in F, n+2E)is given by means of a hierarchy inspired by those of Shoenfield [4] and Grilliot [ 5 ] .D. MacQueen (cf. [6]) has shown that G F , n+2Eif and only if G is recursive in F , n+2Ei n the sense of Kleene [ 2 ] .MacQueen's argument is not unlike that of Grilliot [5]. The hierarchical approach via total objects is equivalent to Kleene's schematic approach via partial objects because of the presence of n+2E. Furictions f,g, h , ... and sets R , S , T , ... are objects of type n+ 1 . Individuals a , b , c , ... are objects of type at most n. Subirzdividuals r. s, t , ... are objects of type less than IZ when n > 0 and are nonnegative integers when n = 0. Fix 1; and g. The hierarchy {So} of sets, defined by induction on u , is designed to define "G is recursive in F , n+2E,g via e" by induction on n. Stage 0. So contains ( 1, a ) for eveiy a, and ( 1, a, m ) whenever ga = m. ( 1, a ) is a n index for So for evely a.

<

THE 1-SECTION OF A TYPE n OBJECT

83

Stage~+l.(2~,a)isanindex forS,+l if(i) (2e,a)$S,,(ii) ( m , a > i s a n index for S , for some integer m, and (iii) there is a functionf recursive in S,, n + l ~ a, via e. Clause (iii) makes sense by induction on n. If n = 0 the clause states f is Turing reducible t o S via e. T h e f o f clause (iii) is denoted by Xb{e}ScJ3"t'EJ"(b). S+ ,l is S , augmented by: (1) all indices for So+l;(2) all quadruples (3e,a,b,rn)if(2e,a)isanindex forS,+, and {e)Su~"+"~"(b) = m ; and (3) all triples (5e,a,rn) if (2e,a) is an index for S , and F(Xb{e}S~~"+'E~"(b)) =m

.

I(e,a)l= u means ( e , a ) is an index for S,. StaKe X, where X is a limit. (7e,a) is an index for S, if (2e.a) is an index for some Ss+l< X and Xb {e)Su,"+lE'u(b) is the characteristic function of a set T of indices such that

S, is U{Ss16<X} augmented by all indices for S,.

f is said to be recursive in F , n+2E,g,avia e, written f=

{ e lF,n+2E,g,a

if (2e, a> is an index for some a n d f i ~ X b { e } ~ ~ ~ " + ' ~ ' ' (G b )is. said to be recursive in F , ni2E, g via e, written

if G(h) is {e}F>"'2Erg,h(0) for all h. R is recursive in F, n + 2 if~ its characteristic function is.ksc(F, is. the set of all objects of type k recursive in F, n+2E.If n+2Eis recursive in F in , = kScF. the sense of Kleene [ 2 ] then k ~ c ( Fn+2E) R is recursively enumerable in F, n+2Eviae if R = { 0 1 ( 2 ~ , ais) an index}. This notion of recursive enumerability coincides with Kleene's [ 2 ] .Thus R is

G.E. SACKS

84

recursive in F , n+2Eif and only if both R and its complement are recursively enumerable in F,n+2E. The principal fact needed in the next section is the existence of a selection operator discovered by R. Candy [ 7 ] ,who had t o wrestle with Kleene's definition. His result follows more readily from the hierarchial definition of this section. Candy's Selection Operator. There exists a recursive function hele* such that if T is a nonempty subset of w recursively enumerable in F , rz+2Evia e, then {e*}F>"'2E(0)is defined and belongs to T.

3. Abstract 1-sections Let A be a nonempty transitive set. A is said to be an abstract 1-section if it is closed under the operations of pairing and union, and satisfies axiom ( I ) and schemas (2)-(3), where a E A and a ( y ) and a ( x , y ) are A 0 formulas of ZF (i.e. formulas whose quantifiers are restricted, cf. Levy [lo]) with parameters in A . ( 1 ) Local countability: (x) (x is countable). (2) A, separation: ( E X ) ( Y ) [ Y E X -.Yea ( 3 ) A. dependent choice:

9(Y)l.

(~)(E.Y)2 (x,Y) -,(E h)(n)9 (hn,h(n + 111, where h : o + A . If A is an abstract I-section, then A is an admissible set as defined by Platek [8], and every member o f A is hereditarily countable. Each hereditarily countable set x can be encoded by a real number Y . Let m be the function that takes a code Y to the set mY encoded by Y . The relation Y is a code & nzY = x Not surprising, since his argument is based on a shadowy hierarchy of computations arising from the Kleene schemas of recursion. The existence of Candy's selection operator was proved by him when n = 0 , by Moschovakis 1161 when n = 1, and by Platek [ 8 ) when I 1 > 1.

THE 1-SECTION OF A TYPE n OBJECT

85

is defined by induction on the rank of x : Y is a code for {m (Y,) I n
-

Proposition 3.1. Let d be a code and P(e),defined by

a(.)

a A 0 formula of ZF. The predicate

is recursive in d , =E. Proof. If sets correspond to codes, then restricted set quantifiers correspond to number quantifiers. For each K C 2- let W K be the set of all sets with codes in K . It is convenient to ignore the distinction between K and W K (e.g. to say K is an abstract 1-section instead of%K is an abstract 1-section) when every member o f K is coded by some member o f K . The next proposition extends a result of Platek [ 8 ] to the effect that the least ordinal not i n m ( , scnE) is XI admissible.

Proposition 3.2. Suppose n > 1 and U is a type n object in which n E is recursive. Then the 1-sectionof U is a countable abstract 1-section.

Proof. It is immediate that W ( sc U) is closed under the operations of pairing and union. To check A. separation let 9 ( y ) be a A. formula and d E scU be a code. It suffices to find a code c E sc U such that ( y ) [ yE m c - y E m d & a ( y ) ]holdsin%(lscti).ThepredicateP(e), defined by

is recursive in U by Proposition 3.1. The desired c is such that {c,ln<w> is an enumeration of all the e's that satisfy P(e). To check A. dependent choice let a ( x , y ) be a A. formula such that

G.E. SACKS

86

holds in W(sc v). Suppose { P } ~ is a code. Let Q, be the set of all n such that {n}’

is a code &

9 ( m { p } O , rn{n}u)

Qp is recursively enumerable in U (uniformly in p ) . Gandy’s selection operator yields a partial function t recursive in U such that t p E Q, whenever { P } ~ is a code. Define g recursively in U by: go = eo ( { e o } u is a code for 0) ,

g(n+1) = tdgrz) .

For each ordinal a let L , be the set of sets constructible in the sense of Giidel [ 1 I ] via ordinals less than a. a is said to be El admissible if L , satisfies the E l replacement axiom schema of ZF. Proposition 3.3. If a is a Eladmissible ordinal, then L , r\ 2w is an abstract 1-section. Proof. Godel [ 1 11 shows L n 2w = L w l n 2 w , where w1 is the least ordinal not countable in L . His argument restricted to L , shows

L , ~ ~ w = Ln 2 w , WP

where LJ? is the least ordinal not countable in L,. It follows L satisfies YP local countability. Godel’s wellordering of L restricted to L IS El over LwP. WP

Let A; be the set of all lightface A; reals. Proposition 3.4. I f min ( i ,j ) 2 1 , then Af is a countable abstract I-section. Proof. If i = j = 1, then the proposition follows from Spector’s boundedness 1 theorem [9] for E l subsets of Kleene’s 0 and Kreisel’s selection operator [ 121 I for KI, predicates of numbers. 1.e. the graph o f t is recursively enumerable in U.

THE 1-,SECTIONOF A TYPE0 OBJECT

ai

Assume max ( i , j ) > 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 <_ T, as S is Turing reducible to T, . The forcing relation I tis defined inductively. (i) p It- 9 if 9 is of rank u, p generates T, for all 6 < 0,and

@,I6

< 0).

9 is true in

(ii) p It- (Ex) a(x) if p IF 9 (m) for some m. (iii) p IF( ~ p9 )( p > 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),<, [Y It- I f m I = 11. I t follows from (a*), (b*), Proposition 4.1 and the validity of the Z1 dependent choice schema in%K that there exist functions X m I p , and hmlu, in W K such that

and ( m )(En)[6 < ,u < un]. Let h = sup {uml m E w } and w = = U { p , I m E w } . Clearly h, w E W K . w generates T,+l with index 2“ thanks to (c) and the fact that 6 < A. Consequently w generates some Th with index 7e. None of the pm’s generate Tk by (d). I t follows from requirement ( 2 ) of the definition of forcing condition and Fact H that w ( T k )is undefined. Thus w has a maximum (namely Tk+l) and so is a forcing condition. But then wI t 17el = -h, an impossibility according to (d).

Theorem 4.4. K is a countable abstract I-section i f and only i f K is the 1section of some type 2 object in which =Eis recursive. Proof. If G is a type 2 object in which 2E is recursive, then lscG is a countable abstract 1-section by Lemma 3.2. Suppose K is a countable abstract 1-section. Let F be generic in the sense of Lemma 4.2, and let G be the recursive join of F and *E. Then K = lscG by Lernmas4.2 and4.3. The following three corollaries of Theorem 4.4 are consequences of Propositions 3.2-3.4.

Corollary 4.5. Suppose n > 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

98

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.

ADMISSIBLE SETS OVER MODELS OF SET THEORY

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

102

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

ADMISSIBLE SETS OVER MODELS OF SET THEORY

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.

ADMISSIBLE SETS OVER MODELS OF SET THEORY

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.

ADMISSIBLE SETS OVER MODELS OF SET THEORY

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:

ADMISSIBLE SETS OVER MODELS OF SET THEORY

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

ADMISSIBLE SETS OVER MODELS OF SET THEORY

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

!Dl
[wrtMo]

ifand only if

21(M<1tYN [wrtAO] .

% I ? Cend

3 andM0 C M ,

112

K.J. BARWISE

aM

?iN and if in4 % ' then ylM In particular, if %rl G1 '31 then '?IN. There are a number of other relations (e.g. =,<) which lift up from (m, 3 ) to but the above is what is needed in the applications we have in mind. We could have considered admissible covers of models of KPU, rather than KP and all of the properties would go through with slight modification. We of KP for pedagogical reasons only. restricted ourselves to models

(aM,aN)

86. lnfinitary methods in the model theory of set theory, revisited: Applications of the admissible cover We first came across the admissible cover in trying to understand what was behind some of the proofs in Barwise [ 19701. The general situation was this. = ( M , E )of set theory and we wanted to conWe had a countable model struct an end extension of % with certain properties. I f M was a transitive set with E = E r M then we were able to exploit the completeness and compactness theorems for LM, the admissible fragment of LU1, associated withM. If % was nonstandard, however, we were forced into some strange considerations. I t is the admissible cover which unifies these two cases and allows the simpler proofs to go through whether m is standard or not. We give two examples of this here. In the first we extend a theorem of Friedman's from standard m to arbitrary m. This result uses only Properties I and 11 of admissible covers, I in order to form sentences like

inAM and I1 to know that if a theory T is an element ofAM then the set of E M with amentioned in T is a set of the form d E , d EM, and hence is bounded in 1137. Friedman's result is the following: Let m=( A , € )be a countable admissible set and let T be a theory of LA which is X I on A , contains KP, and is true in some end extension of 1137. (For example, T might be ZF and it might be false in m.) The conclusion is that has an end extension % with new ordinals which is a model of T but where all the new ordinals are nonstandard. We can extend his result as follows.

a

'm

ADMISSIBLE SETS OVER MODELS OF SET THEORY

113

6.1. Theorem.Let $fl=( M , E )be a countable model of KP with admissible cover Let T be a (finitaiy or infinitary) theory C definable on '21M which is true in or some end extension of 11?7. Then T has a model !J which l is an end extension of %17 with new ordinals but such that no ordinal of % ' is the least upper bound of the ordinals of I f l .

aM.

Proof. Friedman's proof used "supercomplete" theories and the infinitary compactness theorem. We could give an extension of Friedman's proof, but for variety we replace supercomplete theories by the forcing version of the omitting types theorem, Theorem 2.2 in Keisler [ 19731. Let LAM be the admissible fragment of LwIw associated with and let LB be the smallest fragment containing L and the sentence 9 : AM

mM

where e(x) is

Ord('%') = { a E M : a is an ordinal}, and ;is a constant symbol inAM used to denote a. Let @ be the class of formulas of LB which are either in LAM or of the form8(x) or B(y). Let W be the class of all models of T which with new ordinals; i.e. models of T': are end extensions of

T

-

Vx[xEa*VbEaEx=b]

for all a € M

T' is nonempty by the compactness theorem for LAM. The sentence )I can be written as an V V 3 (a) sentence:

So by the version of the omitting types theorem given in Theorem 2.2 of Keisler [ 19731, we will know that 9 is true in some model % E % if we can show that: if p ( c l ... c n ) is a finite subset of @(C) satisfiable in W then for each ck E C one of the following is satisfiable in 9 1 ' 2:

114

K.J. BARWISE

p’(a) = p ( c l ... cn) U {ck
for some a E Ord (B)

Assume that none of the p’(a) are satisfiable in W so that T’ U p ( c l ... cn) e(c,). We need to show that T’ U p ” is consistent. It suffices to show that the following C1 theory of LAM is consistent:

+

T V x [ x € a + - + V b t a E x = b ] forall a € M Ck

>a

all a € O r d ( m )

4 ( C l . . . Cn)

all 4 E ~ ( ... c cn> ~ nA ,

C i > i

all a E Ord (m)

if e(cJ Ep(cl ... cn) . Ck > d

all a E O r d ( B )

d>a

where d is a new constant symbol in C, different from c1 ... cn,ck.But any A finite subset of this theory is clearly satisfiable in any model of 111T U p ( c l ... c n ) ,since then we need only have d > for a in some set X € A M ,X C Ord (m) and any such set is bounded in m . 0

a

For our second example we modify (by strengthening both the hypothesis and conclusion) one of the results in Barwise [ 19701. This result shows how property IX comes into play. A subset of an admissible set is C r if it is definable by a C, formula without parameters. 6.2. Theorem. Let m be a countable model of T, where T is a theoly containing ZFC which is E r on the admissible cover 9lM of B. Let K be a cardinal of m, regular or not, such that

There is a model holds in ‘%I. that :

(n of

T which is an end extension of

PI such

ADMISSIBLE SETS OVER MODELS OF SET THEORY

115

1) no subsets of any h < K are added in % (hence all cardinals X < K of are still cardinals in %), and 2) all cardinals of m greater than K have cardinality K in ‘32 (hence many new subsets of K are added by ‘32 ). Proof. Let m= ( M , E )and let 91, = ( M , , E 1 M,) where Mo = {x EM: k “TC(x) has power 5 ~ ”Then ). m by Levy [1965] Since m, is transitive in m, mo m. Let a, and be the admissible covers of and m, respectively, so that %,o
send

m,<, aM0

in the X 1 definition (in order to write the sentence (iii)) and this set is an element ofM, by the hypothesis A < K =, 2 h IK . If T’ is inconsistent there is a proof in A , of a contradiction from T’, by the completeness theorem for countable admissible fragments. This El property ofA, is thus true inAMo, so there is proof inAMo of a contradiction. This proof can use only 2 K axioms from T’ and each constant a i n these axioms has a EM,. Thus there is an inconsistent subset T, T’which asserts (v) only for 5 K setsa, each of them inMo. But this is absurd since fm itself can be made into a model of any such To by the proper assignment of an element ofM, to the constant symbol c. 0 Actually we see that the theory T can be X 1 using parameters fromM, and the proof still goes through. We could combine the above two theorems to make % have no least upper bound for the ordinals in ‘17. We can also combine Theorem 6.2 with a theorem of Keisler and Morley (Corollary A on page 137 of Keisler [1971]) to get

K.J. BARWISE

116

3

“A is a cardinal

>K

”

implies

so that H, subsets of K are added by 3 without adding new subsets to any h
is needed for the result. For example, if the sentence 2”o = H2 is in T and K = Hy then any end extension of satisfies

which is a model of T and

I K F I = H, must add new sets of integers (the old power set of o is enumerated by a sequence of length an ordinal (Y with ICY( = N, true in %).

57.The construction of the admissible cover and related admissible sets with urelements In this section we will sketch a construction w h c h gives the admissible cover, but we shall not verify all of its properties. On the other hand, we shall carry out the construction in a general setting which gives many other examples of admissible sets with urelements. For a simple such example, consider an admissible set A without urelements, with a C, subsetM ofA. There is a natural admissible setAM overM withA as the collection of pure sets ofA,. The general situation needs the following definition.

ADMISSIBLE SETS OVER MODELS OF SET THEORY

117

7.1. Definition.A structure % = ( N , Rl... R,) is Z1 on a structure

Bm= ( m ; B , F ,...) i f N i s E l o n B m andif thereare El relations R i ...R ; , R [ ... RF on Bm such that for all i l l and allx l . . . x n iE N Ri (xl ...x n i )

-

R f ( x l ... xni)

e l R f ( x 1 ...x,.) where Ri is ni-ary.

7.2.Theorem.Let BW= ( n ; B , F ,...) beamodelof KPUandlet 2 = ( N ,R ... R,) be E on Bm.There is a largest admissible set a = ( % ; A , E N ) s u c hthat themap *: %,-+%m iseverywheredefined: p * = p for P E N (a*), = { b * : b EN a } . For b EN, if bF E N then bF E A . We shall sketch a proof of this for the case where there is just one binary relation R (i.e. 1 = 1, nl = 2). The structure 2 3 is~ a~model of KPU as formulated in some language is an L structure) whereas %,will be a model of K P U as L(E, ...) (where formulated in LO(€), where Lo has just the binary symbol R . To keep things straight we denote this second KPU by K P U , . Choose E l formulas (of L(E, ...)) u(x), @ ( x , y ) $, ( x , y ) so that for all xEMUB

118

K.J. BARWISE

which, of course, is true inBm,. The heart of the proof of the theorem consists of constructing a suitable interpretation I of KPU, in the theory KPU, . Toward this end define predicates of L(E, ...) as follows, using for ( 0 , ~and; ) for (1,x).

x

-

Point (a) 3x [u(x) A a = X] set(a)-gb[a=I; A ~ x ~ b ( ~ e t vpoint(x>>l (x) x & ya 3z(y=; AxEz) R ’ ( a , b ) a 3 x 3 y ( a = i A b=y^ A @(x,y)) R”(a,b)- 3 x 3 y ( a = x A b = j A $(x,y)) The predicate Set (a) needs the second recursion theorem for KPU in its definition, the predicate x & y can be written as a A,, formula. The interpretation I of LO(€) into KPU, is given by the following instructions: replace each positive occurrence of R negative occurrence of R occurrence of x E y quantifier over urelements Vp( ...)

3P(-) quantifier over sets Va( ...) 3a( ...) undetermined quantifiers Vx( ...) 3x (...)

by by by by by by by by by

R’ R” XEY Va[Point (a) +(...)I 3a[Point(a) A (...)I Va[Set ( a ) +(,..)I 3a[Set(a) A (...)I Va[Point(a) v Set(@)+(...)] 3a[(Point(a) V Set(a)) A (...)I .

Equality and the propositional operations are not altered. One must carry this out in a sensible way so that no clashes of variables occur. Denote the translation of a sentence @ of Lo(€) under the above by @I.

7.3. Lemma. For each axiom @ of KPU,,

d is a theorem of

KPU, .

Proof. We run quickly through the axioms. Extensionality: The translation reads: for all Setsa and b , if x & a -x & b for all Points and Setsx, then a = b. Now if Set (a), Set(b) then a = b = go, every x E a, is a Point or a Set and x 8 a t--,x E a, ; similarly for b and b,. Hence a,, = b , so a = b.

zo,

ADMISSIBLE SETS OVER MODELS OF SET THEORY

119

Foundation: If there is a S e t a such that @'(a) then there is a S e t a so that + rk ( x ) < rk 01)so choose a of least rank.

d(a)but for all Sets b g a, l@'(b).Note that x & y

Pair: Let co = { a , b } and c = 2,. If a, b are Points or Sets then Set (c) and x G c iffx = a v x = b. Union: Given a Set a let b = {z : 3 y C% a, z & y } , by AO-separation, and let c = 6. Then z d c iff z C% y I% a for some y . do-separation: We wish to show that given a Set a there is a Set b such that for all x , x E bO x

a A d(x).

If a = 2, let bo = { x E a0 : @'(x)) by 3.5 and let b = J0. (The point here is that d is only a I3 formula but (1 is also a 2 formula.)

@?

AO-coUection: Assume Set (a) and for every x d a there is a y , either a Point or a Set such that ~ ( x J ) where , @ is A. in LO(€). Now the whole statement $

is a I3 formula (since x %!c a + x E TC (a)) so, by I3-reflection, there is a w , C w , such that $ ( w ) holds. Let

a

bo =

u

xBa

{ y E w : (Point(w)(y) v Set(w)(y)) A ( @ ) t w ) ( x , y ) }

and b = So: One easily checks that Set (b) and V x E a 3 y d b @'(x,y). 0 Now when one has an interpretationl of one theory To in another theory T, then any model 2' 3 of T, has associated with it a model 8-I of To in the gives rise via l to usual way. Applying this to our situation the structure a structure 8 -I which is a model. of KPU,. The structu;e'Xj$ has the form

!&n

m %m-I

where

= (%';A',E')

g' is isomorphic to % via the map a^ + a, A' is the set of b E B such that 'XjmkSet@)

K.J. BARWISE

120

and E' = & 1A'. If % is admissible then E' is well founded and %@' is isomorphic to the desired admissible set. In general, however,!ljm need not be admissible but only a model of KPU in w h c h case E' need not be well founded. Thus, we need to apply the following lemma to !By<'.

rn

7.4. Lemma. The well founded part of a model o f KPU is again a model of

KPU.

The proof is like that given in A6 of Barwise [ 19721. Combining these two lemmas, with a natural isomorphism 77 ( ~ ( = ai and ) extend 77: VNr -+ VN via ?(a) = { g ( b ) : b € a } ) gives an admissible set The mapping * arises as follows: Let us work again in KPU,. Define

a* = a. if Point (a) and i, = a a*

= { b * :b

% I

a} if Set (a)

using the second recursion theorem for KPU. This mapping, interpreted, maps !Em-' + maps 9't o 9 and satisfies

$m'

(a*)F = {b*:bE'a}

-'

Restricting this map to the well founded part of % %i and then mapping it over t o % via the natural isomorphism gives the conclusion of the theorem. % To see that 21% is the largest admissible set with such a map *, suppose % ' ' where some other admissible over )37 with a map *: & ! '

%+%El

9

p * = p far p € N (a*)F = {b*:b EN a } .

Consider the map

"

defined in %; by recursion over E N :

p" = ( O , p )

a " = ( I , { b " :b E a } ) . If one chases an x the long way around the following diagram, one sees by

ADMISSIBLE SETS OVER MODELS OF SET THEORY

121

induction on EN that all the maps are defined and that 2, the result of all the ’ composite functions, is x for all x E

9’

I

I t

I

Id

%h

Hence C ‘?lW Findly, if b E N and b, C N then let a = (l,{(O,c): c F b } ) in%m. This concludes the proof of the theorem. When one applies this to obtain the admissible cover, one has 23 = 9 = a model of KP. Most of the properm ties of the admissible cover shouf’d be at least plausible from the above proof. Note that in order to show that the admissible cover is admissible with the covering function, one has to have KPUo formulated in Lo(E,f) wherefis a 1-ary function symbol. In the interpretation Z one then has to replace f(x) = y by f(x) = y where f is defined in KPU, by

f(x) = { j :y EX}Y . References Aczel, P. [ 19701

Implicit and inductive definability (abstract), J. Symbolic Logic 35, p. 599.

Barwise, K.J. [ 1969al Infinitary logic and admissible sets, J. Symbolic Logic, 34, pp. 226-252. [ 1969b] Applications of strictn: predicates to infinitary logic, J. Symbolic Logic 34, pp. 409-423. [ 197 11 Infinitary methods in the model theory of set theory, Logic Colloquium ‘69, North-Holland, 1971, pp. 53-66. Annals of Mathematical Logic 4, pp. 309-340. [ 19721 Absolute logics and L, Barwise-Gandy -Moschovakis [ 19711 The next admissible set, J. Symbolic Logic 36, pp. 108-120. Friedman, H. [ 19721 Countable models of set theories, to appear. Keisler, H.J. [ 197 1) Model Theory for Infinitary Logic (North-Holland, Amsterdam).

122 [ 19731

Kunen, K. [ 19681 Ldvy, A. [ 19651

K.J. BARWISE Forcing and the omitting types theorem, to appear. Implicit definability and infinitary languages, J. Symbolic Logic 33, 446-451. A hierarchy of formulas in set theory, Memoirs of Amer. Math. SOC.No. 57

Moschovakis, Y.N. [ 1973J Elementary induction on abstract structures (North-Holland, Amsterdam) Shoenfield, J. [ 1967J Mathematical Logic (Addison-Wesley, Reading, Mass.)

J.E.Fenstad, P.G.Hinman (eds.), Generalized Recursion Theory @ North-Holland Pu bl. Comp. 19 74 ~

AN INTRODUCTION TO THE FINE STRUCTURE OF THE CONSTRUCTIBLE HIERARCHY (Results of Ronald Jensen) Keith J. DEVLIN University of Oslo

$0. Introduction We shall work in Zermelo-Fraenkel set theory (including the axiom of choice) throughout, and denote this theory by ZFC. We shall adopt the usual, well-known, notations and conventions of contemporary set theory (e.g. an ordinal is defined to be the set of all smaller ordinals, cardinals are initial ordinals, etc.). The paper is entirely self-contained, but some familiarity with the usual definition of the constructible universe, L, in terms of definability, and the proof that L is a model of ZFC + GCH + V = L, will be helpful. The exposition is based, with permission, very strongly on a set of notes written by Ronald Jensen and entitled “The Fine Structure of the Constructible Hierarchy”. Except where otherwise stated, the results are entirely those of Professor Jensen. It is convenient at this point for us to express our appreciation of several dluminating discussions with Professor Jensen on his work in general.

’

Since we wrote this paper, a slightly revised version of these notes has been p u b lished as a research paper. See R.B. Jensen, “The Fine Structure of the Constructible Hierarchy”, Annals of Mathematical Logic, Vol. 4 [ 19721, p. 229. The present paper represents a lengthy discourse on an expansion of the earlier parts of Jensen’s paper, and it is hoped that the somewhat more leisurely pace we adopt here (as opposed to Jensen’s paper) will be of benefit to those not predominantly interested in the set theoretical consequences of the Fine Structure Theory. For those who are so inclined, the notation we use is almost identical to that of Jensen, so this paper should provide a good introduction to Jensen’s. 123

124

K.J. DEVLIN

Previously, Jensen worked, as did most other people, with the usual “constructible hierarchy”. Thus, one defines, inductively, sets La, a E OR, by setting Lo = 0, LA = U,
+

FINE STRUCTURE OF THE CONSTRUCTIBLE HIERARCHY

125

structure of the Jensen hierarchy. A corresponding theory may also be developed for the L-hierarchy, the only difference being that some awkward complications arise because of the above mentioned defects in this definition.

3 1. Preliminaries We shall be concerned with first-order structures of the form M = ( M ,E,A >, where M is a non-empty set and A C M . In general, we shall write ( M ,A ) for ( M ,E, A ). The (first-order) language for such structures consists of the following: (i) variables u j , j E w (generally denoted by u, w , x , y , z , etc.) (Vbl.). (ii) predicates =, E, A . (iii) bounded quantifiers (Vui E uj), (3ui E uj), i,j E w , i # j . (iv) unbounded quantifiers ( h i ) , ( 3 u j ) , i E w . (v) connectives A, v, 1, +,-. Finite strings of variables (or of elements ofM) are denoted by v , X, etc. We w r i t e u E X f o r u l E X A ... A u , E X , w h e r e w e h a v e ~ = a ~...,, a,.Similarly for 3 v , V v , etc. The notions of primitive formula (PFml), formula (Fml), free variable, statement, and satisfaction are assumed known. A formula of this language is Co (orn,) if it contains no unbounded quantifiers. Let n 2 I , and let Q, denote V if n is even and 3 if n is odd. A formula is C,(ll,) if it is of the form 3 x 1 V x 2 3 x 3... Q , x ~ ~ ~ ( V X , ~...XQn+l~,cp) ~ V X ~ where cp is C,. A formula in wluch the predicate A does not occur is called an E-fomulu. kMdenotes the satisfaction relation for M . Thus, kM is the set of all (p, (2)) such that p is a formula of the above language and 2; E M and p holds z in M at the point (2). We generally write k Mcp[~]for (cp, (2)) E kM.‘t=Mn denotes the set of all (cp, (2)) E bM such that cp is En. Let N C M . A set R C M is E f;l(N) (IIF(N)) iff there is a En (n,) formula cp(u,v) and elements a E N such that for all x EM, R(x) q [ x ,u ] . The set of all such R is also denoted by X r ( N ) (II,”(N>>. setXz(N) = nn,M(N). Write C for XF(@ and E,(M) for C p ( M ) . Similarly for II, A. If cp is a formula, cpM denotes the relation { x EM1 cp[x]}. Similarly, and more generally, define cpM [a]for u E M as {x EM1 b~ ~ [ xa]}. , Let F be a class of structures of the form M = ( M , A). A relation R is

-

u,,;x,M(N),A,M(N)=x,M(N)

bM

K . J . DEVLIN

126

uniformly E n ( M) for M E F iff there is a Z, formula q(u, v ) and elements a E n { M I M E F } s u c h t h a t w h e n e v e r M E F , R nM=cpM[a].

52. Rudimentary functions A function f : Vn -+ V is rudimentary (rud) iff it is generated by the following schemata: (i) f(xl, ..., x,) = x i , I Ii In. (ii) f ( x l , ..., x , ) = x i - xi, 1 Ii, j i n. (iii) f(xl, ..., x , ) = { x i ,xi}, 1 Ii,j In. (iv) f ( x l , ...,x , 1= h (81 ( X I ,...,x , ), ...,gk(x 1 , ...,x, 11, where g l , ...,gk, h are rud. (v) f(xl, ...,x , ) = U y E x lh ( y , x 2 , ...,x , ) , where h is rud. For example, the following functions are clearly rud: f ( x )= u x i f(x) = xi u xi (= {Xi ,Xi})

f@)= f@)=

u

{XI

( x ) = { t q } , t X l , ( X 2 , . . . , x , >)I.

And i f f ( y , x ) is rud, so i s g ( y , x ) = ( f ( z , x ) l zE y ) (= U z E y { ( f ( z , x ) , z ) } ) . We say that R C Vn is rudimentary (rud) iff there is a rud function r : Vn + V such that R = {(x) I r(x) # 8). For example, 6 is rud, since .Y~x+-+I-xf!J. We list some basic properties of rudimentary functions and relations. (1)

Iff, R are rud, so is

Proof. By definition, there is a rud r such that R ( x ) ++ r ( x )#

0. Then

= UyEr(x)f(X).

(2)

Let xR be the characteristic function o f R . R is rud if xR is rud.

Proof. By (1).

FINE STRUCTURE OF THE CONSTRUCTIBLE HIERARCHY

(3)

121

R is rud iff 1 R is rud.

Proof. By (2), since xqR(x)= 1 - xR(x). (4)

Let fi : V" -+ V be rud, i = 1, ..., m. Let R j C V n be rud and mutually disjoint, i = 1, ..., m , and such that U E I R j= V n .Define f : V n + V by f ( x ) =fi(x) iffRi(x). Thenfis rud.

By (l), (5)

fi is rud. But f(x)= uEl fi(X).

IfR(y, x) is rud, so isf(y,x) = y n { z l R ( z , x ) ) .

Then h is rud. But f ( y , x )= U Z E , , h ( z , x ) .

(6)

SupposeR(y,x) is rud and (Vx)(3!y)R(y,x).Set the unique z € y such that R ( z , x ) ,if such a z exists.

8,

otherwise.

Then f is rud.

Proof.f(y,x)= U [ y n { z I R ( z , x ) ) ] .

(7)

-

IfR(y,x) is rud, so is (32 € y ) R ( z , x ) .

Proof. Take r rud so that R ( y , x )

r ( y , x )f

8. Then

K.J. DEVLIN

128

(8)

IfRi(x) are rud, i = 1, ..., rn, then so are m

rn

n Ri,

U Ri,

i=1

i=l

(9)

(Trivial).

Let (-)o,(-)l, denote the inverse functions to (-, -). Then (-)*, (-)1 are rud. More generally, let (-);, ...,(-):-1 denote the inverse functions to ( x l , ...,xn).Then (-)", ..., (-):-1 are rud.

Proof. the uniquez E Ux such that ( 3 u l , u z E U x ) ( x = ( u l , u 2 ) Au1 = z ) (XI0 =

6,

if no such z exists.

etc.

(10) The function the unique z E f ( X J > = X(Y> =

6,

u u x such that ( z , y )E x

if no such z exists.

is rud. (By definition.) (1 1) dom and ran are rud.

Proof. dom (x) = { z E U Ux I (3w E U Ux)((w, z ) E x)} ran(x)= { Z E U U X ~ ( ~ ~ E U U X ) ( ( Z , W ) E X ) ) . (12) f(x,y)

= x Xy =

( 13) f(x, y ) = x

UuExUvEy( ( u ,u ) } is rud.

r y = x n (ran(x) X y ) is rud.

( 14) f(x, y ) = x"y = ran (x f y ) is rud.

(15) f(x) = xpl is rud.

FINE STRUCTURE OF THE CONSTRUCTIBLE HIERARCHY

129

Proof. Set h(z) = ((z)l,(z)o). Then h is rud. But clearly, f(x) = x-’ = h”(x n (ran (x) X dom (x))). Recalling our preliminary discussion ( S O ) , we observe that though rud functions increase rank, they only do so by a finite amount. More precisely, by induction on the rud definition * of a given rud function f, we see that there is a p € w such that for all x 1 , ...,x,, rank (f(x ...,x,)) < max{rank(xl), ..., rank(x,)} + p . ~

* Note : In future, we shall often refer to “the rud definition o f f ” ,

or simply “the definition off ”. We mean an arbitrary such definition, the actual choice being irrelevant, and hence assumed made once and for all. We now prove that the rud functions do in fact encompass all of the “simply definable” functions we spoke about in SO. First, let us call a function f : V n + V simple iff whenever cp(z, y) is a Co €-formula, there is a Co €-formula J/ such that k cp(f(x),y) ++ $(x,Y). A useful characterisation of this concept is given by the following:

Proposition.A function f : Vn -+ V is simple iff (i) the predicate x € f(y)is Z ;: and ; so is (VX ~ f ( y ) ) ~ ( x ) . (ii) wheneverA(x) is ,z Proof. (+) By definition. (+) Let f satisfy (i) and (ii), and le‘t cp(x,y) be a To €-formula. An easy induction on the length of cp shows that (p(f(x),y) is equivalent to a Lo Eformula; so f is simple. Using this proposition, and easy induction on the definition off yields:

Lemma 1 . Iff is rud then f is simple.

Now, since there are C x functions which increase rank by an infinite amount, it is clear that the converse to the above lemma is false. However, we do have: Lemma 2. R C Vn is XriffR is rud

130

K.J. DEVLIN

Proof. (+) Let R be Zx.By ( 3 ) ,(7), and (8) above, an easy induction on the Cx definition of R shows that R is rud. (+) Let R be rud. Then xR is rud. SOby Lemma 1, xR is simple. Using our above proposition, an easy induction on the rud definition of xR shows that xR and hence R , is Ex. ~

We require some generalisations of these concepts. Let A C V. We say that a function f is rud in A iff f is generated by the schemata for rud functions and the function xA . Let p E V. We say that a function f is rud in parameter p iff f is generated by the schemata for rud functions and the constant functions h(;r) = p . Lemma 3. I f f is nid in A C V, there are rud functions gl,...,gn such that f is expressible (in a uniform way with respect to the rud definition o f f ) as a composition of gl, ...,g, and the function h ( x ) = A n x.

Proof. By induction on the (nid) definition o f f

A set X is said to be rud closed if for all rud functions f , f “ X n C X . A structure M = ( M ,A ) is said to be rud closed if for all functions f which are r u d i n A , f”M” C M . We say a structure M = ( M , A ) is amenable if u E M + A n u E M . Lemma 4. A structure M = ( M ,A ) is rud closed i f f the set M is rud closed and M is amenable.

Proof. By Lemma 3. Lemma 5. L e t A C V. Zff is rud in A , then f r M n is uniformly X o ( ( M ,A n M ) ) for all transitive, rud closed M = ( M , A nM ).

Proof. By Lemmas 2 and 3. The next result will be of considerable use to us later on. Lemma 6. Every rud function is a composition of some of the following rud functions

FINE STRUCTURE OF THE CONSTRUCTIBLE HIERARCHY

131

Proof. Let Cdenote the class of all functions obtainable from F,, ..., Fg by composition. We must show thatfrud + f € C. For each €-formula cp(xl,..., xn), set

By induction on cp, we show that for all cp, t, E C. (The required result will then be proved using this fact.)

(a)

cp(x)-xiExj,

1
Write F,(y) for F3(x,y).Then t,(u) = ui-' X FL-'(F4(E f l u 2 , u"-j)) so t, E

(b)

c.

Let cpI(x),..., cp,(x) be such that tvl, ..., t

PP

E C.

Let p(x) be any propositional combination of cpl, ..., q p . Since x t, E

-

c.

(c)

y , x U y (= U {xJ}), x f3 y (= x

Let &,x)

be such that

tz E

-

-

(x - y ) )

€

C, we clearly have

C. Let ~ ( x ) 3y?(y,x) or VyF(y,x).

Clearly, t=~,,~(u) = dom (t,(u)) and tvYlp(u)= u n - dom(un - tF(u)). So in either case, t, E C. (d)

v(X)

xi = Xj .

K.J. DEVLJN

132

e.

By (a), (b), t, E But look, k=(u,E) q [ x ] iff (VY E U u ) ~ ~ ~ u " ( " u ) , E ) ~ t Y 7 x l l . Hence t,(u) = u n n tvy,(u U (Uu)), so t, E C?, by (c). Let O(y,x)= y E x j -y

(e)

Exi.

cp(x)zxj€xi,

I<j
-

Let J / ( y , z , x ) = y € z h y = x j ~ z = x i . B y ( a ) , ( b ) , ( d ) , E t J(?But I dx) ~ y w ( Y , z , x ) so , by (c), tp E e.

e.

Hence, for any €-formula cp, tV E I f f : V n+ V , definef* : V + V by f * ( u ) = f n u n .Using our above result, we prove by induction on the rud definition off, that f rud -+f * E C . This easily implies the required result.

(b)

f ( ~ ) =xj

~

xi.

f * ( u ) = f " u " = { x - y ~ x , y~ u }Letcp(z,x,y)-z . E X - y . LetF(u) = t,(u U ( U u ) ) n ( U u X u 2 ) = { ( z , x , y ) ( x , y E u ~ z E x - y } . T h e n f * ( u >= ~ ~ ( ~ ( u >E , 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

FINE STRUCTURE OF THE CONSTRUCTIBLE HIERARCHY

-

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

FINE STRUCTURE OF THE CONSTRUCTIBLE HIERARCHY

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-

FINE STRUCTURE OF THE CONSTRUCTIBLE HIERARCHY

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:

FINE STRUCTURE OF THE CONSTRUCTIBLE HIERARCHY

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

FINE STRUCTURE OF THE CONSTRUCTIBLE HIERARCHY

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<, of the S, suck that: C <%; (i) v1 < v2 -+ (ii) is uniformly E+ for all a. Proof. We use Lemma 22. Set <,, = (4, and by induction:

(i) and (ii) are immediate and (iii) is proved like Lemma 25.

Lemma 28. There are well-orderings <,J of the J, suck that : (i) a1 < “2 <J, <Ja2 ; (ii) <J,+l is an end-extension of<J,; < a ) is uniformly E Jl a ; (iii) (<Jpl (iv) <,J is uniformly x (v) the function pr,(x) = { z 1 z <,J x } is uniformly Z k . (“pr” stands for “predecessors”ofcourse.) +

k;

Proof. Set <, = <w,. (i)-(iii) are immediate by Lemma 27. For (iv), note simply that x ++ 3 v ( x <, y ) . Finally, for (v), note t h a t y = pr,(x) * 3 v [ x E S, A y = { z I z <, x)] (and that <, E J,), and use Lemma 27.

144

K . J . DEVLIN

Lemmas 12 and 26 enable us to prove the following extremely powerful result (due in its original form to Godel, the present version being Jensen’s):

Theorem 29 (Condensation Lemma). Let X < c l J,. Then for some 0 I a, X z Jp. Proof. L e t X < c l J,. Then by Lemma 12, let n : X % W , where W is transitive. We prove by induction on a that W = Jp for /3 = n”(Xn a). Assume, therefore, that whenever v < a and X u < z1J,, the unique isomorphism n u o f X V onto a transitive set W’ yields W’ = J,uft(xvnv). Note that, as (J,I v < a) is EP, v E X n a ++ J, EX.

Claim 1 : For all v E X n a,?(J,) = Jn(,]. To see this, note first that for v E X n a,X f7 J,
For v < a,define rudx(Jv) = the rud closure of X

fl(J,,

U {J,}).

Claim 2 : X = UvEXn, rudx(J,). To establish this claim, note that as X < q J,, X is rud closed, so 3 is obvious. For the converse, let x E X . Then x E J, = U,<,rud(J,), so for some rud functionf, F J a ( 3 v ) ( 3 p EJ,)(x =f(p,J,)). ButX
FINE STRUCTURE OF THE CONSTRUCTIBLE HIERARCHY

145

Note. It is easily seen that we may regard the following as part of the statement of Theorem 29: If Y C X is transitive, then r 1 Y = id / Y . And for v E X n a,n(v) 2 v, and for all x E X, r(x) 2J, X. By an argument well known to all set theorists, it is easily shown that J = UaEOR J, is a model of ZFC. (In fact, setting<J = UaEOR<J,, <J is a J-definable well-ordering of the entire class J , so J satisfies the Axiom of Choice in a strong way.) Using the Condensation Lemma, an equally wellknown argument shows that J k GCH. However, in the next section we will prove (and have already indicated this fact in our preamble) that J = L, so all that the above says is that we can use the Jensen hierarchy in place of the Lhierarchy in order to establish the classical results on the constructible universe.

$5. On the fine structure of the Jensen hierarchy As mentioned in the introduction, a theory similar to the one following can be developed for the usual Lhierarchy, if desired. Central in our discussion will be the concept of a “uniformising function” for a relation, which is a sort of “choice function” for a given relation. Specifically, a function r is said to uniformise a relation R iff dom (r) = dom ( R ) and for a l l x , 3yR(y,x)-R(r(x),x). Let M = ( M , A ) , n 2 1. We say M is E:,-unifomzisabZe iff every Zn(M) relation on M is uniformised by a Xn(M) function. A few moments reflection will reveal that En-uniformisability is a very strong condition to demand of an arbitrary structure M,since in the more obvious cases, the definition of a uniformising function for a given relation would appear to increase the logical complexity by one or more quantifier switches. However, it will turn out that for all a,all n 2 1, J, i s Z,-uniformisable. For n = 1, this will be easy to prove, but for n > 1 , the corresponding argument will only work when J, is E:,-l-admissible, so a more indirect approach will be necessary. We shall outline the approach required after we dispose of some of the more easy results. First, E l-uniformisability. The E1(J,) well-ordering of each J, gives

146

K.J. DEVLIN

us this with little effort. in fact, we have a much stronger result, of importance in applications of E l-uniformisability. Let F be a class of structures M = ( M A , ) , n 2 1. Say F is uniformly Enuniformisable if, whenever p is a En-formula w.p.f. n {MI M € F } such that (PM is a relation on M for each M E F , there is a En-formula $ (w.p.f. n {MIM E F } ) such that for each M E F, $M is a function uniformising $1. Theorem 30. (J,, A ) is El-uniformisable.In fact, the class of all (J,, A ) is uniformly X -uniformisable. Proof. Let p be a X -formula w.p.f. J, such that [cpbx)]‘ , JaA)is a E relation on J,. By contraction of quantifiers, we can, in a uniform way, find a X o formula J/ (w.p.f. J,) such that k ( J , , A ) p b , ~ )c-,3 z $ ( z , y , x ) . Define g by: g(x)= the<J-least w such that k(J,JI)J/((~)o,(~)l,~). Thengis (uniforndy) X l((J,, A ) ) , since

Set r(x)= (g(x))l. Then r is (unifcrmly) E l((J,, A ) ) and clearly uniformises [ d Y ,x>l(J,A).

Remark. We call the above construction the canonical X l-uniformisation procedure. Observe that if R ( y , x ) is a E l((J,, A )) predicate, then the canonical Xl-unifomisation o f R is a function whose El((J,,A)) definition involves only those parameters which occur in the definition of R. Let us take a little time off to examine the above construction more closely. SupposeR(y,x) is a given X I relation, sayR(y,x) * 3 z P ( z , y , x ) , where P is X o . To obtain the Z; uniformisation of R , we first obtain a Z; uniformisation of the T o relation {(w,x)( P((W)~, ( W ) ~ , X ) } and , then simply pick out the requisite component of the result as our required function. And since < is a X well-order of J, the result is also E 1 . However, returning now to the notation of Theorem 30, we see that, if we try to extend this procedure to the case n > 1, we cannot conclude that the function g is En, the problem being the last conjunct in the explicit definition of g. Let \k(w,x)

FINE STRUCTURE OF THE CONSTRUCTIBLE HIERARCHY

147

denote the predicate [l$ ( ( W ) ~ , ( W ) ~ , X ) ] ‘ ~ “ ~F’o. r n = 1, there was no problem, since if \k(w,x)is so is (Vw E t)\k(w,x).However, for n > 1, \k(w,x) is I:,-l, and we can only conclude that (Vw E t)\k(w,x)is I:,-, if (J,,A) is Z,-l-admissible. Otherwise it is merely II, of course, and so the resulting uniformisation of the original X, relation turns out to be En+,. So, in order to establish the general I:,-uniformisation lemma, it is not altogether unreasonable to try and “reduce” all En relations on an arbitrary J, to C, relations on some I:,-l-admissible Jp, for which we have a I:,-uniformisation procedure. In practice, it will turn out that this hint is slightly off target, but in its general tone it is worth bearing in mind. Closely connected with Z,-uniformisability is the notion of a “2, Skolem function”. Let M =( M , A )be transitive and rud closed. By a 2 , Skolem function for M we mean a &(M) function h with dom (h)C w X M , such that for some p EM, h is Z:({p}), and wheneverP E Z r ( { x , p } )for some x EM, then 3 y P ( y )-+ (3E w)P(h(i,x)).(With h, p as above, we say that p is a good parameter for h.) Note that C, Skolem functions need not be (and in general are not) total! As far as existence of I:, Skolem functions is concerned, we can get away with slightly less than might first appear. In fact:

zo,

Lemma 31. Let M = ( M ,A ) be transitive and rud closed. Let h be a Z r ( { p } ) function with dom ( h )C w X M. Suppose that whenever P E I:,,”({x}) for some x E M, then 3yP(y)-+(3i E w )P(h (i, x)). Then M has a X Skolem finction.

,

Proof. S e t i ( i , x ) = h ( i , (x,p)).It is easily seen that h“ is a C, Skolem function for M. Note that in the above, if h is actually I:,: then h” = h. This is used in establishing the following result:

Lemma 32. If (J,, A ) is amenable, then it has a E Skolem function. In fact, for (Ja, A ) which is uniformly I::Jff$” there is a C Skolem function for all amenable (J,, A ) .

Proof. Let ( p i [i < w ) be a recursive enumeration of Fml’l. Let (J,,A) be amenable. By Lemma 9, i=(I; is (uniformly) E:J,A’. Let h = h a J be the

JLJA)

canonical I: l-uniformisation of the I::Jae) relation {(y,i,x)l k $ ~ e ) p i ~ . x ] } .

K. J. DEVLIN

148

(By Lemma 30 and the ensuing remark, h.is thus uniformly E i J a P ) for amenable (Ja, A ) .) By the remark following Lemma 3 1, it is clear that h is a El Skolem function for (Ja, A ) . We refer to h, as the canonical E Skolem finction for (amenable) (J,, A ) . By a similar argument, we have: Lemma 33. If (J,, A ) is amenable and X,-uniformisable, it has a X, Skolem function. The following lemmas indicate our reason for using the word "Skolem" here. Lemma 34. Let M be transitive and rud closed, and let h be a I;, Skolem function for M. Then whenever x E M , x E h " ( o X {x})< c 'M. n

Proof. Set X = h " ( o X {x}). Clearly, x EX. Let P E X F ( X ) , P f @. We must show t h a t P n X # $ . L e t p be agoodparameter f o r h , a n d p i c k y I , ...,y, E X with P E E;({yl, ..., y,}). By definition of X,there are j l , ..., j, E w such that y1 = h(jl,x), ...,y, = h(j,,x). Since h is Z F ( { p } ) , it follows that PEXF({p,x}). Hence, P # @-+ 3 y P ( y ) - + ( 3 i E o ) P ( h ( i , x ) ) + ( 3 y E X ) P ( y ) . Lemma 35. Let M be a transitive and rud closed, and let h be a E n Skolem function for M. I f X C M is closed under ordered pairs, then X C h"(U XX)<x M. n

Proof. Set Y = h"(w X X ) . By Lemma 34, X C Y . Let P E E,"( Y ) , P # @. We must show that P n Y # 0. Let p be a good parameter for h , and pick y l , ...,y m E Y w i t h P E X F ( { y l , ...,ym}).Pickjl ,...,j,Ew a n d x l , ...,x , E X such t h a t y l = h ( j l , x l ) ,..., y, =h(j,,x,).Letx=(xl ,..., x,).Byassumption, x EX. But clearly, ash is ZF({p}), P i s then Ey({p,x}), so P f @ + 3yP(y) + (3iE o)P(h(i,x)) + ( 3 y E Y ) P ( y ) . Corollary 36. Let M,h be as above. Let X C M and suppose h"(w X X ) is closed under ordered pairs. Then X C h"(w X X ) i M. n

FINE STRUCTURE OF THE CONSTRUCTIBLE HIERARCHY

149

Proof. Let Y = h”(w X X ) . Clearly, Y = h”(w X Y ) ,so the result follows by the lemma. Lemma 37 (Godel). There is a bijection @J : OR2 *OR such that Iwa is uniformly C for all a.

@(a,P)2 a,P for all a,P, and @-’

Proof. Define a well-order <* of OR2 by

Let @ : <* OR. By induction on a,@-’ 1 wa is C p (uniformly).

Lemma 38. There is a I: l(J,) map of w a onto (oa)*for all a. Proof. Let Q = (a1@(O, a) = a}.Then Q is closed and unbounded in OR. In fact, Q = (a1@ : a2 ++a}, so wa E Q + wa = a.We prove the lemma by induction on a.Assume it is true for all v < a.

Case 1 : wa E Q. Then W’ 1 a suffices.

+ 1. I f P = 0, then w a = w E Q , so we are done by Case 1. Hence we may assume 0 2 1. Then clearly, there is a X1(J,) map j : wa u p . By hypothesis, there is a E l(Jp) map of wp onto so there is certainly a one-one into wb. Theng E rud (Jp) = J,, so for I: l(Jp) map g of v,y E wa, define Case2: a = P

++

Then f is E1(Ja) and f maps ( ~ aone-one ) ~ into 06. Now r a n 0 = ran(g) E J,, so if we define h by (for v E wa)

i

f-’(v),

h(v) =

if v E ran (f)

(0,O) , otherwise

we see that h is C1(J,) and h : wa-

K. J. DEVLIN

150

Case 3 : lim(a)

A

o a 4 Q. In this case let ( u , T )

= @-'(ma).

T h u s v , T < w a . S e t c = { z l z < * ( v , ~ ) } ( € J , ) . T h u s @ } c m a p s c one-one onto wa, and is I:l(J,). Pick y < a with v, T < oy.Then @-I 1 w a is a C '(J,) map of o a one-one into oy.And by assumption, there is a map g E J, mapping (or12 one-one into wy. SO, settingf((L, 6 )) = g((g(@-'(L)), g(W1(0)))), I , 0 < wa, we see that f i s a X l(J,) map of ( 0 ~ one-one ) ~ into d = g " ( g " ~ )But ~ . d E J,, so we can define a X1(J,) map h on wa by

I

f - ' ( ~ ) , if

h(0)=

(0, O),

e ~d

otherwise.

onto

Clearly, h = o a --+( ~ a ) ~ . The lemma is proved. Using this lemma, we may now establish the following important result: Theorem 39. There is a E l(J,) map of o a onto J, for all a.

be Ep({p}), wherep E J, is the <J-least eleProof. Let f : oa% ment of J, for which such an fexists. Define f o , f 1 by demanding that f ( v ) = (fo(v),fl(v)) for all v E oa. By induction, definef, : o a %(ma)" thus: f o = id 1 ~ a ; f , + ~ (=v ()f o ( v ) , f , .fl(v)). Hence eachf, is Z k ( { p } ) . Let h = h,, the canonical C l Skolem function for J,. Set X = h " ( w X (wax { P I ) ) . Claim 1. X is closed under ordered pairs. To see t h i s , l e t y l , y 2 EX, s a y y l = h ( j l , ( v l , p ) ) , y2 = h ( I 2 , ( v 2 , p ) ) . Let

( v l , v 2 ) = f2(7). Then { ( y 1 , y 2 )is} a E k ( { ~ , p } predicate, ) so by definition of

h , ( y 1 , y 2 ) E X , asclaimed.

So by Corollary 36, X
8 5 a. Since w a c X , we clearly have 8 = a here.

Claim 2. For all i E w , x E X , n(h(i,x))= h(i, n(x)).

FINE STRUCTURE OF THE CONSTRUCTIBLE HIERARCHY

151

To see this, observe first that as h is X p , there is a rud function H such that

x

y = h ( i , x ) *( 3 t E J , ) [ H ( t , i , x , y ) = 11. Now let i € w , x EX. Since X < J,, 1 y = h ( i , x )E X (if defined). Thus, by the above, since x,y E X
n ( y ) ) =l].Sincen”X= J,, thiscan be rewritten as(3tEJa)[H(t,i,n(x),n(y)) = 11. Thus n ( y )= h(i,n(x)),as claimed. Now, f C ( ~ a so ) ~as ,n 1 o a = id 1 oa,n’y=f.And by isomorphism, n‘yis E p ( { n ( p ) } ) .So as n ( p ) IJp , the choice of p shows that n ( p ) = p . So, by Claim 2, if i E a,v E wa, n(h(i, (v, p))) =h ( i , (u, p ) ) , which is to say ntX=idrX.ThusX=J,. Now define i: ( w ~ r+ )~ J, by setting

I

y , if ( 3 t E S,) [H(t,i , (v, p ) ,y )

i(i,U,7) =

8,

11

otherwise.

Thenzis El(J,), andclearlyi”(oa)3 =h“(oX(oaX{p}))=X=J,. Therefore, f 3 is as required by the theorem. Observe that in Lemmas 38,39, the maps constructed generally have parameters in their definitions. Note also that, being total, these maps are in fact A,(J,). Recalling the results of $ 3 , we now investigate those ordinals cy for which J, is an admissible set. Let us call an ordinal a admissible iff a = o p and Jp is an admissible set. Theorem 40. wa is admissible iff there is no E l(J,) map of any y < wa cofinally into we. (Note that such a map, having domain y E ,J would in fact be Al(J,).) Proof. (+). Let y < wa and suppose f : y + wa is El(J,). Then ( V i E y)(3( E oa)(f(.$)= (). If J, is admissible, then by X l-Replacement, ( 3 ~ € w a ) ( ~ i E ~ ) ( 3 ( € ~ ) ( fsofisnotcofinalinoa. (~)=(), (+). Assume w a is not admissible. If cy = fl + 1, then the Z1(J,) map ((wflt n , n > l n E o } maps o cofinally into wa, so we are done. Assume then that lim (a).Since J, is not admissible, there must be a E I(J,) relation R and a ~1 E J, such that (Vx E u) (3y)R(x,y) but for all z E J,,

K. J . DEVLIN

152

l ( V x E u ) ( 3 y E z ) R ( x , y ) .Take y < a with u E J,. By Theorem 39, let f be a C1(J,) map of wy onto J,. Thus f E J,, and u C f " o y . Define g : o y -+ o a

the least T such that ( 3 y E S,)R( f ( v ) , y ) , if f(v) E u

0, if f ( v ) $ u . Theng is a C1(J,) map of o y cofinally into wa. Recalling our discussion at the end of 8 3 , let us call an ordinal a strongly admissible (or non-projectibleadmissible) iff a = 00 and Jp is strongly admissible. Imitating the proof of Theorem 40, we have: Theorem 41. wa is strongly admissible iff there is no Z ,(Jay> 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 -+

FINE STRUCTURE OF THE CONSTRUCTIBLE HIERARCHY

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<,Sn(x) is C1(J,). But if u is transitive, then C,(u) = P ( u ) n rud(u). war). So, by the recur-’ Hence - . -the map Ly * X,(Ly) = L,+l is E1(J,)(y< sion theorem again, we see that (L,l v < war) is l(Ju). Hence L, = L, C J,. For the converse inclusion, it suffices to show that L,, is admissible. (For then, by the recursion theorem, (S, I v < oar)is Cl(L,J, so J,=U,<,,S,CL,,.)LetR beXO(L,,),xEL,,,andassume (Vz E x ) 3yR(y,z). Since (L,I v < war) is El(Ja), we may define a X1(J,) predicate R’ by R ’ ( v , z ) z E x A (3y E L,)R(y,z). Since J, is admissible, t h e r e i s T < w a with(Vz E : x ) ( 3 v < ~ ) R ’ ( v , z ) .Hence(Vz€x)(3yEL7)R(y,z). So as L, E L,, L,, satisfies the X O-Replacementaxiom. Since lim (a), it follows easily that L, is admissible.

uv.,w,

-

Let a,n 2 0. The E,-projectum of a, p:, is the largest p < _ asuch that (J,,A) is amenable for allA E C,(J,) fl 3 ( J ) P’ Roughly speaking, our reason for introducing the X, projectum is this. We have seen that, for example, we can reasonably handle C,(J,) predicates when J, is C,-admissible. This is because Cn-admissibility is a sort of “hardness” condition on J, for C, predicates. For, if we take an arbitrary J,, it may be “soft” for Xn(J,) predicates; we may, for instance, find Zn(J,) subsets of members of J, which are not themselves members of J,, or even E,(J,) functions which project a subset of a member of J, onto all of J,. But if J, is C,-admissible, none of these situations can arise. Thus, we try to isolate that part of J, which is “hard” for E,(J,) predicates, a sort of “Enadmissible core” of J,. One natural way of formalising these ideas is provided by the Z,l projectum. Clearly, J is a reasonable interpretation of the notion P” of a “C,-hard core” of J,. We sh%l eventually give two characterisations of the x,-projectuni which make it appear even more reasonable - if not inevitable. One of these is that p z is the smallest p a for which there is a X,(J,) map of a subset of u p onto J,. Then, since we clearly have, for wa admissible, that o a is strongly admissible iff p: = a,we obtain some justification for our alternative name of “non-projectible admissible” for strong admissibility. I t is convenient, at this point, for us to define an obvious generalisation of the notion of the C -projecturn of an ordinal. Let (J,,A) be amenable. The C,-projectum of (J,,A), p z p , is the largest

<

,

FINE STRUCTURE OF THE CONSTRUCTIBLE HIERARCHY

155

5 a such that (J,,B)

is amenable for all B E C,((J,,A)). is always strongly admissible. Note that by Theorem 43, We shall make strong use of the X,-projectum in proving that every J, is Z,-unifomisable, all n 2 1. Since most of the following lemmas are directed towards this goal, it is worth indicating briefly our strategy. We already know that (J,,A) is Zl-uniformisable for all (J,,A). What we shall do is attempt to “reduce” Zn(J,) predicates to C1((Jpn,-4)) predicates for someA C J which is itself En(J,). To carry out this re%uction, we G need to have at our disposal a Cn(J,) map of a subset (at least) of J onto 4 J,. Thus, what we shall do is to simultaneously prove, by induction on n , a, the following two propositions: p

(P 1) J, is Z,+l-uniformisable (P 2) There is a Cn(J,) map of a subset of o p : onto J,. The proof of (P 1) goes roughly as follows. Let R be a Cn+l(Ja) predicate J, be C (J ). Now, f is a T,(J,) relation, so by on J,. Let f : C u p : assuming x,-uniformisability, f-’cg be ‘‘shrunk” to a X,(J,) map of J, into u p : . This reducesR to a X‘n+l(J,) predicate R’ on J p n . NOWfind a C,(J,) predicateA C J such that R’ is in fact XI((Jpn,&). UnifomiseR’ P: by a C1((Jpn,A)) function, and then reverse the proced&e to recover a Cn+l(J,) uiformising function for R. There is one doubtful point in the above outlbe. Can we in fact find a setA as required. That we can has to be proved as we proceed, so we shall in fact simultaneously prove three propositions, (P I), (P 2 ) , and a proposition (P 3 ) to be formulated precisely later.

-’

Lemma 46. Let n 2 1, and assume J, is X,-unijiormisable. Let y 5 (Y be the J,. Then there is a X,(J,) map of I least ordinal such that ;P(wy) (Xn(J,) a subset of o y onto J,. Proof. By Lemma 33, J, has a C n Skolem function, h. Let h be Z::((p)). We may assume p is the <J-least element of J, for which such an h exists. Let a C o y , a E z,(Jg),a $ J,. Let q be the <J-least element of J, such that a E z+({q}).Define h by z ( i , x ) = h ( i , ( x , p , q ) ) .I t is easily seen that h is a C Skolem function for J, and that ( p , q )is a good parameter for Set X = h “ ( o XJy).Now, t h e r e i s a X I ( J y ) m a p g : wy*J,, soh-gisa Z,(J,) map of a subset of wy onto X . Hence it suffices to show that X = J,.

,-

x.

156

K.J. DEVLIN

Clearly,X ( { n ( p ) } ) En Skolem function for J,, so by choice o f p , n ( p ) = p . But then h , h' are both defined by the same X n formula (with parameter p ) in J,, so h = h'. It follows immediately that a * h = i ,of course. So for i E a, x E J,, n - h ( i , x )=-h *n(i,x) =h(i,x). Thus n 1 X = id 1 X , a n d X = J,.

-

Lemma 46 plays a direct part in the proof of (P 1)-(P 3). The next lemma, however, is only used during the proof of the lemma which follows it, and may, on first sight, appear somewhat uninspiring.

Lemma47.Let(J,,A)beamenable,p=p~JI. I f B C J , isXl((Ja,A)), then E,((Jp,B)) C Z2((Ja,A)). Proof. Case 1. There is a Zl((J , A ) ) map of some y < wp cofinally into wa. or_ Let g be such a map, and let B be Eo((J,, A ) ) such that B(x) t--, 3zB(z,x) for eachx E J Define B ' by B'((v,x)) e ( 3 z E S g ( v ) ) B ( ~ , x )for , v €7, p; x E J p .ThusB IS A,(<J,,A)). And since B ( x ) o ( 3 v E y ) B ' ( ( v , x ) ) , E l ( ( J p , B ) ) C E,((Jp,B')). Thus, we need only prove that Xl((Jp,B') C Z2((Ja,A)).It clearly suffices to prove that EO((Jp,B'))C E2((J,,A)). Let R be Zo((J ,B')). Thus R is rud in B' and some parameterp E J,. By choice of p , ( J p , B ) is amenable, so by Lemmas 3 and 4, there is a Eo(Jp) predicate P and functions fl,...,fm+k, rud in parameter p , such that R(x)

e

-

P(x,fl(x), ...,f,(X),B'nf,+,(x), ..., B'nf,+,(x)). HenceR(x)3 ~ 1..., , 3ykIv1=B'nfm+l(x)A . . . A y k =B'nf,+,(x)AP(x,f,(x),... ...,f , ( x ) , y ...,y k ) ] .Now P is certainly L o(Ja), and f l , ...,fm+k are rud in

parameterp, so it suffices to show that the function b(u) = B' n u is E2((J,,A)). It is in fact n,((J,,A)), because: y = b(u) t--f Vx [x E y * x E u A B'(x)] , aud B' is Al((J,, A ) ) . Case 2. Otherwise. As before, we must show that Zo((Jp,B)) C Z2((J,,A)). Again as before, this reduces, by the amenability of (Jp,B), to proving that the function b(u) = B n u is Z2((Ja, A ) ) on J,. Now, we clearly have

FINE STRUCTURE OF THE CONSTRUCTIBLE HIERARCHY

y = b(u) * (Vx E y ) ( x E u A B(x)) A (vx E u ) (B(x)

+

157 X

Ey)

Now, the second conjunct here is Ill((J,,A)). We show that the first conjunct is Zl((J,,A)), which is sufficient. It reduces to showing that (Vx Ey)B(x) is Xl((J,,A)). But look, we know that Case 1 fails to hold, so this is proved just as in Lemma 14. The next lemma is the key step involved in proving, by induction, the as yet unformulated (P 3). Lemma 48. Let (J,,A) be amenable, p = pAA. Suppose there is a Z,((J,,A)) map of a subset of u p onto J,. Then there is a B C J,, B E Zl((J,,A)), suck ((J, ,A ) )for all n 2 1. that Z (, (J,, B)) = P (J, ) n Z Proof. L e t u C u p , a n d l e t f : u * J , be Z,((J,,A)).PickpEJ, such that f i s C:Ja’A’({p}).Let (qli < w )be a recursive enumeration of Fml’I. Set C B = ( ( i , x ) l i € ~ A x E J ,A k(Jk,A)pi[x,P]}.

Now, (J,,A) is amenable, and hence rud closed, so by Lemma 9, B E Cl((J,,A)). And of courseB C J,. Commencing with Lemma 47, an easy induction shows that for all n 2 1, c Z,+I((J,>A)). For the converse, letR(x) be a X,+l((J,,A)) relation on J,, n 2 1. Assume, for the sake of argument, that n is even. L e t P be a Cl((J,,A)) relation such that, f o r x E J,, R(x) +-+ 3y1Vy2 ... Vy,P(y,x). Definepby p(z,x) * [z,x E J, A P ( f ( z ) , x)] . By choice off, any x E J, is XiJe’A’({p,v}) for some v < u p , so by definition of B, pis rud in B and some parameter v < u p . In particular,pis Al((J,,B)). Again, D = dom(f) is rud in B and some parameter T < u p , so D is also A,((J,, B)). But for x E J,, R ( x ) +-+ (32, E D)(Vz2 E D ) ... (Vz, E D)F(z,x), which is thus

~,((J,,fw

q(J,m.

We are now ready to formulate (P 3) and prove our promised uniformisation theorem. Let a,II 2 0. A C , master code for J, is a setA C J ,, A E Z,(J,), such that whenever m 2 1, C,((JpB,A)) = 9(J ,) n Z,+,&). Qff

158

K . J . DEVLIN

Theorem 49. Let a, n 2 0. m e n : (P I ) J, is En+l-uniformisable. (P 2 ) There is a Xn(J,) map of a subset of u p : onto J,. (Unless n = 0 , when it is X l(J,).) (P 3 ) J, has a master code.

xn

Proof. We prove the theorem (for all n ) by induction on a. For (Y = 0, it is trivial. So assume (Y > 0 and that (P 1)-(P 3) hold (for all n ) for all 0 < a . We prove (P 1)-(P 3) at a by induction on n. Case I : n = 0. (P 1) is already proved (Theorem 30). (P 2 ) p : = a, so (P 2 ) is already proved (Theorem 39) (P 3) since p: = a , A = 6 is a Eo master code for J,.

Case 2 : n = m t 1, m

2 0. Let p = p:

for convenience.

We first prove that p is the least ordinal such that some Xn(Ja) function maps a subset of w p onto J,. To this end, let 6 be the least such ordinal. Suppose first that 6 < p . Then B = {t E w61 g 4 f(g)} is a En(J,) subset of J,, so by definition of p , (J,,B) is amenable. Thus, as 6 < p , B = B n w6 E J, CI J,. S O B = f(g) for some E w 6 , whence g Ef([) ++ [ E B * $f(E), which is absurd. Hence p 5 6. Suppose p <6. By definition of p , this means that for some En(J,) set B C J , , (J,,B)is not amenable. Since (Jl,B>must be amenable. 6 > 1. I f 6 = y + 1, then since there is a E l(J,) map of o y onto w6, there i s a Xn(J,) map of a subset of wy onto J,, contrary to the choice of 6. Hence lim (6). I t follows, since (J,,B) is not amenable, that there is T < 6 with B f3 J, @ J,. By induction hypothesis, J, is En-uniformisable. So as 7 < 6. Lemma 46 implies that Y(w7)n En(Ja) C J,. But there is 3 J ~ so, this implies Y(J,) n E ~ ( J , )c J,. In particular, 11 E J,, h : R 0 J, E J,. Hence for some 0 < a, B n J, is Jp-definable. Let p be the least such, and let r be least such that B n J, is Zr(JP). By definition, ( J * , B n J,) PP is amenable, so if T < p i , then B n J, = (B f3 J,) fl J, E J r C Jp, contrary PP to the choice of p. Hence T >_ p‘p. By induction hypothesis, there is a X,(Jp) m a p g from a subset of wpi onto JP. And since B n J, E JP+l and B n J, 4 J,, t 1 > 6 , or p >_ 6. Hence there is a E,.(Jp) map g’ from a subset of u p ; onto U S . Then f - g ’ is a Xn(J,) map of a subset of u p ; onto J,. But we have established that p i 5 T < 6, so this contradicts the choice of 6. Hence 6 =p. (P 2) follows immediately from the above result of course.

FINE STRUCTURE OF THE CONSTRUCTIBLE HIERARCHY

159

We turn now to (P 3). By induction hypothesis, let A be a Ern master code for J,. Set 77 = p r for convenience. By the above, let f be a ZJJ,) map of a subsetofup ontoJ,.Bychoice ofA, f ’ = f t(f-l”J,)isaC1((Jq,A))map of a subset of wp onto J,. By choice of A , it is clear that p = p t = P 1, , ~ . Finally, of course, (J,,A) is amenable. So, we may apply Lemma 48 to (J,,A) to obtain a Zl((Jv,A)) setB C J, such that C,((J,,B)) = 3(J,) n Z,+l((Jq,A)) for all r 2 1. By choice ofA, B E Zn(Ja) and Z,((J,,B)) = 3(J,) n X,+JJ,) for all 1-2 1. Hence B is a Enmaster code for J,. Finally we prove (P 1). Let B be, as above, a C, master code for J,. Let R ( y , x ) be a Zn+l(Ja) relation on J,. Define, with f as above, a relation R on J, by a r , x > - [ Y , X E J, A R ( f ( Y ) , f ( % ) ) l -ThenE is Z*+l(JJ and hence Zl((J,,B)). Let? be a Zl((J,,B)) function uniformising R. Sincefis Z,(J,), so is f But J, is C,-uniformisable, by induction hypothesis, so we can 1etf”be a C,(J,) function uniformising f-l. Set r = f a ? f It is clear that r is a Zn+’(Ja) function which uniformises R. The proof is complete. N

-’.

-’.

The above results give us two (intuitive) equivalent formulations of the

E n-projecturn:

Theorem 50. Let a , n 2 0. Let 6 be the least ordinal such that some Z,(J,) function maps a subset of w6 onto J,. Let y be the least ordinal such that 3 (wy) n C,(J,) Q J,. Then 6 = y = p z . Proof. That 6 = p i was actually proved during the proof of Theorem 49. Since we now know that J, is Z,-unifonnisable, Lemma 46 tells us that 6 5 y. Assume 6 < y. Now by definition, let u C w6, and let f : u J, be Zn(Ja). Let Z = (51 [ ($f(l)}.Then 2 C 0 6 and Z E Z,(J,), so by definition of y, 2 E J,. Thus Z = f([) for some 5, so [ E f([) * ( 4 f([), which is absurd. Hence 6 = y.

There is, of course, a concept which, for A, predicates, plays the role that the En projectum plays for X, predicates. And, as might be expected, there is a corresponding ‘‘total function” or A, equivalent of Theorem 50 for this concept . Let a,n 2 0. The A,-projectum of a (sometimes called the weak Enprojectum), Q:, is the largest Q a such that (J,,A) is amenable for all An(J,) setsA C J,.

<

160

K.J. DEVLIN

Thus the A,-projectum of a represents the "hard core" of J, with regards to A, predicates on J,. Clearly, 77; 2 p t . We do not, however, necessarily have equality here. For example, let Q be the first admissible ordinal > w . Then it is easily seen that 77; = a, whereas p; = w . Corresponding to Lemma 46, we have:

Lemma 51. Let n 2 1, and let y be the least ordinal such that 3(oy) n A n ( J a ) J,. Then there is a Xn(J,) (and hence A,(J,)) map o f w r onto J,. Proof. Let n = rn + 1, n 2 0. Since Xm(J,) C A,(J,) C Z,(J,), Theorem 50 implies that p: 6 y 5 p';y".Theorem 50 also implies that there is a X m ( J a ) map of a subset of u p : onto J,. So, we can clearly define a C,(J,) map of w p z itself onto J,. This reduces our problem to showing that there is a Z,(J,J map of o y onto u p : . As a first step, we have the: Claim. There is a X n ( J a ) map g from w y cofinaliy into u p ? . LetA be a E m master code for J,. By hypothesis, let b C wy, b E A,(J,), b @ J,. By choice o f A , b is A ~ ( ( J ~ ~ , ASuppose )). b is in fact defined by:

whereBo,B1 areXO((J m , A ) ) .Then pol

there can be But ( J p E , A ) is amenable, and hence rud closed, so as b 4 J PF' no r < u p : such that

Define g : o y + u p : by

Clearly, g is Z n ( J a ) and cofinal in u p : , proving the claim.

FINE STRUCTURE OF THE CONSTRUCTIBLE HIERARCHY

161

We now prove the lemma. Since p : Iy, there must be a Xn(J,) map f from a subset of o y onto wp; (Iwa), and of course such anfwill then be E1((Jpm,A)). Definey: ( w ~2 )onto --.up! as follows. L e t f b e given by f(v) = 7 , - 3yF0,,7,v),where F is E0((Jp,,A)). Set ff

-

f (v,7)=

8 , if

(3YESy(,))F(y,8,4

0, otherwise.

Then S i s X1((JpmlA)), and hence Xn(J,).

u p : , as g cofina? m u p ; .

AndSclearly maps ( ~ 7 onto ) ~

Since we have (by Lemma 38) a X1(Ja) map of wy onto lemma follows.

the

Corresponding to Theorem 50, we have: Theorem 52. Let a, n'> 0. Let 6 be the least ordinal such that some X,(J,) (and hence A,(J,)) function maps 0 6 onto J,. Let y be the least ordinal such that P ( o y ) n A,(J,) Q J,. Then 6 = y = Q:,

Proof. Suppose y < q:. Let B C oy,B E A,(J,),B 4J,. Then wy n B = B 4Jqn, contrary to (Jgz,B)being amenable. Suptose now that Q: < y. Then there isA C J,, A E A,(J,), such that (J,,A) is not amenable. In particular, y > 1. Suppose y = 5' + 1. There is then a Xl(J,) map of w( onto wy, so by Lemma 51 there is a Zn(Ja) map,f, of 05' onto J,. Then Z = { L E wE I L 4f(~)}is a A,(J,) subset of w [ . Clearly, Z 4 J,, so this contradicts the choice of y. Hence lim(y). Thus as (J,,A) is not amenable, there must be T < y withA n J, 4J,. But by choice o f y , T < y -+A n J, E J,, so for some 0 < a ,A n J, is J,-definable. Let 0 be the least such.ThenA n J , E P ( J , ) n A m ( J e ) f o r s o m e m E w , a n d A nJ,$Je. Thus by Lemma 5 1 there is a E,(J,) mapf of WT onto J,. (Actually the hypotheses of Lemma 5 1 require that we have a A,(Jo) subset of OT not in Je , whereas we have only exhibited a subset of J, with these properties. However, since there is available a X l(J,) map of W T onto J ,, this point causes no problem.) Since 8 < a , f E J,. But A n J, 4 J,, andA n J,E Je+l, so 19 2 y, and there is thus a map f ' E J, of OT onto wy. By Lemma 5 1, again, this gives us a Xn(Ja) map k of 07 onto J,. Then, clearly, K =

162

K . J . DEVLIN

{ 1 I i @ K(1)) is a A,(J,) subset of W T not lying in J,, contrary to T < y. Hence y = 77:. Now, by Lemma 5 I, we have 6 5 y.Suppose 6 < y. Let f : w6%J, beZ,(J,). LetZ={vlv$f(v)}.ThenZE3(w6)n An(J,) J,. But this contradicts the choice of y. Hence 6 = y. ~

Remark. Lemmas 46 and 5 1 can be regarded as much sharper versions of the following, much earlier theorem of Putman: L,. Then contains a Suppose 9 ( y ) n well-ordering of y of order type a.(For y >_ 0.) Putman actually proved this result for the case p = w , but his proof works in the general case. The methods described above have, of course, many uses, We give just one, very general, example, showing that (in certain circumstances) it is possible to carry out Lowenhein-Skolem arguments for non-regular ordinals a which can generally only be done when a is actually a regular cardinal. More precisely, the following theorem, is well-known: Theorem 53. L e t K be a regular cardinal. L e t y < K 2 u p , and suppose that Y C Jp, I Y I < K . Then there is X < Jp such that Y U y C X a n d K n X E K .

To prove this, one simply forms an a-chain Xo< X I < ...4X , 4 ... 4 Jp of elementary submodels of J p , takingXo as the Skolem hull of Y U y in Jp, and Xn+l as the Skolem hull of X , U sup ( K f' X , ) in Jp, and then X = U n < w X n is the required submodel of Jp. By construction, K n X is transitive, and hence an ordinal, and since K is regular, I XI < K , so K nX E K . It should be observed that K being regular is a necessary condition for the above procedure to work (in general). However, providing we can, in some way, ensure that for each n , sup ( K n X,) < K , then we can, of course, get by with just cf(K) > w . The theorem below shows that, in certain cases we can do just this, providing we relax our demands somewhat. Let n 2 1, a 5 wp. We say that a is E,-reguZar at p iff there is no X ,(Jp) map of a bounded subset of a cofinally into a. For example, by Theorem 43, wa is strongly admissible iff wa is X 1 regular at a.

FINE STRUCTURE OF THE CONSTRUCTIBLE HIERARCHY

163

Theorem 54. Let n 2 1, wp 2 a 2 1. Suppose a is X,-regular at p. Let Y C Jp, w 2 1 Y 1 < cf (a),and let y < a- Then there is an X
Proof. Since cf (a)> w , it clearly suffices to prove that, under the stated hypotheses, there is X i r , Jp such that Y U y C X and sup (a n X ) < a. Let h be a E n Skolemfunction for Jp (by Theorem 49 and Lemma 33). Since w 5 I Y 1 < cf(a), we may, without loss of generality, assume that Y is closed under ordered pairs. Furthermore, let @ be the function defined in Lemma 37. Since a is X:,-regular at 0, a is certainly strongly admissible. Hence, by Lemma 37, { .$ E a I @ ' I t 2 C .$}is unbounded in a. It follows that we may y 2 is also, without loss of generality, assume that arty2 C y. Recall that zip.

Let X = h t t ( oX (Y X 7)). Then we claim that X i s closed under ordered pairs. To see this,letxl,x2 EX, sayxl =h(il,(yl,vl)),x2=h(i2,(y2,v2)). Lety = (y1,y2)E Y a n d v = @((vl,v2))Ey. Clearly {(xI,x2)} is t: P ( {p ,( y, v ) } ) , where p is a good parameter for h. Thus for some i E w , (x1,x2)= h ( i , ( y , v ) )E X , as required. So, by Corollary 36, X < z H Jp. And of course, we clearly have Y U y C X . We show that sup (a nX) < a. F o r y E Y , iEw,definehi,y: C y + a b y h . (v)l.h(i,(y,v)).Thushi,, 'Jn is Xn(Jp), and so as a is En-regular at 0, sup (hi,y y) y ( i , y ) < a. Since I Y I < cf(a), it follows that supyEy y ( i , y ) N- y( i ) < a . Since cf(a) > w , we conclude finally that supjEwy(i) < a. But clearly, supiEw y(i) = sup(& n X ) , so we are done.

-

The above lemma may be used to prove that if V = L and K is a regular uncountable, non-weakly compact cardinal, then there is a Souslin K-tree. (Jensen.)

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

DEGREE THEORY ON ADMISSIBLE OKDINALS Stephen G . SIMPSON Yale University and The University of California, Berkeley

0. Introduction The study of recursive functions on the ordinal numbers was initiated by Takeuti in the late 1950’s. Takeuti’s concept was generalized by several authors to that of a-recursive functions on admissible initial segments a of the ordinals. An intensive study of the generalized concept was begun by Sacks in 1964 [ 111. Sacks’ unstated program was to suitably generalize all the theorems of ordinary recursion theory to &-recursion theory. A typical theorem of ordinary recursion theory is the Friedberg-Muchnlk solution to Post’s problem: there are two recursively enumerable subsets of w such that neither is recursive in the other. A recent paper of Sacks and Simpson [ 131 generalizes the FriedbergMuchnik theorem as follows: for any admissible ordinal a, there are two arecursively enumerable subsets of a such that neither is a-recursive in the other. (The Sacks-Simpson proof is of some methodological interest in that it involves a rather delicate downward Lowenheim-Skolem argument within La, the a f hconstructible level.) Our object in the present paper is to survey the recent work in a-degree theory for arbitrary admissible ordinals a. (An a-degree is an equivalence class under the relation: A is &-recursive in B a n d B is wecursive in A . ) The flavor of a-degree theory can be savored only in the proofs; we therefore give detailed proofs of a few theorems. The reader must understand that in this survey The preparation of this paper was partially supported by NSF Contracts GP-34088X and GP-24352. The main result of Section 4, due to Sacks and the author, was presented by the author in an invited address to the 1971 Summer School in Mathematical Logic at Cambridge, England, sponsored in part by the North Atlantic Treaty Organization. 165

S.G. SIMPSON

166

paper we are leaving many important matters out of account, e.g. (in decreasing order of importance): 1. reasons for generalizing recursion theory ; 2. the lattice of a-r.e. sets ; 3 . alternative notions of relative a-recursiveness ; 4. non-regular and non-hyperregular a-r.e. sets 5 .

’

A detailed outline of the paper follows. In Section 1 we review the basic notions of a-recursion theory such as adegree, a-r.e. sets, and the a-jump of a set. We also discuss two important auxiliary notions, regularity and hyperregularity, which have no counterpart in ordinary recursion theory. In Section 2 we prove the following generalization of Friedberg’s Completeness Theorem: an a-degree b 0‘is regular if and only if it is the jump of a regular, hyperregular a-degree. We then give a necessary and sufficient condition on a that there exist an a-degree d such that every b 2 d is regular. We show in particular that d exists if a is countable. In Section 4 we reprove the main theorem of Sacks-Simpson [ 131 in the following strong form: for any admissible ordinal a there are a-r.e. degrees a , b such that a 4 b, b $ a , a U b is regular and hyperregular, and

>

(a u b)‘= 0’.

In Section 5 we sutvey recent work of Manny Lerman, Richard Shore, and others. Their work is further evidence that most if not all theorems of ordinary degree theory can be generalized (not straightforwardly!) to ar-degree theory. In Section 6 we discuss briefly the connection between a-recursion theory and Jensen’s “fine structure” theory.

1. Fundamental notions We employ von Neumann’s notion of ordinality; thus an ordinal is identified with the set of smaller ordinals. Our set theoretical notation is standard.

*3 See Kreisel [ 2 ) especially Section 3.

See Machtey (81;Chapter 1 of Simpson [ 171; Lerman-Simpson [ 7 ] ; and Lerman

[4ii

See pp. 157-158 of Kreisel [ 21 and pp. 39-44, 92-94 of Simpson [ 171. See Chapters 2 and 3 of Simpson [ 171.

DEGREE THEORY ON ADMISSIBLE ORDINALS

167

In particular Ux (union of x ) , x n y (intersection of x andy), x X y (Cartesian product of x andy), x r y (x restricted toy), x”y (the range of x r y ) , x - y (set theoretical difference o f x andy), x C y ( x is a subset ofy), x E y ( x is an element ofy), 8 (empty set), (x,y) (ordered pair), dom ( x ) ,rng(x) (domain and range of x ) ,x :y + z ( x is a function from y into z ) have their usual meanings. Throughout this paper a is a fixed but arbitrary admissible ordinal. Lower case Greek letters denote ordinals less than a, i.e. elements of a; except for /3 and A which may denote limit ordinals less than or equal to a. Upper case Roman letters denote subsets of a. A set X 5/3 is unbounded in /3 if U X = 0; otherwise it is bounded below p. We sometimes write unbounded for unbounded in a and bounded for bounded below a. A partial function from a into a (i.e. a function whose domain and range are subsets of a ) is a-recursive if its graph can be enumerated via an equation calculus resembling Kleene’s but allowing infinitary bounded quantifications such as (3x < 6 ) where 6 < a. (For details see [ 121.) A subset of a is arecursive if its characteristic function is a-recursive. A subset of a is arecursively enumerable (abbreviated a-r.e.) if it is the domain or range of an a-recursive partial function. Every a-r.e. set is the range of a one-one arecursive function whose domain is an ordinal less than or equal to a. . A subset of a is &-finiteif it is a-recursive and bounded. A basic principle of a-recursion theory is: i f f is an a-recursive partial function and K C dom (f) is a-finite then f ”K is a-finite. In the usual way one defines an a-recursive function k : a X a+ (Y such that: (i) k ( y , q ) = 0 implies y < q ;and (ii) {y I k ( y , q) = 0) ranges over the a-finite sets as q ranges over a. We fuc the notation K , = {y 1 k ( y , q ) = 0 ) and call q a canonical index for K if K = K , . The projectum of a, denoted a*, is the least /3 such that there is a one-one a-recursive function f : a + /3. It is easy to see that a* is equal to the least /3 For any limit ordinal a the notion of a-recursive partial function can be defined. One can then define a-finiteness as above and prove: a is admissible if and only if principle (*)holds. Note that iff is a-recursive and 7 < 01 then h [ f ( ( g , y ) )is a-recursive.

S.G. SIMPSON

168

such that not every a-r.e. subset of is a-finite. I f K is an a-finite set, the acardinality of K is the least /3 such that there is an a-finite one-one correspondence between K and p. An ordinal less than a is an a-cardinal if it is equal to its own a-cardinality. An a-cardinal 0 is regular if every a-finite subset of 0 of order type less than is bounded below p. An a-cardinal is singular if it is not regular. I t is easy to see that if a* < a then a* is the largest a-cardinal. As in ordinary recursion theory one defines an a-recursive function r : a X a* + a such that: (i) K,,(,,,) CKr(7,,) E T whenever u < T < a ; (ii) u{Kr(u,E)Iu < a} ranges over the a-r.e. sets as E ranges over a*.We fix the notations Rz = Kr(u,E)and R E= U { R : / u < a}. We call E < a* an index for R i f R = R,. As in ordinary recursion theory, one can a-effectively compute an index for an a-finite set K from a canonical index for K but not conversely. We now discuss relative a-recursiveness. For A , B 5 a we say A is a-manyone reducible to B (abbreviated A 5 &Iif)there is an a-recursive function f :a + a such that for ally

A is a-recursive in B (abbreviated A < a B ) if there is an index E < a * such that for all y, 6

Clearly A 5 maB implies A defined by B ' = {E

5, B but not conversely. The &-jump o f B is

< a* I( 3 g)( 371)[(t;, 77) E R E& K E 2 B & K,

nB = (31) .

I f f is a partial function f r o m a into a, we say f is weakly &-recursivein B (abbreviated f 5 w&?) if there is an index E < a* such that for all y, 6

f ( r )= 6

c--,

(W(377)[(Y,6 , t;, 77) ER E

KEC_ B

K , nB = 61 .

For such an e we write f = [ E ] and call E an index for f from B. We say A is a-recursively enumerable in B ifA is the domain or range of a partial function weakly a-recursive in B.

DEGREE THEORY ON ADMISSIBLE ORDINALS

169

IfA C a we denote by cA the characteristic function o f A . Caution: A I,B implies cA I , $ but not conversely. However, the following facts are easily verified. 1.1 5, is transitive, and f i , , A B implies f B. 1.2 B I,, B' but not B' 2, B. 1.3 A 5, B if and only ifA' B'. 1.4 A is&-r.e. in B if and only ifA I,, B'.

<,

Two sets A and B are said to have the same a-degree if A -, B i.e. A 5, B and B 5 A . Lower case boldface lettersa, b , ... are used to denote a-degrees. Fy If a , b are the a-degrees of A, B respectively then we write: a 5 b for A 5, B; a U b for the a-degree of A @ B where

0 for the a-degree of 0 = 0; a' or jump (a) for the a-degree of A ' . Thus the adegrees are an upper semilattice under U; moreover the jump operator is well defined, and increasing on a-degrees. We now discuss two important auxiliary notions due to Sacks.

<,

Definition. A

C a is a-regular ifA n 0 is a-finite for all p < a.

Definition. A 2 a is a-hyperregular iff "0 is bounded whenever 0< a, f :p+a,f5,,A. Definition. An a-degree is regular (resp. hyperregular) if it contains an aregular (resp. a-hyperregular) set. Clearly every subset of o is w-regular and w-hyperregular. The existence of non-a-hyperregular sets is what makes a-recursion theory more intricate than ordinary recursion theory. For instance, if a* < a then not even every a-r.e. set is a-regular (and conversely). This means that the difficulties associated with non-a-regularity can arise even in arguments concerning a-r.e. sets. Such difficulties are alleviated somewhat by the following theorem. 1.5 (Sacks [ 111). To every a-r.e. set A there is an a-regular a-r.e. set B such that B 5, A.

170

S.G. SIMPSON

A short proof of 1.5 can be found in Chapter 2 of [ 171. 1.6 (Sacks [ 111). For A E a the following are equivalent : (i) A is a-regular and a-hyperregular. (ii) f "K is a-finite whenever f < w a A , K C:dom ( f ) , K a-finite. Proof. That (ii) implies (i) is immediate from the definitions. To prove (ii) assuming (i), suppose f i , , A , K E dom(f), K a-finite. Let E
swa

Thus f " K is a-finite. The end of a proof is indicated by m. We close this section with some useful technical lemmas concerning E2 functions. Following Rogers [ 10: pp. 301-3071 we define the aurithmetical hierarchy. Thus a relation on a is C, if it is a-recursive, ",I if its complement is C,, C,l+l if it is the projection of a H n relation, and A n if it is both En and n,. In particular a relation is C, if and only if it is ct-r.e. A partial function f is said to be En if its graph is En. In particular f is C, if and only if it is a-recursive. Caution: the bounded quantifier manipulations described in Rogers [ 10: p. 31 11 do not generalize to a-recursion theory except in very special circumstances. However, as in ordinary recursion theory, one has: 1.7. Lemma. Let f be a partial function from a into a. Then the following assertions are painvise equivalent: I t can be shown thatA ture ( L , , t , A ) i s admissible.

C a is a-regular and a-hyperregular if and only if the struc-

DEGREE THEORY ON ADMISSIBLE ORDINALS

171

(i) f is 2,; (ii) f is weakly a-recursive in 0'; (iii) there is an a-recursivefunction f ' : a X a -+ a such that for all y

In (iii) the limit is taken in the discrete topology as u approaches a. As usual,

x =y means that x is defined iffy is defined in which case x = y . Thus in (iii)

f ( r ) =6 iff(3a)(VT>_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
< a * ,Dp is dense.

Proof. The proof splits into cases following the definition of Do. A more conventional way of writing

CB 1 u

€1 (7)= 6

B is: [ e J (7)= 6.

S.G. SIMPSON

174

Case I. Define an a-recursive function

m : CondXa* +a*+l by m ( p ,E ) = the least y < a* such that p does not decide [ E ] (7); m ( p , ~=) a* if n o such y exists. Fix p < a* and define an a-recursive partial function O from Cond into Cond by B(p) = the least q (in the canonical a-recursive well ordering of Cond) such that p 5 q and m ( p ,E ) < m ( q , E ) for some E < 0. Then, given a condition p , define an a-recursive increasing sequence of conditions by qo = p ; q , = U {qEl[ < q } if q is a limit ordinal; 4[+1= O(q5)if 8 ( q E ) is defined; to= the least such that O(qE)is undefined. We claim that to< a. Suppose t o the contrary that to= a. For each E < 0 define

u

Then the It’s are simultaneously a-recursive and a = { I , / € <@}. Furthermore m ( q o ,E ) ... m ( q [ ,€) m(q[+,) ... a* so each I , has order type less than or equal t o a*. Let H be the set of E < such that I E has order type a*. Then H i s an a-r.e. subset of /3 < a*. Hence H is a-finite. It follows at once that U { I E (E E H } is a-finite. Let y < a be an upper bound for u {I,( E EH}. Let J , = I , n { 6 I y 5 6 < y + a*}. Then (J,I E < 0)is an a-finite sequence of a-finite sets each having order type less than a*. Furthermore u {J,I E <0} = ( 6 I y 5 6 < y + a*}.This contradicts the Case assumption that a* is a regular a-cardinal. The claim is proved. Thus q E ois a condition extendingp. For each E < p it is clear from the construction that qt0 decides [ E ] (y) for each y < m(qto,E ) ; furthermore if m ( q E oE,) < a* then qto ( [ E ] (m(qEo, E ) ) undefined). Thus qtOE Do. Since p is arbitrary, Do is dense.

< <

<

< <

+

Case II. Fix p < a* and define an a-recursive function

m : CondXa*+D+l by m ( p , E ) = the least y
DEGREE THEORY ON ADMISSIBLE ORDINALS

175

a = U {I,le < 0). Furthermore m(qo,E ) 5 ... 5 m(q5,e) 5 m(qE+l, E) I ... 5 (3 so each I , has order type 5 0.Hence there is an a-recursive partial function i from a subset of (3 X 0 onto a (namely i ( t , E ) = the tthmember of ZE).This is impossible since fl < a*. Thus qt0 is a condition extending p . As in Case I we note that qEoE Dp hence Dp is dense. =

Remark. From the above proof for Case I we can extract the following combinatorial lemma: Assume that (3 < a* and that a* is a regular a-cardinal. Let (Z,I v < (3 ) be a simultaneously a-recursive sequence of a-finite sets each of order type less than or equal to a*. Then U {Z,l v < (3) is a-finite. One can extract a similar but weaker combinatorial lemma from the proof for Case 11. Corollary. There is a A, function q : a* X Cond -+ Cond such that for all p E Cond and 0< a*,

Proof. Define q((3,p)= qEowhere q t o is as in the proof of Lemma 2.2. Using Lemma 1.7 it is easy to see that this q is a A z function. A set A 5 a is said to be generic if A meets Dp for each (3 < a*. The following Lemma and its proof show that our notion of genericity is not so special as it appears. (It can be shown thatA E a is generic if and only if (L,,A) is admissible and every X 2 sentence true in (L,,A) is forced by some condition p C cA . However, this remark does not simplify the proof of Theorem 2.1 .) 2.3 Lemma. Let A and

5 a be a generic set. Then A is a-regular and a-hyperregular,

A’-, A

@

0’.

Proof. For each y < a one can find E < a * such that p decides [c] ( 0 ) if and only if l h ( p ) >_ y. Hence for each y < a there is (3 < a* such that lh(p) 2 y for all p ED,. It follows that A is a-regular.

S.G. SIMPSON

116

SupposeA is not a-hyperregular. Let K be the least ordinal such that for some e < a*, K E dom([eIA) and { [eIA(y)I y < K } is unbounded. Obviously K < a since A is not a-hyperregular. It is also clear that I( is a limit ordinal. We claim that K is a regular a-cardinal. Suppose not and let ( K I p~ < n) be an a-finite sequence such that n < K = U {K,,] y < n}. For each y < 7~ let h(y) be the least u such that for all y < K , , there exists ( g , ~ , y6,) E R : with K g E A and K7, n A = @.Clearly h ( p ) is less than a since K~ < K . Also h Sw,A since A is a-regular. Hence {h(p)i y < n} is bounded since n < K . It follows that { [elA(y) I y < K } is bounded, a contradiction. The claim is proved. We now split into Cases as in the definition of the Do’s. In Case I we have K
<,

and

The existence off is a consequence (via 1.6) of the fact that A is a-regular and a-hyperregular. The existence of g follows directly from the genericity of A . We are now ready to complete the proof of Theorem 2.1. Let B be an aregular set. Thus, for each y < a,cs y is a condition. If p and q are conditions we write

r

i.e. p * q is the concatenation of p and q . Let X = A2 - cf(a). Recall Lemmas 1.7, 1.8, and 1.9 which state several useful properties of A. Let F(resp. G) be a A2 function from h onto an unbounded subset of a (resp. a*). Define p,=U{p,Ip
if

v=Uv;

DEGREE THEORY ON ADMISSIBLE ORDINALS

111

where q is the A, function mentioned in the Corollary to Lemma 2.2. For each v < A, { F ( p )I p < v} is bounded below a. Hence only a-finitely much of B is used in the definition of ( p pI p < 2v). Hence, by 1.7 and 1.9, (p,l p < 2 v ) is a-finite. Thus (pvI v < A) is an increasing sequence of condi). there tions. Moreover U {Ih(p,)I v < A} = a since F(v) 5 l h ( ~ ~ . + ~Hence is a unique set A C a such that cA = {pv(v < A}. For each v < A we have p2v+2E D G ( u ) ;henceA is generic. It follows by Lemma 2.3 thatA is aregular and a-hyperregular, and that

u

I t remains to show that

This is an immediate consequence of the following claim. For each v P ( v ) be (a canonical index for) (p,I 1.1 < 2v).

Claim: P is weakly a-recursive in B @ 0’ and in A

@

< A let

0’

We shall give an informal, “Church’s thesis” type of argument for the claim. To compute P(v) from B CB 0’. First compute y = U { F ( p ) I p < v}. By Lemmas 1.7 and 1.9 this computation uses only an a-finite amount of information about 0‘. Then use 0’ and the a-finite sequence cB ry to compute P O , p 1 , ..., p p , ._.( p < 2v) in order. Again by 1.7 and 1.9 this uses only afinitely much information about 0’. To compute P ( v ) from A @ 0’. First, compute (F(,u)l,u< v ) and ( C ( p ) I p < v ) as above. Next, start generating nu,the set of all (canonical indices for) a-finite sequences of conditions ( p l l p < 2 v ) such that U { p ; I 1-1 < 2 v 1C cA and

S.G. SIMPSON

178

for all p < v. Clearly n, is a-recursive in A . As you generate n,, examine simultaneously all the members of n, trying to find one with the property that

for all p < v. This involves computing pieces of the A , function q . Do this arecursively in 0', taking care to dovetail the examination of the various members of n, so that you don't spend too much time on any a-finite subset of Il,. (By Lemma 1.9 the examination of any single member of n, will eventually terminate having used only a-finitely such information about 0';however, there is no guarantee that the examination of an a-finite subset o f n , will so terminate.) Eventually you find a sequence ( p l l p < 2 v ) in n, with the desired property. Then P ( v ) = (pL1 p < 2v). More importantly, you have used only &-finitely such information aboutA @ 0'. We have informally described computation procedures showing that PI,, B @ 0' and P<,, A @ 0'. The proof of Theorem 2.1 is complete. m One can use the same method to prove the following generalization of a theorem of Spector (see Rogers [ 10 : p. 2671): 2.4 Theorem. There exist regular, hyperregular a-degrees a, b such that

a u b = a ' =b ' =0'.

3. Upper segments of regular a-degrees An interesting corollary of Friedberg's jump theorem is this: there is an wdegree d such that every w-degree 2 d is the jump of some w-degree (namely d = 0').We now ask: for which admissible a does there exist an a-degree d such that every a-degree 2 d is the jump of some regular, hyperregular adegree'? By Theorem 2.1 this is equivalent to asking: for which admissible a does there exist an &-degreed such that every a-degree 2 d is regular? We write I a 1 for the (von Neumann) cardinality of a i.e. the least /3 such that there exists a one-one correspondence between a and /3. We write cf(a)

DEGREE THEORY ON ADMISSIBLE ORDINALS

179

for the cofinality of a, i.e. the least h such that there is a function from h onto an unbounded subset of a. 3.1 Theorem. Assume the Generalized Continuum Hypothesis. Let a be an

admissible ordinal such that I a I is a regular cardinal. Then the following assertions about a are painvise equivalent. (i) There is an a-degree d such that every a-degree 2 d is regular. (ii) To every a-degree a there is a regular a-degree b 2 a. (iii) Every X C a of cardinality less than I a1 is a-finite.

Before proving 3.1 we give two lemmas in which 1 a 1 is not assumed to be regular.

3.2 Lemma. Let a be an admissible ordinal such that Assertion 3.1 (ii) holds. Then every X a of cardinality less than cf(a) is a-finite. Proof. Let X 5 a have cardinality < cf(a). By 3.1 (ii) there is an a-regular set B such that cx Swa: B. Let E
Since I XI < cf (a), {h(y)I y E X } must be bounded below a,say by 7.Then for all y we must have y E X iff cB T k [ E ] (y) = 1. Hence X is a-finite.

r

3.3 Lemma. Assume the Generalized Continuum Hypothesis. Let a be an admissible ordinal such that Assertion 3.1 (ii) holds. Then cf (a) = cf ( I a I ).

Proof. The hypothesis 3.1 (ii) clearly implies that the set of regular a-degrees has cardinality 2'"'. On the other hand, the set of regular a-degrees clearly has cardinality at most I a I cf (&). If cf (a)< cf (la I) then by the GCH we would have 2'O' <_ lalcf(")= la1 contradicting Cantor's theorem of cardinal arithmetic. So cf (a)2 cf ( I a I). For the other direction recall Konig's theorem of cardinal arithmetic which

180

S.G. SIMPSON

says I a I C f ( l o l ' )> IaI. Hence there is a non-a-finite set X C 1 a1 of cardinality cf ( 1 a I). By Lemma 3.2 X cannot have cardinality less than cf (a). So cf( 1 @I) 2 cf(a). Proof of 3.1. The implication (i) 9 (ii) is trivial since any two a-degrees have an upper bound. The implication (ii) * (iii) is a special case of Lemma 3.2 in view of Lemma 3.3. To prove (iii) * (i) first note that (iii) implies cf ( a ) = I al. Let (?,I u < I a t ) be an increasing sequence of ordinals such that a = u{y,I I.' < 1 a ( } .Let g be a function from 1 a I onto a. Define

and let d be the a-degree of D. Given B

5 a define

Then D and B* are a-regular in view of (iii). Furthermore it is easy to see that

Thus every a-degree 2 d is regular.

9

Corollary. Let a be a countable admissible ordinal. Then there is an a-degree d such that every a-degree 2 d is regular. Remarks. 1. It is not hard to construct (in ZFC) an admissible ordinal (Y such that la1 = cf(a)= w 1 but not everyconstructibly c o u n t a b l e X C a i s a finite. Such an a falsifies some attractive conjectures.

-

2. I f I a1 is not regular then we conjecture that the equivalence 3.1 (i) 3.1 (ii) still holds. We can prove in ZF + V = L l o that 3. I (ii) is equivalent to l o The set-theoretical hypothesis V = L can be expressed in mecursion theoretic language as follows: for every infinite cardinal K , every subset of K bounded below K is K-finite.

DEGREE THEORY ON ADMISSIBLE ORDINALS

the assertion that there is a function f : I a1 '%' for each y < I aI.

181

a such that f r y is a-finite

4. Incomparable a-r.e. degrees In this section we generalize to a-recursion theory a strong form of the Friedberg-Muchnik theorem. 4.1 Theorem.Let a be an admissible ordinal. Then there are a-r.e. degrees

a, b such that (i) a I b and b 4 a ; (ii) a U b is hyperregular; (iii) (a u b)' = 0 ' . (By Theorem 1.5 a U b is necessarily regular.) In the proof we shall define a-recursive functionsA', B', f ( u , E ) , and ( E
u

( f ( ~ )E,) €A+--+3 q [ ( f ( ~ E) ,,Q ) E R , & K , n B = @] and

for each E < a*. From this and the a-recursive enumerability of A and B it will follow that A 4,B and B 4,A . At the same time we shall define a certain auxiliary a-recursive function P ( U , E ) whose purpose will be to preserve certain computations so as to make A @ B a-regular and a-hyperregular and (A @ B)' =, 0'. The definition of p ( u , e) will closely parallel the key definitions in Section 2 whose purpose was llkewise to make a certain set a-regular and a-hyperregular. (Thus for instance there will be a split into Cases I and I1 exactly as in Section 2.)

182

S.G. SIMPSON

Remark. The construction and proof below are arranged so that the reader not interested in conclusions 4.1 (ii) and 4.1 (iii) can skip certain parts. Specifically, such a reader should (a) ignore the functionsy(u, y, E ) , m(u,E ) , and p ( a , f); (b) replace the below definitions off(< u, E ) andg(< u, E ) by

and

(c) regard Lemma 4.6 as an immediate consequence of Lemma 4.2; (d) disregard entirely Lemmas 4.3, 4.4, and 4.5. Before describing the construction we must define some important functions and parameters which do not depend on the construction. (1) Let h be the A 2 cofinality of a. By Lemma 1.8 h is equal to the A , cofinality of a* so let G : h -+ a* be a A 2 function whose range is unbounded in a*. For each v < X define H ( v ) = u {G(p)I p < v}. Thus we have

O=H(O)< ... I H ( v ) < H ( v + I ) < ...
E

< a* satisfiesH(v) 5 t < H ( v + 1) for a unique v < A.

Remark. Thus H partitions a* into A “blocks” each of size less than a*. The use of H was suggested by R. Shore and has the effect of making the present proof somewhat simpler than that of Sacks-Simpson [ 131. Shore has also put this “blocking” idea to good use in his own work (see Section 5).

(2) Let G : a X X -+ a* be an a-recursive function such that C ( v )= lim, G(.u, v) for all v < h. (Such a G exists by Lemma 1.7.) Define H(o,v)= U{G(u,1.1)Ip
Furthermore, for each u < a we have

O = H(u, 0)

< ... i H ( u , v ) I H ( u , v + I ) < ...
DEGREE THEORY ON ADMISSIBLE ORDINALS

183

(3) Let L : a +. h be an a-recursive function such that L-'({v}) is unbounded in a for each v < A. We can now present the construction. As a preliminary to stage u of the construction we set

(l-Y)y
( 0 9

Y

3

f )

=0

if a* is a regular a-cardinal,

m(u,E) = (l-Y),<,Y(O>Y>f) =

otherwise ;

if ( c p , ~ ) € A < ~ ,

f (<0,E ) =

max {p, p ( u , E ) }

otherwise

where

g(< 'J,4 =

if ( J / , E ) E B < '

max { 9,p ( u , E ) }

otherwise

where

Note: In the definitions of y ( o , y , ~ )and m ( u , ~the ) usual convention con~ cerning the bounded p-operator is observed. Namely, ( ~ L X )...~is <defined

S.G. SIMPSON

184

to be the least x less thany such that ... if such a n x exists a n d y otherwise. Stage u of the construction is now described as follows. Put u =L(u>where L is as in (3) above. Define AO=A
where (4)

~ ~ ~ [ ( ~ ( < u , E ) , E , ~ I )nB
Define

d o , €7=

u+ 1

if E ' > _ 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 = B<"U

{(g(u,E),E)

IH(u,v) < e < H ( u , u + l ) & (5)}

where (5)

3 7 7 [ ( g ( u , E ) , E , 7 ) ) E R ~ &nAu= K~ @].

Define

f ( 0 ,El)

=

u+ 1

if E ' > H ( u , u + I ) & B Z~B < O ,

u+ 1

if

f ( < u, E')

otherwise.

E'

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

H(p) changes value at stage u ,

This concludes stage u and completes our description of the construction. We now prove a series of lemmas leading to the conclusion that the construction works. For each v < X put

Obviously the 1,'s are simultaneously a-recursive. Our first lemma corresponds to the Friedberg-Muchnik observation that in their construction each require-

DEGREE THEORY ON ADMISSIBLE ORDINALS

185

ment is injured only finitely often (see Rogers [ 10: p. 1661). 4.2 Lemma. Each Z, is a-finite.

hoof. By induction on v. The induction hypothesis tells us that each Z,,, < v, is a-finite. Since v < h = A 2 - cf(a) it follows by Lemma 1.9 that U
p

V ( E )= the least u 2 u1 such that (g(u,E),E) E B u

We claim: (6) if E < H ( v ) and c p ( ~ ) is defined theng(u, E ) = g(p(f), E ) for all u > l p ( E ) . Claim (6) is proved by induction on u > ( P ( E ) . The induction hypothesis tells us that g(< u , E ) = g(cp(e), E ) since (g(cp(E),E ) , E ) E B<'. But g(a, E ) = g(< u, E ) since H ( v ) = H(u, v) and u 2 uo. Our cp is an a-recursive partial function and dom (9)5H(v) < a* Hence cp is a-finite. Let u2 >_ u1 be such that rng(cp) 02. We now claim: (7) for each E < H ( u , v+l) and u 2 u2, f(u, E ) = f(< u , E ) . To prove claim (7), suppose u 2 u2 and E' H(u,L(o) + 1) and B' # B". Hence L(u) 5 v and there must be an E such that H(u,L(u)) 5 E < H(u,L(u) + 1) and (g(u, E ) , e ) E B u - B<". Then E
s

$ ( E ) = the least D 2 u2 such that Cf(<

u,E),

~

E)

€Au

Again $ is an a-finite function, and we let u3 >_ u2 be such that rng($) C 03. Again we make two claims: (6!) if E $ ( ~ ) ; ( 7 f' )o r e a c h e < H ( u , v + l ) a n d u > _ u 3 , g(o, E ) = g(< u, E ) . The proofs of (6') and (7') are similar to the proofs of (6) and (7). It follows from (7) and (7') that I,, E u3 hence Z,,is a-finite.

S.G. SIMPSON

186

4.3 Lemma. Let v and u3 be as in the proof of Lemma 4.2. Suppose E < H ( v + l ) a n d y < r n ( u , E ) , u 2 u 3 . Then y ( u , y , ~ ) = y ( ~ , y , ~ ) a n d y < rn(~,E ) for at1 T 2 u. Proof. Let us write C7 = A T@ B T .It suffices to show that CTn y ( u , y , e ) =

6“ ny ( u , y, E ) for all T 2 u. Suppose not and consider the least T 2 u such

that this fails. We havey(T,y,E)= y ( a , y , ~ )hencey(u,y,E) E . Hence there must be an E‘ 5 E such that (f(< T , E ‘ ) , E ’ ) E A T - A<‘ or ( g ( T ,E’), E ’ ) E BT - B“. This is impossible since~2u~.m 4.4 Lemma. For each

E

< a*

{ y ( u , y , ~ ’ ) IT < a & y < m(u,e’) & E’

<E }

is a-finite. Proof. Let v and u3 be as in Lemma 4.3 where E
4.5 Lemma. C i s a-regular and a-hyperregular and C’ = a 0’.

Proof. Suppose C is not a-regular and a-hyperregular. Let K be the least ordinal such thatf”K is not a-finite.for somef<, C, K C:dom(f). As in the proof of Lemma 2.3 we can show that K is a regular a-cardinal. In particular K 5 a*. Note that {$I K E5 C} is a-r.e. since Cis. Hence there is E < a* such that for all y, 6

DEGREE THEORY ON ADMISSIBLE ORDINALS

187

Thusy(y, e) = lim, y(u, y, e) exists for all y < K . Furthermore

The proof now splits into cases. If a* is a regular a-cardinal then K Im(e) obviously. If a* is not a regular a-cardinal, then K
dY,$ K7 C C' * h ( y ) E C' and

K,

fl

C'= 4 ++k(y)

4 C' .

By Lemma 4.4 there is an a-recursive function t : a* + a* such that for all

€
From the existence of h , k , and t it follows that C' I , 0'. 4.6 Lemma. For each e < a* there exists u such that f(u, g(u, e) = g(7, e) for all 7 L u.

E)

= f(7,E) and

Proof. Immediate from Lemmas 4.2 and 4.4. Thus f ( e ) = lim, f(u, E) and g(e) = lim, g(u, E ) are a-finite. Define A = {A"lu < a} and B = ( B u (u < a}.It is perhaps amusing to note that

u

u

S.G. SIMPSON

188

clauses (4) and (5) of the construction have played no role in the proofs of Lemmas 4.2 -4.6. 4.7 Lemma. For each E < a * ,

and

Proof. First suppose ( ~ ( E ) , E ) E ALet . u be such that ( f ( ~ ) , f ) € A "- A'". It follows that f(< T,E)= ~ ( T , E=) f(e) for all T 2 u. Put u = L(u). Then H(o,v) 5 E < H ( o , v + 1) and s o H ( ~ , p=) H ( p ) for all p 5 v, T 2 u. Since A " + A < " w e m u s t h a v e g ( u , ~ ' ) = u + 1 foraIlE'2H(u,v)=H(u). Furthermore there must be an 77 such that ( f ( ~ ) , e , q )E R F and K , n B'" = @.Note that K, 77 < u. We claim that K , nB = @.Suppose not, say (g(u', E'), E') E K , n (B" - B<"'), u' 2 u. Then g(u', E') < u hence g(u', E') = g(u, E') and ~ ' < H ( u , v ) = H ( u ' , v ) . Put v'=L(u'), thenH(u',v')<E' u a contradiction. Now for the converse suppose ( ~ ( E ) , E , ?E) R Z and K , n B = 0.Let u be a stage such that f(< u, E) = f ( ~ and ) ( f ( e ) ,e , q ) E RZ and H(u,L(u))5 e < H(u,L(u)+ 1). Then ( ~ ( E ) , E ) E A S " A. We have proved the f(e) part of the lemma. The g(e) part is similar. 4

6

<

4.8 Lemma. A

<

4, B and B $,

A

Proof. In the first place ( 5 I K EC B } is a-r.e. since B is. Hence A imply the existence of an E < a* such that for all 7

But y = ( f ( ~ ) , e )witnesses the contrary. 4 The proof of Theorem 4.1 is complete.

<, B would

DEGREE THEORY ON ADMISSIBLE ORDINALS

189

5. Further results During 1970-1972 the theory of a-degrees has developed rapidly. Two of the prettiest theorems are to be found in the Ph.D. thesis of Richard Shore.

5.1 Theorem (Shore). Let A be an a-regular, a-r.e. set and let B be a non-arecursive, a-regular, a-r.e. set. Then there are hyperregular a-r.e. sets A O ,A , such thatA = A , U A , , A O n A , = 0,B $ , A , , B e , A , .

Corollary. Any non-zero a-r.e. degree is a nontrivial join of two hyperregular a-r.e. degrees. 5.2 Theorem (Shore). Let a and c be a-r.e. degrees such that a < c. Then there is an a-r.e. degree b such that a < b < c. If a is hyperregular then b can be made hyperregular. Theorems 5.1 and 5.2 generalize two famous theorems of ordinary recursion theory due to Sacks. They are known respectively as the Splitting Theorem and the Density Theorem; for an exposition see Shoenfield’s degree monograph [ 141. The proofs of 5.1 and 5.2 are remarkable applications of Shore’s “blocking” device duscussed in the Remark on page 182 above. We offer the following analysis of the proof of 5.1. A typical goal or requirement of the construction is cB [elAi where i < 2, f < a*.In the special case a = w , Sacks satisfies this requirement by preserving (with priority e) those computations which make cB and [elAi look equal on long initial segments of a.(Because of these preservations, Sacks is able to argue at the end that cB = [elAi would imply that B is a-recursive.) In the case of an arbitrary admissible a,this will not work, because the preservations associated with an infinite, a-finite set of requirements can get out of hand if A 2 - cf(a) < a*. Instead Shore breaks up the set of a* requirements into A*..- cf(a) blocks each of size less than a*.A typical block Bv,i is {cB [ E ] I H(v) 2 f < H ( v + 1)) where i < 2 , v < A2 - cf (a).(See page 182.) Shore satisfies the requirements in block Bv,i by preserving with priority v those computations which make {y < a1 cB(y) and [eIAi(y)look equal for some E, H(v) 2 e < H ( v + 1)) contain long initial segments of a.Thus the block of requirements Bv,j is handled as a single requirement, so the preservations associated with Bv,i do not get out of hand.

*

+

190

S.G. SIMPSON

In another line of investigation, M. Lerman and C.T. Chong have attempted to generalize the theorems of Lachlan's Lower Bounds paper [3] to a-recursion theory. The results of this attempt are as follows:

5.3 Theorem (Lerman). If a and b are a-r.e. degrees with a n b = 0 then

aub
Lerman and Sacks, with good reason, call an admissible ordinal a refractory if (i) there is a largest a-cardinal, call it N; (ii) A 2 - cf (a)< H;and (iii) there is n o I;2 function f : a I s ' 0 where 0 < N. For example let a be the first constructibly uncountable admissible ordinal having constructible cofinality w ; then a is obviously refractory. 5.5 Theorem (Lerman-Sacks). Suppose a is not refractory. Then there are nonzero hyperregular cy-r.e. degrees a, b such that a n b = 0.

Of course one conjectures that the hypothesis of non-refractoriness can be eliminated, but this does not seem easy to do ll. Another area where the results to date are fragmentary is the question of minimal a-degrees. An a-degree m is said to be minimal if m > 0 and there is no a-degree a with m > a > 0. 5.6 Theorem (MacIntyre). Let a be an admissible ordinal such that I a I is regular and eveiy X E cy of cardinality less than I a I is alfinite. Then there exists a regular, hyperregular, minimal a-degree. In particular, minimal a-degrees exist if a is countable. (For a = o this is an old theorem of Spector.) MacIntyre has conjectured that regular, hyperregular, minimal a-degrees exist for all admissible ordinals a. The proof of 5.6 is not a priority argument. By exploiting priorities i la Shoenfield [ 14 : pp. 54-56] and the "a-finite injury method" of SacksSimpson [ 131, one enlarges the class of admissible ordinals a for which minimal a-degrees are known to exist. Namely, I ' Recently D. Posner has shown that for every admissible ordinal (Y there exist regular, hyperregulara-degreesu, b such that u n b = 0.

DEGREE THEORY ON ADMISSIBLE ORDINALS

191

5.7 Theorem (Shore). Suppose A 2 - cf (a)= a Then there is a regular, hyperregular, minimal a-ciegree m < 0’. One can probably adapt the proofs of 5.6 and 5.7 to get finite distributive lattices as initial segments of a-degrees. However, to eliminate the special hypotheses on a seems a difficult problem indeed. This completes our survey of the current theory of a-degrees. Apart from what we have reported here, most questions concerning a-degrees are virgin territory. It is not always easy to appropriately generalize the statement of a theorem of ordinary degree theory to a-degree theory, much less the.proof. One obstacle is that the admissibility of a (i.e. of the structure (L,,€)) does not imply admissibility of the expanded structure (L,,€,C) where C C a, even if C is a-r.e. and a-regular. Therefore “relativization” to C (cf. Rogers [ 10 :p. 2571) may be difficult or impossible. This suggests a typical question: for an arbitrary admissible ordinal a,are there X 2 setsA,BCa such thatA 3, B @ O’andBe, A @ Of? For a = w (indeed whenever A, - cf (a)=a) the answer is obviously yes by relativizing the Friedberg-Muchnik theorem to 0’. For aibitrary admissible ordinals a, the answer is probably still yes but the proof may require an “infinite injury” argument. Carl Jockusch has proved the following theorem of ordinary degree theory: there exists a degree d such that for all b 2 d there is u < b such that there is no c with a < c < b. Jockusch’s proof is unique in degree theory in that it employs the powerset axiom of ZFC (via a result of Paris concerning GaleStewart games). Therefore, the following question is of exceptional interest: for which admissible ordinals a does Jockusch’s theorem generalize to adegrees?

6. Appendix: the fine structure of L The consmtctible hierarchy is defined by recursion on the ordinals as follows: Lo = {P}; L,,+l = { X C L,IX is first order definable over (L,,E) allowing parameters from L,,}; L, = U {L,I u < A} for limit ordinals A; L = U {L, I u an ordinal}. Ronald Jensen [ 11 has made a detailed study of the fine structure of L. (Jensen then uses his results to settle a number of open questions in set theory under the assumption V = L.) There is a close connection between some of Jensen’s ideas and some of the ideas in a-recursion

192

S.G. SIMPSON

theory. We now discuss this connection briefly. First, a glossary comparing some of Jensen’s terminology to some of ours. (The equivalence proofs would be tedious but straightforward.) 6.1. Let a be a limit ordinal. Then L, is an admissible set iff

ordinal.

a is an admissible

6.2. Let a be an admissible ordinal and A C a. Then (i) A is Xn(L,) iffA is En in our terminology; (ii) A E L, iff A is a-finite; (iii) The structure (L,,E,A) is amenab2e iffA is a-regular.

6.3. Suppose a is admissible andA I a is a-regular and letf be a partial function from a into a. Then ( i ) f i s CI(L,,A) i f f f l w , A ; (ii) the structure (L,,E,A) isadmissible iffA isa-hyperregular; ( i i i ) B C a i s A,(L,,A)iffc, &,A. 6.4. Suppose a is admissible andA B C a is A (La, A ) iff B < a A .

Cr a is a-regular and a-hyperregular. Then

A decisive role in Jensen’s theory is played by the notion of En projection. For a an admissible ordinal and n 2 1, the En projecturn of a is defined to be the least /3 such that there is a C,(L,) function from a subset of /3 onto a. Thus the El projectum of a is just what we have been calling&*. Jensen’s main theorem reads as follows. 6.5.Theorem.For n 2 1, the C, projecturn of a is equal to the least 0 such that not every C,(L,) subset of /3 is a-finite. The proof of 6.5 is essentially model theoretic rather than recursion theoretic in nature. One needs a delicate refinement of the Skolem hullcondensation method first used by Godel to prove the GCH assuming V = L. The reader is invited to study Jensen’s proof in [ 11. For a admissible and n = 1, Theorem 6.5 is trivial and we have used it repeatedly in the present paper. For n = 2 Theorem 6.5 is non-trivial but has been used elsewhere in a-recursion theory, e.g., in [7] and in Chapter 3 of

DEGREE THEORY ON ADMISSIBLE ORDINALS

193

[ 171. The interesting point is this: Theorem 6.5 for n = 2 can fail if one looks at admissible structures of the form ( L a , E , A ) . Namely, one can construct an admissible ordinal a and an a-regular, a-hyperregular setA C a such that (i) not every A2(La,A) subset of w isa-finite;(ii) there is no Z 2 ( L a , A ) function from a subset of w onto a. This means that some arguments of a-recursion theory, e.g., the proof of Theorem 3.2 of [7], do not generalize straightforwardly to recursion theory on admissible structures (L,,E,A). This is a sad but amusing state of affairs which deserves to be investigated further.

Bibliography [ I ] R.B. Jensen, The fine structure of the constructible hierarchy, Annals of Math. Logic 4 (1972) 229-308. [ 21 G. Kreisel, Some reasons for generalizing recursion theory, in: R.O. Gandy and C.E.M. Yates, eds., Logic Colloquium '69 (North-Holland, Amsterdam, 1971) 139-198. [ 31 A.H. Lachlan, Lower bounds for pairs of recursively enumerable degrees, Proc. London Math. SOC.16 (1966) 537-569. [ 4 ] M. Lerman, Maximal a-r.e. sets, Trans. Amer. Math. SOC.(to appear). [5 J M. Lerman, Least upper bounds for minimal pairs of a-r.e. crdegrees (to appear). [6] M. Lerman and G.E. Sacks, Some minimal pairs of a-recursively enumerable degrees, Annals of Math. Logic 4 (1972) 4 15-442. [7] M. Lerman and S.G. Simpson, Mximal sets in a-recursion theory, Israel J. Math. (to appear). [S] M. Machtey, Admissible ordinals and lattices of a-r.e. sets, Annals of Math. Logic 2 (1971) 379-417. [9] J.M. MacIntyre, Minimal or-recursion theoretic degrees, J. Symb. Logic 38 (1973) 18-28. [ l o ] H. Rogers, Jr., Theory of Recursive Functions and Effective Computability (McGraw-Hill, 1967) 482 pp. [ 11J G.E. Sacks, Post's problem, admissible ordinals, add regularity, Trans. Amer. Math. SOC.124 (1966) 1-23. [ 121 G.E. Sacks, Metarecursion theory, in: J.N. Crossley, ed., Sets, Models, and Recursion Theory (North-Holland, Amsterdam, 1967) 243-263. [ 131 G.E. Sacks and S.G. Sinpson, The a-finite injury method, Annals of Math. Logic 4 (1972) 343-367. [ 14) J .R. Shoenfield, Degrees of Unsolvability (North-Holland, Amsterdam, 197 1) 111 pp. 1151 R.A. Shore, Minimal a-degrees, Annals of Math. Logic 4 (1972) 393-414. [16] R.A. Shore, Priority Arguments in a-recursion Theory, Ph.D. Thesis, M.I.T. 1972. 1171 S.G. Simpson, Admissible Ordinals and Recursion Theory, Ph.D. Thesis, M1.T. 1971.

J.E.Fenstad, P. G.Hinman (eds.), Generalized Recursion Theory @ North-Holland Publ. Comp., 19 74

MORE ON SET EXISTENCE Frangoise VILLE University of Orleans

6 1. Standard sets of a theory: definition and general results The structure of the standard sets is, by definition, the structure ( V ,Ev) of the cumulative hierarchy; it is extensional and well founded. The transitive closure of an element a of V is denoted by [ a ] . Let us suppose that (T) is a theory formulated in a language k,containing at least the binary relation symbol E. Realisations of k have thus the form vM,EM, ...>; i f m is an element ofM, [mIEMdenotes the transitive closure of m with respect to €*. To avoid confusion, membership in the metalanguage will be denoted E.

Definition 1 . Let 311 be a realisation of L. An element a of V is called a standard set of% iff there is an ma in M such that:

ma is said to represent a in %. If ( M ,E M ) is extensional, a standard set is represented by at most one element of M .

Definition 2. The element a of V is called a standard set of the theoxy (T) iff a is represented in every model of (T). S(T) denotes the collection of all standard sets of (T).

Note that (S(T),iEv I' S(T)) is a transitive substructure of CV, EV). When (T) C k , (where L, = V, = H, = the collection of well founded 195

196

F. VILLE

hereditarily finite sets) and so, each formula of (T) is of finite length, it can be shown by a compactness argument that every standard set of (T) is hereditarily finite: thus S(T) C L,. If (T) is a denumerable set of formulas (of denumerable length) C Lu1,, then one easily deduces from the LowenheimSkolem theorem that every element of S(T) is hereditarily countable. A brutal generalisation of these facts would be: “LetA be standard and admissible, and let (T) be a theory in the language LA (associated withA, as in Kunen, Banvise etc.), then S(T) CA”. But this cannot hold, as shown by the following example: TakeA = Lw,k, and the following theory (T): Language of (T): - the binary relation symbols E and =, - for each integer n , a constant symbol c,, - two distinguished constant symbols c, and c,. Axioms of (T): - extensionality - Vx x 4co and for each integer n: vx (x Ec,+l -(x = C n V X Ec,)) - vx (x Ec, +-+ Wn&,X = C,) - vx (x E cw + x Ec,) - for each integer n : c , ~Ec, iff n & W

c, + c w

iff

n$W ,

where W E Il - Xi,for example the set of Godel numbers of recursive well orderings. and S(T) = o U { W } U {a}; but as is well known Clearly (T) C

W q! Luck, so S(T) 1 not I I ~ .

LWck.Note that the theory (T) above is 1

n: U 2: but

The principal aim of this paper is to show that S(T) C LUfk is obtained by bounding the complexity of (T): (*) Provided that (T) has an extensional model, if (T) is an L k-r.e.theoryof wF ‘LwFk> then s(T) Lwfk.

Below we shall use o1to denote the least non recursive ordinal which we called above. So LW1will be the set of all sets constructible with recursive

oik

MORE ON SET EXISTENCE

197

order. We shall also make the following assumptions on (T): - there is at least one extensional model of (T) - (T) is a theory with equality In the following(T) will be an Lu1-r.e. theory of LL . wi

92. Properties definable on S(T) In this paragraph we assume that D is a subset of S(T) which satisfies the following condition: (D) Every element of D is defined in every fnodel of (T) by a formula of that we shall denote by @,(x). So that for every a in D, for every model 311 of (T) there is an m, in 9?2which @.,(x)) is true in %. represents a in 5% such that Vx (x E ma

-

Definition 3. a) A subset X of D is LU1-definable on D in a given model '% of (T) iff there is a formula G W ( x ,x l , ...,x,) of LLWl with n + 1 free variables, and parameters a l , ...,a, in nZ such that X = {a E V : (ma,a l , ..., a,) satisfies 4% in% 1 n D. b) A subset X of D is LUl-definableon D in (T) iff it is LU1-definable on D in every model of (T).

Examples: a) for each element a of S(T), a n D is defined on D in each model 371 of (T) by the formulax E x l with parameter ma. b) If L, E S(T) and L, C D, then o is definable on D in (T) as well as the graphs of + and X, and every arithmetic subset of w; here L,, or more precisely, its representative mL,, can be used as a parameter. Below we shall need the following analysis of LU1-definability on D in terms of the consequence relation in (T): Theorem 1. Under the condition (D), if X C D is LUl-definableon D in (T) with only one free variable then there are formulas $ (x) and $ 2(x) in LL, such that for e v e y a in D:

F. VILLE

198

Theorem 1, which can be proved via the omitting types theorem for denumerable admissible languages, is fundamental for the proof of (*). The case for L, is treated in chap. 6 of [Kr.Kr]. We shall distinguish two cases according to whether (T) is an extension of Kp ( 5 3 ) or not (84).

$3. (T) is an extension of KP In this case S(T) which obviously contains every element of L, will provide, in every model of (T) all the parameters we need. Every model of (T) is extensional; then by the remark following Definition 1, every element of S(T) has at most one representative in every model of (T). Every model of (T) is admissible; then if S(T) has no infinite element, S(T) = L,. If, on the contrary, there is an infinite element in S(T) then L, E: S(T) and we have LW1C S(T) (cf. [8]p. 243). Note that in each case: -

Lemma 1. mere is an L, l-recursivemapping Q -+ @,(x)from LU1into XL W1 such that for every a in L, n S(T), @&) defines a in evety model of(T). This is an easy consequence of the extensionality of every model of (T). Therefore LW1satisfies condition (D) of $2. Lemma 2. Let a subset X of L, L, recursive.

be L, l-definable on L,

As in Theorem 1 we prove that there are every a in LU1 :

For the definition of KP see [S] p. 232.

in (T). Then X is

$2 in LLW 1 such that for

MORE ON SET EXISTENCE

199

The applications: a 3x($,(x) A $l(x)) and: a + 3x(,(x) A J / ~ ( x )are ) LU1-recursiveby Lemma 1; therefore, consequence in (T) being LUl-r.e., X is an LU1-recursivesubset of Lwl:

Theorem 2. w1 $ S(T). Let us first prove one lemma about model of KP.

Lemma 3. Let % be a model of KP and R a linear ordering of w which is A l dejinubb in%. For evep ordinal a of %if there is an isomorphism f of h , E M I’ a>onto an initial segment of the ordering R then f belongs t o m .

f is an isomorphism from (a,EM r a>into an initial segment of R (which we shall denote < R ) iff f satisfies the following equations, I ( f ,a):

f(0) = a

iff a is the least element in the ordering < R , andforallX
f ( h )= b iff b is the least element of R greater than any f ( 5 ) for ( < h in the ordering < R . Now we want to prove that we have in% : for every ordinal a, there is one and only one f such that Zcf,a). Note that there is a ZlforrmlaIs(f,a) which describes the relation f is a fitnction of domain a ~ Z ( f , ain) CYfC(we allow parameters) since R is A l in%. By induction on ordinals we establish that 3 !f I s ( f ,a) holds in m for every a. The uniqueness is easily established, so w have to establish the existence; we proceed by induction:

- if a = $9 thenf= $9 and is the only fsatisfyingIs(f,a) in%. - if a f (?J then by induction hypothesis there is a unique f , say f,, which

satisfiesIssCfE,E)in %(for t < a). Thus,

- if a is the successor of 0, we put f , = f p U (0, b ) where b E w is chosen as described in I. As f p belongs to W ,f a obviously belongs to311 . - if

a is a limit ordinal, one can apply X l-replacernent, since Is E Z and, by induction hypothesis

F. VILLE

200

Thus there is an f in CM, such that: W / =(f = f, : (
Proof of Theorem 2. Let Q be a recursive linear ordering of w with an initial segment IQ of length w1 (for example the ordering in Candy’s [GI). Since Q is a recursive subset of w 2 ,Q can be defined in every model of (T) by a A o - . formula, using o as a parameter. belongs to (is Suppose o1E S(T); and let W be any model of (T): represented in) 371. By the choice of Q,there is an isomorphism f from (wl,EM r w,) into (ZQ, Q ) . Since the hypothesis of Lemma 3 are satisfied, fbelongstoW.ZQ can thenbedefinedby 3 $ € w 1 ( E , x ) E f w i t h x a s the only free variable and the two parameters w1 and f. So Ze is defined on w in (T) and, by Lemma 2, ZQ is L,, -rec; and hence A:, since ni over w or over L, is equivalent to LW1-r.e.(cf. [BGM]). But this contradicts the fact that ZQ is a ll: subset of w which is not xi. Thus we cannot have w , & S(T). Theorem 3. S(T) C LU1. Lemma 4. Let % be a model of KP and a be a subset of L which belongs to W.Then there is an ordinal a of W such that a C L,.

Since W is admissible, the function od which associates to each constructible x its order [ is expressible in 311 by a formula Od which is Z over CM and verifies %k V x ~ a 3 !Oyd ( x , y ) . By C , replacement there is a b in% such that

%k

b = {Od(x) : x EM a} .

Take a to be the least upper bound of b ; it is an ordinal of 71’2and a C L,. Proof of Theorem 3. Assume a is an element of S(T) included in LW1 ;the example a) following definition 3 shows that a is LWI-definableon LU1 in (T)

MORE ON SET EXISTENCE

20 1

and Lemma 2 shows that a is LW1-recursive. In every model 7 R of (T) there is, by Lemma 4, an (Y such that a C La, and this a does obviously not depend on cPZ; therefore a € S(T) and by Theorem 2, a < wl. So a is an LUl-rec. subset of LU1 which is LU1 bounded, i.e. a E LW1. An induction shows that every element a of S(T) belongs to L, 1 . Theorem 4. S(T) = L, or S(T) = L,

1.

Either S(T) has only finite elements and S(T) = L, ; or there is an infinite element in S(T). In this last case we saw that LWtC S(T); by Theorem 3 S(T) C LW1,thereforeS(T) = Lwl. Theorem 4 of Chapter 8 in [Kr.Kr] can now be generalised to JL,~. Definition 4. A subset X of S(T) is uniformly LU1-defined in (T) iff there is a with a single free variable, such that for every model 717of formula CP of JL (T)x= { a E ~ w h ~ @ ( a ) } . Theorem 5. Every subset X of S(T) which is uniformly LU1definable in (T) is L, -finite. By Theorem 4 and Lemma 1 every element a of S(T) is defined in (T) by formula $&) such that the mappinga +$Jx) is LW1-recursive.If X C S(T) is uniformly LW1-definablein (T), it is a fortiori LUl-definable on L,, in (T), and by Lemma 2 it is L, l-recursive. Suppose qh is a formula of JL which w1 defines X uniformly in (T). Let c be a constant symbol not occurring in (T). The set of axioms

is Lwl-r.e. and is obviously inconsistent. By compactness there is a b E L, such that

is consistent. Therefore

202

F. VILLE

and, as c does not occur in (T)

This means that X is bounded by b, which implies X

&

Lwl.

94. (T) is not an extension of KP It is convenient to introduce auxiliary theories (Tf) and ( E ) , formulated with an additional type of variables. (Ti) is needed to formalize the metamathematical notion of admissibility; let us call “metauniverse” the universe of models of (Tf). (E) relates standard sets of (T) to “real standard sets” i.e. standard sets of metauniverse. We assume that (T) is consistent with extensionality. Language of (Tf): - its variables, of a type different of the types in (T) are denoted X , Y ,Z , ... - two binary relation symbols E‘ and =‘. So the atomic formulas of this languages are of the form X E’ Y ,X =’ Y ... . The axioms of (Tf) are those of KP formulated in this new language. Language of (E): - variables of the type of the arguments of E (the distinguished relation symbol of (T)) and of the type used in (Tf). - a binary relation symbol E , as well as the symbols E and E’. The new atomic formulas have the form x E X . The axioms of ( E ) are: (1) 3 ! X V x ( V y y 4 x -+x EX) (2) V x ( V y E x 3 ! Y y E Y + 3 ! X x E X ) (3) VxVyVXVY(x E X A y E Y +. ( X E y *X E’ Y ) ) (4) V X ( V Y E’ x 3 ~ EyY+. iix x EX). Let (T’) be (T) U (Tf) U (E). Every realisation of the language of (T ’) has the form:

MORE ON SET EXISTENCE

203

Definition 4. A standard set of (M U M’, ...) is a standard set of the meta-universe, i.e. of ( M I , E&,,=;Mt).

Lemma 5. Let 311i= (E) and (M, E M )be extensional, Then the restriction of EM to the well founded part Mo of M is the graph of a function which maps representatives of standard sets in (M,E M ) onto representatives of standard sets in (M‘,E&~). We prove that for all x in M o 5% 3 ! X(x E X ) by induction along €*rMo with the help of axioms (1) and (2). Thus there is a function, say cp, defined every where on Mo whose graph is EM l- M o X M . By Axiom (3) this function embeds ( M o , €M PMo) isomorphically in (M’,€&t); by Axiom (4), writing € M for EM r M o

Therefore if ma represents the set a in ( M , E M ,= M ? , @(ma)represents a in (MI,

€ht,=L*).

Lemma 6. (TI) is consistent. Let 311 = ( M ,CEM, ...) be an extensional model of (T) and Mo be its well founded part w.r.t. EM.Define by induction along EM, (i.e. € M 1 M o ) a function $ from Mo into V by:

$I is every where defined on Mo. Put: M i = $[Mo] Mf = the least admissible containing M l El = e V I M f

= identity in Mf E l = the graph of $ 311’ = ( M U Mf, E M , El, =1, El, ...) . 311’ is a model of (T’) since: - (M, EM, z M )is by hypothesis a model of (T) - (Mf ,E 1, =1 ) is admissible and therefore a model of (Ti) - %’ satisfies (E) by definition of $ (cf the proof of Lemma 5). =

F. VILLE

204

Lemma 7 . Every standard set of (T) is a standard set of (T’). As (T) has an extensional model, (T) U {Extensionality} is consistent; it is consistent with ( T f ) U (E). Let (T1) be (T) U {Extensionality}. By Lemma 5 every standard set of (T1) is a standard set of (T’): therefore as (T) C (T1), every standard set of (T) is a standard set of (T’).

Theorem 6. S(T) C LU1.

(T‘) is an Lwl-r.e. theory of .CLw 1 such that (T‘) 3 Kp; by Theorem 4 S(T‘) is equal to either L, or LU1.By Lemma 7, S(T) C S(T’). Therefore S(T)

c LW1.

This result evidently generalizes the results of [GKT] which establishes if a theory (T), formulated in the language of finite types, i n n : then subsets of w which appear in every a-model of (T) are hyperarithmetic and every collection of subsets which appears in every w model of (T) is Lwl-finite. The only restriction we make on (T) is its consistency with extensionality. How to avoid this condition will be shown in a forthcoming paper.

References J. Barwise, Infmitary logic and admissible sets, JSL 34 (1969) 226-251. J. Barwise, R.O. Gandy and Y. Moschovakis, The next admissible set, JSL 36 (1971) 108-120. [GI R.O. Gandy, Proof of Mostowski’s conjecture, Bull. de 1’Ac. Pol. des Sc. VIII 9 (1960). [GKT] R.O. Gandy, G. Kreisel and W.W.Tait, Set existence, BuIl. de l’Ac.Po1. des Sc. VIII, 9 ( 1 960). [Kr. Kr] G. Kreisel and J.L. Krivine, Elements of mathematical logic (North-Holland, Amsterdam, 1967). [B] [BGM]

PART I11 INDUCTIVE DEFINABILITY

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

INDUCTIVE DEFINITIONS AND THEIR CLOSURE ORDINALS S t a AANDERAA

*

IBM ThomasJ. Watson Research Center, Yorktown Heights, New York Abstract. We shall prove some theorems about inductive definitions and their closure ordinals. As corollaries we obtain the following results. I A; I < I < I Z; I < 1 Al-monl = I A : l = I ~ ~ - m o n l < l n : l < IA:-monl.WealsohavethatInAI#IZ;:,I,InAI+ IZA-monl, IxAlZ l a - m o n l for all n. Moreover if n 2 2, then IAA-monI = IAAlS IZA-monI < IC;l< IALl-monl and a A - m o n I = I A A l I InA-monI < I AA+l-mon I. Moreover, assuming the axiom of constructibility we have IAL I = I XA-mon I < Ix; I < Ink1< IAA+l I, for all n 2.2. However, assuming that both Projective DeterIXil-monl < minacy (PD) and the axiom of Dependent Choices (DC) hold, then I & l = 1 IZ:~I< I I I ~< ~ I AI Z ~ +=~NIzi+l-mon~ I < I I1’ I ~<~~Z~i +1~ +~~. I c o m p l e t e p r o o f sa r e g i v e n for the most important results. Many less interesting results are stated without proofs, or with a sketch of a proof.

Inhl<

1. Introduction Let w denote the set of nonnegative integers which coincide with the ordinals less than w .

Definition 1. By an inductive definition (abbreviated i.d.) we shall mean a mapping r : P(w) + P ( o ) , whereP(w) = {XlXE w } = the set of all subsets of w . (“Inductive definitions” is abbreviated i.ds.) Definition 2. An i.d.

r is monotone iff X C Y E w implies r ( X ) E r ( Y ) .

Definition 3. Let I‘ be an i.d. and let A be an ordinal number. Then we define I‘h to be a set of integers, defined by transfinite recursion as follows:

rO=O. * On leave from Institute of Mathematics, University of Oslo, Blindern, Oslo 3, Norway. 201

208

For

S. AANDERAA

X > 0 we have

Definition 4. Let r be an i.d. The closure ordinal of is the least ordinal X such that I'h = We shall let I rl denote the closure ordinal of r, and r'w= rIr'is the set defined by r. Remark I . 1 rl exists for every i.e.

r, and I

is a countable ordinal.

Definition 5 . Let w w be the set of all functions of one number variable, i.e., the set of all total functions mapping w into w. Subsets of the product space

x=X,

x ... xx,

( X i = o or

X i = P ( o ) or Xi= w w for all i = 1 , 2 , ...,k ) ,

will be called pointsets. Sometimes we think of these as relations and we write interchangeably x € A o A ( x ) . Remark 2. It turns out that it is convenient for us to permit Xi= P(w). In this way we deviate from the definition of pointsets in Moschovakis 1970 and 1972. For n , m E w, let ( n ,rn) be a coding of pairs, say ( n , m )= l/2(n2 + 2nm + m 2 + 3m + m )as in Rogers 1967, p. 64. We shall usefo,f1,f2, _..to denote some functions throughout the paper according to the following definition.

fi

Definition 6. For each i E 0, is a mapping of w into w defined as follows: fi(x)= ( i , x ) ,x E w, and& is also a mapping o f P ( o ) intoP(w) defined as follows: fi(S)= { f i ( x )1 x E S } . Moreover is a mapping of P(w) into P(w) defined as follows:

fcl

, Let a E w w . Write (a)ifor the function h x a ( ( i , x ) ) i.e. (a)j(x)= a ( ( i , x ) ) .

E w w and

INDUCTIVE DEFINITIONS AND THEIR CLOSURE ORDINALS

209

We do not need the full axiom of choice in this paper. At some places we need a weak axiom of choice:

Dependent Choices (DC). For each A C w w X w w we have

This follows from the axiom of choice. In some cases a still weaker axiom is sufficient:

Countable Choices (CC). For each A

5w X w w we have:

The axiom of Dependent Choices implies the axiom of Countable Choices. The axiom of Determinacy (AD) also implies CC. Note that the axiom of CC has to be used in order to prove that the following prefuc transformations are permissible:

... vo31...+ ... 3lVO ... ... 3OV' ._. + ... v 1 3 0 ... (See Rogers 1967, p. 375 and Exercise 16-2, p. 446.)

Definition 7. A point class C is a class of pointsets, not necessarily all in the same product space. I t turns out to be convenient for us to represent each i.d. F by the pointset

instead of the pointset {(A,r(A))iA)cWwxWw. We shall identify an i.d. with its representation. Hence

r €C? means

210

S. AANDERAA

Given a pointset C? we usually in this paper pay attention only to the subclass(WwXw) n C o f e . Definition 8. ZA is the class of all pointsets definable from recursive relations by EA prefix. Similarly for HA. Definition 9. Let for all S E w ) .

r be an i d . , then fi = (woX w ) r (i.e. k(S) = w -

Definition 10. Let 6 be a class of pointsets. Then 8 = {x - A 1A A E C}. (Here x is a Cartesian product as in Definition 5.)

-

r(S)

E x and

Remark3. L e t r b e a n i . d . T h e n ? € l l f , i f f r E Z f , a n d f I f , = E f , , c f , = n f , . Definition 1 1. Let C be a class of pointsets. Then the supremum ordinal for C is I C I = sup { I r I I r is an i.d. and l? is an i.d. and E C } . Definition 12. LetA C w and l e t H be a pointset such that H E X , X X 2 X ... X X , where the Xi’s are as in Definition 5. Then = d(x1t+x2,...,x,-~) GI, x 2 , ..., x k b l , A ) € H I if Xk = P ( w )and & = H otherwise. Let C be a class of pointsets. Then H A = {# I H € C } . is defined as in Remark 4. Note that if C = then C A = ZAA, where Rogers 1967. (See pp. 409,374 and 304.) In the same way, if C= n A then eA=nAA. Definition 13.Let C be a class of pointsets. The ordinal h is called a terminal ordinal for C iff there exists an A C w such that h = ICA 1. The set of terminal ordinals for C is denoted by ter(C). Definition 14. Let C be a class of pointsets. The ordinal h is a transit ordinal for C iff there exists an A S o such that h < ICAI but I r I # h for all i.ds.

rEe.

Definition 15. Let A and I’ be i.ds. Then A is one-one reducible to

r up to h

INDUCTIVE DEFINITIONS AND THEIR CLOSURE ORDINALS

(notation: A

2I 1

‘r) iff there exists a one-one recursive function f such that E fl)).A is one-one reducible to r at

(Vt)(Vx)(t
everyordinal(notation: A < T I ’ ) i f f A I t r w h e r e h=(sup{lAl, Irl})+ 1.

Definition 16. Let A and I‘ be i.ds., and let h be an ordinal; and let g be a mapping of ordinals into the ordinals such that g ( t ) is defined if r; < A. Then A is one-one reducible to the g-l contraction up to h (notation A 5 $ I?) iff there exists a one-one recursive functionf such that (Vt)(Vx)(t

<7

7

Definition 17. Let A and r be i.ds., then A is one-one reducible to a reorganization of r up to h (notation A 5 = r) iff there exists a one-one recursive functionfsuch that the following two conditions are satisfied, for all ordinals 8,r;, where r; < A. f-l(re) AE * f - l ( r e + l ) ~ 5 + 1 1. 2. f-l(re)- A’ # 0 * re+l- re 6. A is one-one reducible to a reorganization of r (notation: A 5 T = r),iff A < = r where A = IAI.

c

c

+

t

Remark 5. Let A and r be i.ds. and let h be a one-one recursive function. Then “A 5 T r via h” small mean that A 2 r where the recursive one-one functionfmentioned in Definition 15 can be chosen to be equal to h , i.e., ~ ‘ ( f l=)AT for all g <_ I A I Similarly . for the following expressions: < r via h”, “A 5: r via h” “A <1 hrvia /I” “A II’<-g r via h”, “A 5” 1-z and “A 5 ; s I’ via h”. N_

Definition 18. Let be a class of point sets, and let I’be an i.d. Then r is inductively Ccomplete iff r E C and (VA) (A C o X o & A E C * A 5 7 r). Definition 19. Let e be a class of pointsets. Then the spectrum of C (denoted: s p ( e ) ) is the following set of ordinals: {lrlIF€ e and F is an i.d.1. Remark 6. We shall from mow

XI

assume that the axiom CC holds.

212

S. AANDERAA

2. We shall prove the following results.

Theorem 1. Let C = Z A or C = Il;. Then there exist an i.d. r which is inductively e-complete and such that Ah S1I’” for each i.d. A in and for each ordinal h Hence IF1 = I el E Sp(C).

e,

Remark 7. It is proved in Aczel and Richter (1972) that I Xi I f IrIiI; and they have later obtained a lot of results including proofs of theorems 1-4 in this paper.

e

Theorem 3. Let = II; or let = E : for some n. Suppose I I L I c I. Let r and A be inductively C-complete and inductively C-complete, respectively. Let g(h) = h t 1 if h is a successor ordinal, and let g ( h ) = h otherwise. Then A < T < g r a n d riiAl+l A.

e

C!

sg

Definition 20. Let K be a set of ordinals. An ordinal X is apoint of accumulation of K iff (Vt) ([ < A * ((31) ([
Theorem 4. Let C be a class o f pointsets equal to J I A or EA for some n 2 1 Then Acc(tra(C) U ter(C?)) = Acc(tra($)U ter(8))

and tra(C)

-

Acc(tra(C?) U ter(C)) = t e r ( 6 )

-

Acc(tra($) U ter(6)).

Remark 8. T r a ( I l f ) = 6. Hence T r a ( I l i ) # Ter(Ti) = Ter( Ei).We d o not know whether Tra (Xi)= Ter (Ill), but we would like to state that as a conjuncture. We do not know if it is consistent with set theory to assume tra(nA) - ter(ZA) f 6 or tra(E;) - ter(II;) f 0 for some n. Theorem 4 will not be proved in this paper.

INDUCTIVE DEFINITIONS AND THEIR CLOSURE ORDINALS

213

To state the next theorem we need the notions C-norm and Prewell-ordering(C).

- -

Definition 21. Anorm on a set p is a function 4 : p +. ordinals; we call a C-norm if there are relations s a n d i n . C and @,respectively, such that (VY ( y € p =2. (VX)( x Y 3Y [x E P ?Lc! 44x1 5 4(Y )I 1).

5


i

Definition 22. Prewellordering ( C )

every pointset p in C admits a C-norm.

Theorem 5. Let r and f. be i.ds., and let r E C and € C?, where C = C,! or c = n,!,for some n 2 1. Suppose r(s)- s @ 3 F(s> - s c w. Then there exists an i.d. A E C such that I' 5; 5, A and IA I = I f I + 1, where g is as in Theorem 3.

+ D =+

Theorem 6. Let C = C,!, or C = ordering (C). Then I C I < I 6 I.

Ili for some n > 1, and suppose Prewell-

Corollary 3. Suppose that every element in Godel, then l E i l < In,!,Iforalln 2 2.

w is constructible in the sense of

Corollary 4. Assume Projective Determinacy (PO) and the axiom of Dependent Choices(DC)hold. Then l ~ ~ i - l l < l C ~ i - l l a nlCiil
2. Proof of Theorem 1. Let C = or C = n,!,for some n 2 1. Then there exists a pointset E E C? such that E S w X w XP(w), which enumerates all pointsets in P(w XP(w)) n C in the sense that if A E C and A C w XP(w), then there exists number i such that

A(x,S) -E(i,x,S)

for all x E w and S S w

.

S. AANDERAA

214

We shall write x E Ai(S)for E(i,x,S). Then we have that for each i.d., A E C there exists an i E w such that Ai = A. Let r be an i.d. defined as follows: ) EAz(f;l(o)} (1) r(s) = {XI ( 3 Y ) ( 3 z ) ( x = f z ( Y &Y

u

= ;=o fi(4 (S))). (Recall thatf,(x) = (z,x), see Definition 6.) Then r E C , and by induction on h we can easily prove that (Vx) (x €A; - f z ( x ) E r,”). This proves that I‘ is C-complete since if A is an i.d., and A E C, then A = Ai for some i and Ai 57 r via fi. To prove the last part of the theorem we shall prove a lemma.

(fL1

Lemma 1. Let C = Ek or C = l l L for some n. Let r be an i.d. in e. Then there exists an i d . r0in C such that I’57 roand I” I I?; for each ordinal A. Proof. Let robe defined as follows:

We shall now complete the proof of Theorem 1. Let r be defined by (1) as before and choose roE C such that r 5; ro and rh< I ri for all ordinals h which is possible according to Lemma 1. Then ro= Ai for some j E o.Given i E o , we have A t I1 < 1~$7L1I-”. Hence <_ r” for all ordinals h. This completes the proof of Theorem 1.

rh

4 5

3, Proof of the Theorems 2 and 3. T o prove these theorems, we shall prove a lemma.

Lemma 2. Let C = E or C =n;,and let Aand I’ be i.d.5 such that A E and r E C. Then there exists an i.d. Z = ZA*r, such that Z E (? and

INDUCTIVE DEFINITIONS AND THEIR CLOSURE ORDINALS

2 15

Proof. We shall first outline the general idea of the proof, before giving the details of the construction. The main problem we have to solve is to construct E E 6 from A E 6 and I' E C?such that rhis coded in Z h in some way. First we note that E Suppose that we can recover rhfrom 2'. We cannot get r(rh)directly, but we can get f'(rh) = w -I"+'.If 4 < h then o - rhC w - fi. Hence we cannot save w - I" directly. But we can use the elements Ah as indices. Hence we can try to choose Z such that Z h = UElhAEX ( w - r E ) .We shall use a somewhat more complicated construction. Let I , = {x t 3 1 x E A,}. In order to generate the indices I,, code AX as fo(Ah),and I" is coded as UE-<, (IEX(w-I't)). In this way, we can go on simulating both A and r as long as A generates new indices. (In the proof of Theorem 5 we shall use another construction to obtain new indices.)

e.

In order to obtain IEl= I A l + 1 if 1A1= Irl we addfi(0) = ( 1 , O ) when FA+'= rh.Moreover, in order to obtain I' Lir"' 2, Z, we add ( 2 , ~ to ) when x E I". We shall now give the details of the construction. U >constructZfrom 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
S. AANDERAA

216

Moreover,

IZI = 1Al except when I A l = lrl. If lrl
( 1,O) E Elr1*' - Elr'. Hence

follows easily and the proof of Lemma 3 is complete.

To prove Theorem 2, suppose I ZA I = In; I for some n. Choose A E Xi and such that IAl= ICAl and l r l = IIIAI. Apply Lemma 2, and we get ZEZ,!, such that ! E l = IA(+1 = lZ,!,ltl, whichisacontradiction.Thisproves Theorem 2.

r En:

To prove Theorem 3, let Z = rl; or C = Xi and suppose

I6 I 5 I C I.

Then

161 < I C I by Theorem 2. Let r and A be inductively e-complete and induc-

and Z2 = Zrl* as in tively C-complete, respectively. Construct Z, = Lemma 2 . Then El E 6! and Z2 E C and A 5; Z, 5; A and r <\A'+1 SgZ,. Hence 5;""' 5, A. Moreover, r <; E , 5; r and A 5;
4. Proof of Theorems 5 and 6 and their corollaries

c

Let C = XA or C =HA and let r and satisfy the assumptions in Theorem 5. The proof is very similar to the proof of Lemma 2. The difference between the constructions in the proofs is the generation of indiceslA.We shallnowletZA= { ~ + 3 1 ( 3 . 9 ( ~ < h & x € f ' ( r ~ )We ) } shall . as before now - I , # flif h < I I'l. As before, we let code I , asfo(ZA). Then (1,O) - Alrl. The details of the construction are as follows. Let as beforefi(y) = ( i , y )(see Definition 6) and letH(S) = { u I ( 3 u ) ( ( v + 3 , ~ ) ~ S & ( O , ~ ) E S ) } , a stheproofofLemma2. in LetA(S) be defined as follows:

INDUCTIVE DEFINITIONS AND THEIR CLOSURE ORDINALS

217

Then A is an i.d. and A E 6. Moreover, we can easily prove by induction A. i. rh= H ( A ~ ) ii. F(H(A~)) c rh+l iii. rX+l - r h # @ * F(H(A~))- H ( A ~ # >0 iv. rh= ~ Y ' ( A ~ + ' ) v. = f F 1 ( A h )if h is alimit ordinal. Hence / A / =/ P I + 1, since lrl I IAl a n d ( l , O ) E A l r ' + ' -Air'. Moreover, by iv. and v. above we have r 5; 5, A. This completes the proof of Theorem 5.

rh

We shall now prove Theorem 6. We need one more definition.

Definition 23. An i.d. r is single-valued iff x E r(S)andy E r(S)implies x = y . The set of single-valued XA i.d.'s is denoted by '"X; and '"IT: denote the set of single-valued IT: i.d.'s.

Theorem 6 is going to be an immediate consequence of Theorem 5 and the following lemma.

Lemma 3. Let E = X,!,or C = n,!,for some n 2 1, and suppose Prewellordering (C). Let r be an i.d. in C . Then there exist i.d.3 i; in and in (?,such that f' is single-valued and i f r(S) -S # $ then $9# p(S) = f(S) C r(S)- S.Moreover, i f F(S) - S = @, then l?@) = w and f'(S) = $.

e

Proof. Let r'(S)= r(S)- S . Then I" E C . Consider the set A = { ( x ,S ) I x E r ' ( S ) } . By Prewellordering ( C ) we have that there exists C-norm 6 of A . Then there are relations I in C and in 6 such that if T c w a n d y E w , and i f y E r"(T), then ior all x E w and all S C_ w we have

2

Then we define

f as follows.

Then fi E C . Given S,and suppose r'(S)# 6. Let hs be the least ordinal h such that h = @((x,S)) and x E r'(S)for somex. Let xs be the least x E w

218

S. AANDERAA

such that A, = @((x,S)). Then r(S)= {x,}.

-

If I?'@) = 0 then r(S)= 6. Hence

r is single-valued. We now define r as follows x € i.(S)

(Vy) ( y E F(S) * y = x)

Then r(S) = F(S) if F(S) # 0 and f(S) = o if f'(S) = follows immediately.

0. Lemma 3 now

Theorem 6 follows from Theorem 5 , Theorem 1 and Lemma 3 , by choosing r € C such that lrl=I C I. Then we obtain a A € C such that IAl= Irl+l= I el t 1 by Lemma 3 and Theorem 5. This proves I CI < ICI, and the proof of Theorem 6 is complete.

Remark 9. We may obtain a slightly stronger result than Theorem 6 by using a weaker assumption than Prewellordering (C). It is enough to assume that there exists a norm on A and a relation R S o X w X P(o) such that if ( y , S ) € A (i.e.,y E r ' ( A ) ) then we have for all x E o

In this case we can define

Then r(S)- S #

i. as follows

fI * fI # e(S) - S C r(S).

We shall now show that Corollaries 1-4 follow from Theorem 6. We have that Prewellordering ( H i ) and Prewellordering ( E i ) (see Moschovakis 1970, p. 3 3 ) . Hence In: I < I Xi I and I I $ l < Inil by Theorem 6. This proves Corollaries 1 and 2. According to Moschovakis 1970, p. 3 3 , the arguments of Addison 1959a, suffice to show that if' every real number is constructible in the sense of Godel, then for each k > 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 < I , all a = (a,, ..., ak-1) with each aj < y,all fl, u , 7,,$,and { < y, and all A E ORD: (0) (( 0, k 1, n )>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 < p 3 7 > 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,
ORDINAL RECURSION AND INDUCTIVE DEFINITIONS

22s

Proposition 3.5. For any a < w there is an Esuch that for all y, a < y, and f :{el,(a,f> = {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 , where in general g 1; (u) = 0 , ifu>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 ) <, I,>, >,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,<,, - f ( a )= 0 v a = y ) . Our next goal is to define a p.0.r. T-predicate for ordinal recursion. First, we need to study p.0.r. inductive definitions (such as [f]).If r is p.o.r., 1, i f a E r c , to be p.0.r. we want the function F , defined by F(<,a)= 0, i f a e r c ,

Lemma3.12.IfGisp.o.r.andFisdefi:nedbyF(<,a,p,~,f)SUP( E , h o e F ( o ,a , p , Y,f))for lim (8or = 0, and F(E+1, a , p , 7 ,f) = G(ff,p,Y,(XP.F(<,P,p,Y,f)) r;;,f),thenFisaZsop.o.r.

<

Proof. F can be defined by recursion on the lexicographic ordering of ORD X p .

ORDINAL RECURSION AND INDUCTIVE DEFINITIONS

2 27

Definition 3.13. For f =(fo, ...,J - - ~ ) ,Tz(g,y,~ , f iff ) T E nt[f]

-

Proposition 3.14. T‘ is p.0.r. for all 1.

n Proof. 0Y [f]is an inductive definition over p = s ~ p ~ < ~ ( y , . . . aY[f] , y ) , is seen to be p.0.r. by inspection of Definition 3.1. It follows from Lemma 3.12 that T is p.0.r. Proposition 3.1 5. (a) For all recursively regular y, all a, p < 7,all Q < a,and all y-recursivef, {a),(a,f) -0 iffYt,o
u

Proof. We can show by inspection of Definition 3.1 that u Q i I f ] is closed under 52, If], and therefore equals E[f] ; (a) anzla?f;?(b) follows from this. If y is not regular inf then we have a functional {a}, and a , p with s~p,<~(a},(u, a , f )=y; it is then not difficult to construct a functional ( c } , w i t h ( ~ , a , @ , O ) E ~ ~-[ 52;. f]

Corollary 3.16. RR is p . 0.r. n r2h-r Proof. Let h(y) = supn (y, ...,y). h is p.0.r. and RR(y) iff “7
4. Recursive analogues of large cardinals The class of recursively regular ordinals is intended as a recursive analogue of the class of regular cardinals; w1 obviously corresponds to N,.

Definition 4.1. (a) w, is the a’th ordinal which is recursively regular or a limit of recursively regular ordinals; (b) a+=leastp [B>a A RR(p)], the “next” recursively regular. Notice that w o = w , (a,)+ =

and for limit ordinals a supp<,wp= w,.

Definition 4.2. (a) a is recursively inaccessible (RI (a)) iff RR ( a ) and a is a limit of recursively regulars;

228

D. CENZER

(b) a is recursively Mahlo (RM (a))iff RR(a) and every normal weakly arecursive function f from a to a has a recursively regular futed point ( f is normal iff strictly increasing and continuous at limit ordinals). The following is easily verified (part (c) using Propositions 3.15 and 3.16.)

Proposition 4.3. (a) RI (a) i f f RR (a) A a, = a; (b) Rh4 (a)+ RI (a): (c) RI and RM are p.0.r. We could defined properties like recursively hyper-Mahlo and hyper inaccessible, but our interest here is in stronger notions of recursive largeness.

Definition 4.4. (a) a is absolutely projectible t o p iff there is a functionfarecursive in parameters 6 < p mapping a I- 1 into 0; (b) a is projectible to /3 iff there is a weakly a-recursive functionf mapping a 1-1 intoo; (c) a is non-absolutely projectible (NAP(&)) iff RR (a) and there is n o /3 < a such that a is absolutely projectible to u p ; (d) a is non-projectible (NP(a)) iff RR(a) and there is no 0< a such that a is projectible to up. The following concept turns out to be very useful in the study of recursively large ordinals and particularly relevant to the two notions of projectibility.

Definition 4.5. For all ordinals a and all Q and < a: (a> p is a-recursive in u iff there is a a-recursive function F such that /3 = F ( a ) ; (b) p is a-recursive iff there is an index a such that {a}, = p. We derive the following equivalences for projectibility .

Proposition 4.6. For all recursively regular ordinals a and all /3 < a,P closed under >; (a) a is absolutely projectible to iff V r
ORDINAL RECURSION AND INDUCTIVE DEFINITIONS

229

Proof. We give the proof for (a); (b) is similar. (-+)Suppose we have t< and F such that A T , F ( T , ~is) a projection of a top. Given ~ < alet , u = F ( r , k ) ;t h e n r = l e a s t f < a . F ( { , t ) = u , andT isa-recursivein(u,t) (+) Suppose V r < a 3u
Definition 4.8. (a) a is y-stable in f iff a is closed under all functions yrecursive in f. (b) a is stable in f iff a is -stable i n k (c) a is stable (S(a))iff a is stable in $9. We need the following lemma and proposition to prove that stable ordinals exist. See Cenzer [7], pp. 47-49, for a proof of Lemma 4.9.

Lemma 4.9. For any ordinal y, any D E y, and any f from D to D such that D is closed under all functions y-recursive in f, i f IT is the collapsing function mapping D 1-1 onto an ordinal in an order-preservingfashion, then for any a,p E D and any y-recursive functional F , F ( a ,f ) --p implies F(n(a),f ) = @). Proposition 4.10. For any ordinal y, any a
D. CENZER

2308

(b) For any f from w to w , there is a countable 6 stable inJ We can use a different technique to find ordinals stable in functions from ORD to ORD. Proposition 4.12. For any f : ORD -+ ORD and any a, there is an ordinal (Y such that 0 is stable in f:

0>

Proof. Let Do = ( F ( 5 ,f ) : 5< a and F ism-recursive) and uo = SupD,. For all n < w, let D,, = (F(6,f) : [ < u, and F ism-recursive and a,+1 = SUPD,,~. is stable in f . Then 0 = Sup,<,u, Definition 4.13. (a) 6 2 = the a’th ordinal stable in x A ; (b) 6 * = least 0 [p > 6 A S@)] ; ( c ) 6 , = 6,.0 We point out that F is the least non-00-recursive ordinal and for all n , is the least ordinal which is not w-recursive in 6,. Proposition 4.14. (a) For any 0and any y > 0,if0 is y-stable, then 0is recursively Mahlo ; (b) 0 is non-projectible iff 0 is a limit of ordinals which are 0-stable; (c) the relation RS, defined by RS(a,P) i f f a is 0-stable, isp.0.r.; (d) S is not m-recursive.

Proof. (a).Given a normal weakly 0-recursive function f , let Sup (a,Xu .f ( u ) ) , for lim (a); g(a)= f ( a ) , otherwise. Theng is y-recursive, g(a) - f ( a ) for all 0 <0, andg(0) =p. But then a. = least a < y [RR (a) A g(a) = a ] < 0 by stability, so that f has a recursively regular fixed point. (b) If 0 is non-projectible, then for any a <0,a, = Sup { F ( k ) , 5 <_ (Y A F 0-recursive} is 0-stable and a < ool;if 0 is projectible to a < 0,then by Proposition 4.6, 37 < 0 {F(LJ,y) : a < a } = p . It is clear that there can be no 0-stable ordinal greater than max(a,y). (c) This follows directly from Proposition 3.15. (d) IfS were m-recursive, then 6 , = least 6 .5(6)would be w-recursive, contradicting its stability.

i

ORDINAL RECURSION AND INDUCTIVE DEFINITIONS

231

Corollary 4.15. (a) NP(a) implies Rh4 (a); (b) there are non-projectibles less than 6,. Given any p.0.r. relation R on ORD, p = least a . R(a) is -recursive, further, p = least a < p+ R(a),and so p+-recursive. For example, w1 = least a < w 2 [RR(a) A a # 0 1 . We have the foilowing.

.

but

Proposition 4.16. If a is any of the following: wl, the least recursively inaccessible, the least recursively Mahlo, the least non-absolutely projectible, the least non-projectible, then a is a+-recursive. Definition 4.17. (a) a(*) = a ; = (a("))+; (b) y, = least a. a not a(n)-recursive. Since for any stable ordinal 6 , 6 is not dn)-recursive,it is clear that the y, exist and are all less than 6 .

,

Proposition 4.18. For all n > 0 , (a) RR(yn) and RI (7,); (b) VT < 7, r is y,-recursive; (c) VU, yn i u < y r ) , u is not yF)-recursive.

.

Proof. We give the proof for n = 1. (a) If 7 R R ( y 1 ) , then y1 = sup,&'(u, 61,for Some 0, 6 < 71 and 71recursive F ; but then 0, t a r e y recursive ( 0 is p+-recursive and P' < y t ) and li so is F. This implies that y1 is y,-recursive, a contradiction. If -7RI (yl), then y1 = 0+,for some 0 < y l ; but then y1 = least y < y; [RR(y) A P < 71. (b) T < y1 implies r r+-recursive, but since RI (yl), r+ < yl. (c) If u is 7;-recursive and y1 5 u < yt , then y1 = least y < y t . V r
Proposition 4.19. (a) For all n > 0, y, = least y. 7 is $")-stable; (b) yi is the least non-absolutely projectible ordinal.

Corollary 4.20. (a) For all n > 0, RM (7,); (b) the least recursively Mahlo ordinal is less than y l .

D. CENZER

232

The following characterization of y, will be useful.

Proposition 4.21. For all n > 0, yn is the least ordinal y such that Va < [{a}+)(y) J-, 3a < Y {aIa(,>(4 41.

.

Proof. (2) Suppose {a}yt(yl)J.. By 4.18c,

< (PQ

P = least pP<,,; [ T ’ ( ( P ) ~ ( ~, ) 2 (a,(p)l, , (010)) (012 is less than yl. (Notice that u T+ iff V $ 5 u [RR($)+ $ 5 T ] is p.0.r.) Let (y = ( P ) ;~

<

anda
(I)Given y < y l , we know that y = {a} + for some a < w. Let 7 {b)o+(U) =least tEe0+. {a}o+= A $ = u ] . Clearly, {b},,+(y)4 and for any a < 7,{b}a+(a)T. The proof for n > 1 is similar. Corollary 4.22. For any n > 0 and any i < w : (a) i f { a } , p ( ~ N ~ i, ) then 3a < Y, {a),(,)(a) (b) if {aIa(n)(a)N i and if {a},p’(yn)

.

I

= i; 4,then {a),lt”)(yn)N i.

In section 6 we make use of the foregoing material to prove that In: I = y l . First it is necessary to study the class of IT: relations with reference to ordinal recursion.

5. H! relations In this section we prove the following two generalizations of the KreiselSacks [I41 result that forA 2 o , A En: iffA is wl-semirecursive.

Proposition 5.1, If Q C w X P(w) is IT;, then there is a p.0.r. functional F with rg(F) E (0, I} such that for all m and A : Q ( m , A ) iff 30 .F(U,m, XA) 1 iff 30 < w;’. F(u,m,x~)1. Proposition 5.2. The relation K , defined by K ( ( a , m , n ) , A )iff {a} A(m)11 n, a 1 is ni. We want to code ordinals into the natural numbers. Let {a}A be the a’th function partial recursive in A .

ORDINAL RECURSION AND INDUCTIVE DEFINITIONS

233

Definition 5.3. (a) W(@)iff @ is the characteristic function of a non-strict well-ordering of a subset of w ; (b) Field (@)= {s : @(s, s) = 1); if W(@),14 I is the ordinal isomorphic to the well-ordering defined by @ and, for s E Field(@), Is I, is the image of s under the map from Field(4) to 141; s 5, t iff Isl, 5 Itl@iff @ ( s , t )= 1 .

Lemma 5.4. (a) Wis nk; (b) there are Z relations M and M’ such that for all Cp and all $ such that W ( $ ) , M ( @ , $ ) i f fW(@)A I@ILI$I,andM’(@,$)iffW ( @ )A I@I
l@l<_IJ/I,a~dL’(@,$)iff~(@) A W($)A l 4 l < l $ l .

hoof. (a) W ( @ )iff Cp is a linear ordering A V8 3 p -+ (O(p+l)<, q p ) ) . (b)M(@,$) iff 38 V m Vn . @ ( ( m , n )=) $(@(m),O(n)));M’similar. (c)L(@,$) iffW(4) A W($) A-M’($,@);L‘ similar. The following lemma is an easy relativization of the standard result; see Shoenfield 1221, p. 184, Cenzer 171, pp. 29-3 I .

Lemma 5.5. (a) For all A E w, {a: W({a}A)}is IIi -A complete; (b) (Boundedness Principle) For any A 5 w and any V C {a: W ( { a } A ) } , there is a u such that for all u in V, L( ( u } ~ ,{ u } ~ ) . We generalize Spector’s results [24] on’n: monotone inductive definitions; see Cenzer 171, pp. 32-34 for proofs.

hopositjon 5.6. I f K is a n: relation such that for all A , FA, defined by F,(B) = {m:K ( m , B , A ) } , isa monotone inductive operator, then: (a) the relation P, defined by P(rn,u,A) i f f m E F$u}A1, is ni; (b) the relation S, defined by S ( m , A ) i f f m E Fw$, is ni. ( c ) f o r e v e r y A ~ w i r,A i s w f ; (d) the relation Q, defined by Q ( m , A ) i f f m E G ,is ni. The following lemma completes the preparation for the proof of Proposition 5.1.

Lemma 5.7. If Q is a IIi relation on w X P(o),then there is a IIy relation K

D. CENZER

234

such that for all A , r,, defined by r,(E) = {m: K(m, B ,A)}, is a monotone inducfive operutor such that for all m, Q(m, A ) iff (m, 1) E c . Proof. Suppose Q ( m , A ) iff V @ 3 p- R ( m , ( p ) , A ) , where for all s,t.R(rn,s,A) A s C t +R(rn,t,A).Let K((m,s),B,A) iff ( r n , s ) € B v R ( r n , s , A )v V p . ( m , s * ( p ) ) E B , a n d l e t FA k d e f i n e d f r o m K as above. I t is easily seen that for all m , s, and A , (m,s) E FA iff V I ) ( s L I ) +3p.R(m,$(p),A),sothatQ(m,A)iff(m, 1 ) E & ( 1 beingthe empty sequence). Proposition 5.1. I f Q 5 w X P ( w ) is H i , then there is a p.0.r. functional F with r g ( F ) (0, 1 ) wch that for all m and A : Q(m,A) $ f 3 0 . ~ ( u , m , x , ) = 1 i f f 3 u < o f . F ( o , m , X A ) = 1.

c

Proof. Let K and r, be defined from Q as in Lemma 5.7. Then Q ( m , A )iff 3 a . m E rs iff 3 0 < wf in E,;'I where the latter equivalence follows fiom Proposition 4 . 6 ~Since . K i s n y , i t is p.0.r. by Proposition 3.1 1 . I f we s e t F ( u . m , X A ) = 1 i f f ( r n , l ) E r ; , thenFisp.0.r. byLemma3.12.

.

To prove Proposition 5.2, we code up the ordinal recursive functions on nf- A fashion.

wf in a

Definition 5 . 8 . N ( u , m , A ) iff W ( { U } ~A) I{u}'l = m .

The relation N is easily seen to be arithmetic.

Proposition 5.10. For all 1 > 0, the relation R', defined by R'(m,$,A) iff nz E K; 141, is n

i.

Proof. Kfi [$I is defined by a ni - $ , A monotone inductive operator f l A [@] which parallels SZwA [+I. From a slight generalization of Proposition 5.6b it follows that Kfi [$I*, which equals R, [$Iw:' is rI; $,A uniformly in @ , A . We give some cases of the definition of a, [$] : (0) GO.k,i.n), ~ 4 ..., ~ uk.. ~ . u ) E QA[+](B)iff ~ ( { u ( ~A ... } ~A )~ ( { z. ,i >~A > A N ( u , n , A ) ; -

ORDINAL RECURSION AND INDUCTIVE DEFINITIONS

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,<, H ( u , @) y m , if @(m,rn) = 1 Let G(m,@)2 if @(m,m)= 0. Then H and G a;e a-recursive and I@I = suprn< w G ( m ,@)< a by regularity. (b) The graphs of G and H above are p.0.r. in a, 4.

.

D. CENZER

236

Corollary 5.13. (a) For all A C w , w.;' is the least ordinal # w recursively regular in A ; (b) for any recursively regular a > w and any A C w , if x A is weakly arecursive, then m.;' a.

<

Proof. Suppose a is recursively regular in A and w < a < 0.;'. Then there is a well-ordering @ w-recursive in A of type a. But 9 must be weakly a-recursive in A , since a > w, and then by 5.12~1I@I < a. (b) Any well-ordering @ recursive in A must be weakly a-recursive, but then by 5.12a. I@I<cu.

6. Hi inductive definitions In this section we prove Theorem 6 . l . ( a ) l n i l ~ y ~ ; (b) for all A 5 w ,A E n; i f f A is y -semirecursive. Definition 6.2. Given an f operator r, we say that F and I , rg C (0, l}, are p.0.r. functionals associated with r if for all rn,A,O,r, (a) m E r ( A ) iff 3 0 . ~ ( orn,, x A ) 1 iff 3 a < w.;' F ( U , rn, x A ) = 1 (as in 5.2) (b) 1(7, m ,p) = Sup (7, A 5 I ( 5 , rn, p)), if lim ( T ) or T = 0; 1(r+l,m,p)~Sup(p,ha.F(o,m,hn .Z(~,n,p))).

-

.

.

We see by inspection that if rpis defined by m E r p ( A )iff 3a < p AF(o,m,XA)= 1, then m E Fi iffI(.r,rn,p) = 1. For sufficiently large p (e.g., p = H I ) , rP= I'. We get a better bound on p by looking at the rr individually. Lemma 6.3. For any ni inductive operator r with associated I , and for all ordinals r: (a), For all rn arid all /3 2 w T ,+(m) I ( m ,T , @ : (b), 0;' Iw , + ~ ; Proof. We break the proof into four parts: (1) (a)o and ( l ~are ) ~trivial.

ORDINAL RECURSION AND INDUCTIVE DEFINITIONS

231

(2) (a), A (b), (a),+, by Definition 6.2. (3) (a), + (b), by Corollary 5.13b. (4) [lim (7)A Vu < 7 (a),] + (a), by Definition 6.2. +

.

Proposition 6.4. For any IIi inductive operator

r, I r I 5 yl.

Proof. For any y, N ( y ) implies by 6.3 that for all m , Xry(m)= I ( y , m , y) and m E 1'7+1 iff 30 < y', F(o,m, Xp ,I ( y , p , y) = 1. Given m E PI+', choosea so that {a},(a)=least u < a [ F ( o , r n , X p ~ ~ ( a , p , aI )ARI(a)] )~ Then {a}y;(yl)$, so by Proposition 4.21, there is an a < y1 such that {a},+(a)$, so that m E E We next define a n : inductive operator A, such that 1 A, 1 = y l . Recall from Proposition 5.2 that the relation K ( ( a , rn,n),A) iff {a}&(m) = n is n,. 1

Proof. The proof is broken up into four parts: (1) and (b)o are trivial. (2) (a), A (b), (a),+, by the definition of A, and the regularity of w , + ~ . (3) (a), (b), for all T 5 y l , as follows: for any T 5 y1 and any $ < o,, there exists some a < w and u < o,such that = {a}o. Choose b so that {b},(rn,n)-Oiff {m}, 5 {n},. Then { ( r n , n ) :(b,m,n,O)EA,} is aprewell-ordering of length a,.Refine this to a well-ordering by choosing a least member of each equivalence class to obtain a well-ordering recursive in A, of A type w,; it follows that o1 2 w , + ~ .On the other hand, it follows from Proposition 3.15 that A, is p.0.r. in w, and therefore weakly w,+,-recursive; then by Corollary 5.13(b), 5 (4) (lim (7)A Vo < T. (a),) -+ (a), is trivial. -+

-+

4

Corollary 6.7. / A , 1 = y,.

Proof. ]A, I 2 y1 by Proposition 6.4. By Proposition 4.18(b), for any 7 < y l ,

D. CENZER

238

there is an m such that {mjYl= wT+l, so that {mj,?. Let b be as in (3) of the proof of Lemma 6.6 above: then ( b , m , m , 0 ) E A:] - A i , so that I A I I ~ Y ~ . = This completes the proof of Theorem 6.1(a). We prove 6.l(b) (that y 1-semirecursive over P ( w ) )as follows: For any inductive operator r, by Lemma 6.3(a), r = { m : 3 0 < y 1 [ I ( o , m , o )=-1 A RI(o)]} and is therefore yl-semirecursive. Any A _C w such that A r is also y1-semirecursive. (2)Suppose that m € A iff {a}Yl(m).l. Choose b so that {b},l(m)= 0 {a}yl(m).Then by Lemma 6.6(a), m E A iff ( b ,m , 0 )E

(c)

-

A,.

7. A functional GT with

= y1

Recall that for any partial functional F , wf is the least ordinal not equal to I @ 1 for any well-ordering @ recursive in F. I t is easily seen that for any F, there is an a such that 0;” = w,, although w, need not be recursively regular, as indicated by the remark following Lemma 7.8. Hinman has defined the functional E f from ww to {0,1).

The following results are proved by Aczel [ 2 ]: #

Theorem 7 . 2 . (a) w y l = 12;L-monI; (b) fi)r all A C w , A E X2;f-monijyA is Ef-semirecursive.

Grilliot first pointed out that, using Ef, one can prove that I Ei-monI = I C l I. Definition 7.3. G r ( f ) = n iff

u(O)} f + ( f ( 1 ) ) = n. (WJ

I n this section we prove the following theorem. #

=A

Theorem 7.4. ( a ) w y l ni I = y, ; (b) tor all A C w ,A E n f iff A is Gy-semirecursive i f l A is y -semirecursive.

,

ORDINAL RECURSION AND INDUCTIVE DEFINITIONS

239

We define recursion in Gf as in the definition of ordinal recursion (3.1); it is possible to consolidate the first seven clauses into five, since we are doing w-recursion. Clause (8) is replaced by the following to define Q [ G a (8)# for any f € o w if , for all s and t , f(s) = t implies ( b ,s,m,t )E A and Gf(f) e n , then ( 6 , k , b ) , m , n ) ER{Gf](A).

Lemma 7.5. (a) Gf is consistent, that is, for any f and g in owand any n < w , ( f C g A ~ f ( f=) n ) Gf
v(O)}

Proof.(a) G i v e n f C g a n d J- '(f(l))=n.we b,)+ g( 1) =f( I), and 04 2 a{. (b) This follows directly from (a).

haveg(0)-f(O),

#

Defmition7.6. (a) {a}'] ( m ) - n i f f ( a , r n , n ) E a [ G f ] ; # (b) feww is Gf-recursive iff there is an a < w such that f = {a}G1. We need the following definition and lemma to prove that wy? 2 y1

Definition 7.7. F,(a, rn) = fl iff {a},(m)

= fl

A

<w.

Lemma 7.8. For any recursively regular a, with w < a < yl, wFa = a. Proof. (5)Since Fa is a-recursive, it follows as in Corollary 5.13(b) that

wp5a.

(2)As in (3) of the proof of Lemma 6.6, we have for any w , < (Y a well-

-recursive and therefore a-recursive. But ordering @ of type c3, which is then for some a , @ = A t . Fa(a,t ) and is therefore recursive in F,. We remark that for non-regular limits a of recursively regulars, if one sets N n iff 3 u < a . {a},(rn) = n , then = QI as in Lemma 7.8.

F,(m)

wp

Proposition 7.9. w f f >_ y l . Proof. Suppose w$+ = 7 < yl. Let e ( a , m, t ) = n iff ( t = 0 A n = a ) v ( t = 1 A n = m ) v 3 c , p ( t = (c, p ) A { c }G#1 ( p ) = n). Then 0 is Gf-recursive and for all a,m, w ~ t ' e ( a ~ m = 7. , rHence ) for all a,rn,

D. CENZER

240

C f ( h t . 0(a,m,t)) -F7+(a,m). But then F,+ is Gf-recursive and by Lemma 7.8, wyp 2,'-I a contradiction. Our next goal is to show that IR[Gf] I 5 yl. We need to be able to comfrom the graph off in order to construct associated functionals F pute and Z analoguous to those of 56. The following lemma is a rather technical application of Proposition 5.12(b); we refer the reader to Cenzer [7] for the proof.

4

Lemma 7.10. The relation Ro, defined by RO(a,@) i f f there is an f E ow such that @ =Xgraph(f) and of=a, isp.0.r. We now define a p.0.r. functional F associated with R [ G f ] . Lemma 7.1 I . There is a p.0.r. functional F with rg(F) 5 (0, 1) such that for all ordinals 0 and all A C w, if(wf)+ <_ p, then for all s, s E R [Gf] ( A ) iff F(P,s,x'4)= 1. Proof. F is defined by cases and the only interesting one is, of course, case (S)', the application of G f . Let 1, if 3 s , t ( p = ( s , t ) ~ O ( ( b , s , m , t ) ) =1; J (b, p, ( m ) , O )= 0, otherwise. Then for a n y b , m , f , a n d A , i f f ( s ) = t i f f ( b , s , m , t ) E A , then h p J ( b , p , ( m ) x, A ) = Xgraph(f). We now define F(P, ((8,k , b ) ,m, n ) ,0) 1 iff O(((S,k,b), m , n ) ,0 ) = 1 V g t , u < p 3 a , t [ R o ( uh, p .J ( b , p , ( m ) , e ) )A 4 < u+ A 0 ( ( b ,O,m,a))= 0((b, I , m , t ) ) =1 A T o ( ( , t , ( a , t , n ) ) JI.fR , ( u, hp . J ( b , p , ( m ) , x A )then ), u 5 of, so taking p 2 (wf)+ is sufficient.

-

We next define a p.0.r. functional 1 associated with R[Gf] Definition 7.12.1(7+1 , m , b) = F ( 0 ,m, hn .Z(T,n,p)); Z(T,m , p) = Sup ( 7 , h u . Z(u, m, p) ) , if lim ( I - ) or I- = 0. Proposition 7.1 3. For all m , n

< w, all limit ordinals a, all r = a + n , and all

B2w,+Zn: (a), m E R ' [ G ~ ] iffI(I-,m,P)= 1;

ORDINAL RECURSION AND INDUCTIVE DEFINITIONS

24 1

Proof. As usual, we break the proof into four parts: (1) (a)o and (b)Oare trivial. (2) (a), A (b), -+ (a),+l by Lemma 7.1 1. (3) (a), -+ (b), follows from Corollary 5.13(b). (4) (lim (7) A Vo < r .(a),) (a), is trivial. j .

Corollary 7.14. (a) For all recursively inaccessible ordinals a, all b, m < q and all f E uwsuch that f ( s ) = u iff (b,s,m, u ) E f2*[G?], f is a-recursive; (b) for all recursively inaccessible a and all m < u, rn E P + l [Gfl i f f I(a+I,m,a+)= 1.

Proof. (a)f(s) =(least p < a!. Z((P)~, ( b ,s , n ~ , ( p ) ~ ) , ( p=) ~1)A

( P I 2 2 U(p),+w)l0 .

[Gf] as above, uf 2 a, so that as in (b) NOWwhenever f comes from Lemma 7.1 I , taking 02 a+ suffices. Proposition 7.15.(a) lCl[Gf]I < y l ; (b) for all f E ow,i f f is Gf-recursive, then f is Yl-recursive; (c) uf? i 71.

Proof. (a) Suppose 9 E Q 7 1 + l [Cf] - WI[ G t ] ; the difficult case is 9 = ( ( 8 , k ,b ) , m , n ) .By Corollary 7.14(b), for all recursively inaccessible a, q E Cl0l"[Gf] iff 3g,u,a,t
alca]

D. CENZER

242

Combining ( c ) with Pioposition 7.9, we have Corollary 7.16.

=yl.

Finally, we obtain the other directions of (a) and (b) of Proposition 7.15. Proposition 7.17. (a) I!2[Cf] I = y1 ; (b) f o r all f E w w ,f i s Gy-recursive iff f is yl-recursive; ( c )fiir all A 2 w . A is GT-semirecursive ijj'A is y I-semirecursive. Proof. (a) Suppose 1Q[Gf] 15 OL < y l ; without loss of generality we assume that OL is recursively inaccessible. Then it follows as in Proposition 7.1 5(c) c# that w I 1 5 1y < y l , contradicting Proposition 7.9. (b) By Corollary 7.16, we have w y p = y l ; as in the proof of Proposition 7.9, it follows that the function F is Gf-recursive. Now for any partial y l r: recursive function f~ w w ,for some a , f ( m ) = {a},,(m) = {a}r:(m) =Fr:(a,rn) for all m. (c) This follows directly from (b). This completes the proof of Theorem 7.4.

8 . Extensions of the class of II; operators Riclitcr [ 191 defincs a method of combining operators as follows. Definition 8.1. (a) [To, r, ] ( A ) =

r O ( A ) ,if r o ( A )# A ;

The fullowing results, due to Richter, are of interest Theorem 8.2. (a) I [KIf,II!] 1 is the least recursively inaccessible ordinal, 1 [ rI:),@, n:] I is the least recursively hyper-inaccessible ordinal, arid similarly .tor I [nl.n0. 0 0 ...,n:] I ; (b) 1 [n1,nl] 0 0 1 is the least recursivelj'Mahlo ordinal, I [ n ~ , H ~ , II isI the ~] least recursivelis hj>per-Mahloordinal, arid similarly f o r I [I$,

..., ny]1.

ORDINAL RECURSION AND INDUCTIVE DEFINITIONS

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

ORDINAL RECURSION AND INDUCTIVE DEFINITIONS

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 <. 'y F ( a , rn, A n . I ( y , n , y ) ) = 0. Hence rY+l=l?Y iff V m < o [ I ( y , m y) , = 0 + 3a < y+ F(o,m, hn .I(y,n, y)) = 11 iff 37
-

This gives half of Theorem 9.4(b). Since, as we pointed out in 88, Xi 2 (nil', we also have the following.

ORDlNAL RECURSION A N D INDUCTIVE DEFINITIONS

24 I

Three rather technical lemmas are needed for the proof of Theorem 9.1. We refer the reader to Cenzer [7] for proofs of these lemmas.

Lemma 9.9. For any non-recursively-inaccessibleordinal CY < y, (a) ifa is a limit of inaccessibles, then a is a'-recursive; (b) if there is a largest recursively inaccessible p < a, then @'-recursive in p. Lemma 9.10. There is a l l i relation K * such that for all a < w , for all a, and all A 5 w such that A is a prewellordering of length a, K *(a,A ) i f f {a}oi+(")&.

r

Definition 9.1 1 . For any operator r, let be defined by r ( A )= { ( m , n ) : m , n E r({ p : ( p ,p ) E A } ) A ( n , n ) $ A } U A .

Lemma 9.12. For any inductive operator r, any ordinal r, (a) F7 = {(rn, n ) : m , n E r7 A (least u rn E ru)I(least o n E r u)} ; (b) 11'1. (c) i fr is X i ( l l i ) , then so is r.

.

Irl=

.

We are now ready to prove Theorem 9.1, that 1 = least u - va < w [ ( V r< u . {al7+(7)&)-+ ta3,+(0)-11(9For any u < I Xi 1, we find an index iisuch that V r < u . { i T } 7 + ( ~ ) . 1 , but {T}u+(u)T. The proof splits into two parts. (a) Let a be recursively inaccessible. We have a X{ operator I? with associated F a n d 1 and an rn E POrt1- roi. It follows from Lemma 9.6 that there is an iff M ( r ) v rn 4 r7+l. index Tisuch that for all r , {if}7+(r).1 (b) It follows from Lemma 9.9 that there is an index a such that for any a! < 1 E! 1, at least one of the following holds: (i) {a}&+(&) 0 and RI(a); (ii) {a}&+(.) c and { c } ~ = + a; (iii) {a}a+(a)= ( P , b ) ,/3 is the largest recursively inaccessible ordinal less than a!, and {b}&+(P) =a. It is important to note that the function h a . {a}&+(.) is 1-1 on the set of non-recursively-inaccessibles less than I Xi 1. If (i) holds, we apply (a) above. If (ii) holds, choose ? so {T}r7+(r) i = least [ < r+[{a}7+(r)# c ] . If (iii) holds, apply (a) t o p to find 6 such that V r < p a {6}7+(r)& but {z}p+(p)T; then let 0, if { d j 7 + ( r )< w v ( { U } ~ + ( T ) ) f ~ b; GI,+(7) {6}7+(( {a},+(r))o), otherwise.

1

-7

--

24 8

D. CENZER

(2)Let u = I X;I and suppose that for some a, {a}7+(r).1for all r < u but {a}u+(u)T.Let r be a Xi operator such that lrl=u, and define r, as follows,

usingLemmas9.10and 9.12: m E r l ( A ) i f f m E r ( A ) v ( m = O A-K*(a,A)). rl is Xi and 0 E r'"' - r'y, so that lrlI > u = I X I, a'contradiction. (Notice that since ?' contains only pairs, 0 $ ?.) This completes the proof of Theorem 9.1; we have as a corollary to the proof the following.

Proposition 9.13. I Ei I is recursively inaccessible. Proof. If not, then part (b) (ii-iii) of (I)will provide a contradiction to the conclusion in (2). We now complete the proof of Theorem 9.4.

Proposition 9.14. (a) For any u < 1 Ei 1, i f u is u+-recursive, then u~~pectrum(~i); ( b ) f o r a n y u < Ill:I,uESpectrum(lli). Proof. (a) Given u 1 {c>,+, choose b so that { b } 7 + ( =least ~) {
-

For the proof of Theorem 9.2 we need the following lemma.

Lemma 9.15. There is a relation K such that for all a < w , all a, and all A f w such that A is a prewellordering of length a , K(t,A ) iff To(&,a, t). Proof. Since T is p.o.r., this is an easy consequence of Lemma 9.10. -

We now prove Theorem 9.2, that Xi = I I-semirecursive. Let u = I X f 1. (g)Given a Xi operator r, we have by Lemma 9.6 and Theorem 9.1,

ORDINAL RECURSION AND INDUCTIVE DEFINITIONS

249

m E F i f f 3 a < u ( R I ( c u ) A I(cu,m,a)== 1). (2)Let r be a universal operator and let Fl be defined by rl(A) = A u ((0,s): s E r ( { p : (0, p ) a })} u {( 1, t ) : K(t, { p : (0, p ) € A } ) } . We have 1 I'l 1 = u and for all a , m, and n < w , {a},(m)N n iff ( 1, (a,m, n))E

q.

10. Za relations, stable ordinals, and H l-recursion In 94 we defined, for A 2 ORD, the ordinal 6f and showed in Corollary 4.1 1 that forA C W , ~< ? K, and Hl is stable in A . It follows that, for any A and B in P(w), B is m-semirecursive in A iff B is H l-semirecursive in A iff B is 6f-semirecursive in A . Recall that 6:-A is the least ordinal not isomorphic to a well-ordering A: in A . In this section we present the following results, parallel to those in 95 regarding ni and uf. Proposition 10.1.If Q 5 w X P(w) is Xk, then there is a p.0.r. functional F with rg ( F ) C { 0,1} such that for all m and A : Q(m, A ) i f f 3a .F(CY, m,XA) N 1 i f f 3a!< Hi.F ( a , m),'X N 1 iff

3a<61A .F(cu,m,xA)% 1.

Proposition 10.2. The relation K,, defined by K2((a,m,n),A ) i f f

id,p(m,XA)'n,

is xi.

Proposition 10.3. Forall A

C w , Sf

= 6;-A.

The proof of Proposition 10.1 is basically an adaptation of Shoenfield's [21] Absoluteness Theorem to ordinal recursion theory with a set parameter; the proof follows:

Proof of Proposition 10.1 : The second and third equivalences follow by stability; we prove the first. Suppose we have Q(m, A ) iff 34 V $ 3 p .R(m,$(p), $ ( p ) , A ) , where R is recursive and p
3

a]

D. CENZER

250

.

30 3g(g: o + u ) Vp C(m,&), xA) = 1, where G is p.0.r. with rg (G) 2 { 0, I}. The direction (-+) of the last equivalence involves lettingg code up a y e n @ and f and taking u = ( 0 , ~ for) a given T. Let h ( u ) = Sup (a, An so that for any g: w u and any p , &) < h(u). As in Lemma 5.7 there is a p.0.r. relation K such that for all ordinals u and all A C w , ro,A, defined by 7 E ro,A(q iffK(.r, u , A , ~satisfies , ( m ,I ) E iff Vg(g: w - + u ) 3 p .G(m,g7p),xA)-0.ThenQ(m,A)iff 3 u ( m, I ) @ iff 3 u 3 T [ r;,L1= r;,AA ( m ,I ) ri4 1. (a)i+l= r ( 4 A (m,1) ; if r(ol),/l (cu),P Let F(a, m , xA) = 0, otherwise.

.(m))

-+

.

rA

rA

r-t;Ap

F i s easily seen to be p.0.r. and Q(rn,A) iff 3a. F(a,m,xA) = I. This completes the proof of Proposition 10.1; next we prove Proposition 10.2. ProofofProposition 10.2: Since Sf is stable in A , K2((a,m,n ) ,A ) iff {a}H,(m,XA)=niff 3 8 .{a},(m,xA)=n iff 3B3u,u(N(u,m,B) A N ( u , n , b ) A ( a , u , u ) E K ; [ x A ] ) ,whereNandKB are taken from Definitions 5.8 and 5.9. It is clear from Proposition 5.10 that this is a L; relation. The proof of Proposition 10.3 depends heavily on the Novikoff, Kondo [ 131, Addison Uniformization Theorem: (See Shoenfield [21], p. 188.) Theorem 10.4. For any nfrelation P there is a Il relation Q such that for all m ,4, $, and A : (a) Q(m,$,$ , A ) -+P(m,6, $,A);

(b)3$.P(m,4,$,A)tt3!6.Q(m,4,$,A). Corollary 10.5. (a) I f Q is II;, then for all m, $, and A : 3$.2(m,$, $ , A ) i f f 3 $ E A ; - A .Q(m,$, $;A); (b) If Q is Xi, then for all m,$,and A : 3$.Q(m,O,$,A)iff34€ A i - A . Q ( m , ( a , $ , A ) .

Proof of Proposition 10.3, that Sf = 6; - A for all A E w : We show that S!, - A is stable in A . Suppose {a}-(a, xA)$ for same a and s o m e a < 6 i - A ; c h o o s e Q , E A i - A such that 1 ~ 1 = a . T h e n 3 B 3 u , u ( l ~ l ~ = I { u > ~ I A ( a , u , u )E K h [ x A ] ) applying ; Corollary 10.5 there is a B which is

(I>

ORDINAL RECURSION AND INDUCTIVE DEFINITIONS

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

ORDINAL RECURSION AND INDUCTIVE DEFINITIONS

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

ORDINAL RECURSION AND INDUCTIVE DEFINITIONS

Proposition 11.17, pl is the least ordinal such that for all a < w , {a}p*(P,S)J. 3a
.id,*(a,s>.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-n}=B,. We need the following rather technical lemma.

D. CENZER

256

Lemma 12.6. There is an index s such that for all ordinals 0,ifall a < SP are Sp-recursivein S (for example, any 05 pl) or 0= 0, then for all a < cs)s&, X B P ) =S ( 4 B

Proof. As in the proof of Lemma 6.6, there is a well-ordering { d } P (d for short) of length 6 P such that for b in the field of d , {b}, ( S ) N Ib I d . Choose P c so that for all a and 7,{c},(a,S) = 0 iff {a},(S) is stable. Recall from Proposition 5.12 the relation W , such that Wo(u,@) iff W(@)A I@1 = u. Choose 1, if wo(a,d) or 3a [ ( c ,a , 0 ) E Bp A s s o that { ~ } s p + ~ ( " ' x ~ p ) ~ w,((Y,d { b : d ( b , a ) = 1 A a.itb)l; 0, otherwise.

ro

Proposition 12.7. There isa Xi relation K* such that K * ( ( a , m , n ) ,B P )iff {a),,+,(m,S)-n, for alla,m,n, ando.

Proof. Begin with K 2 from Proposition 10.2 and apply Lemma 12.6 and the recursion theorem (Proposition 3 , 5 ) . Definition 12.8. T(A) = A U {s: K*(s,A)}. The following proposition is easily verified and completes the proof of Theorem 12.l(a) for n = l .

Proposition 12.9, (a) For all a
= Sa+l;

~

=pl-semirecursive: It is now easy to see that We have { ( a , m , n ): {a},,(m,S)-n} by Proposition 12.9. (2)For any Xi operator r with associated1, by Lemma 12.5, m E r i f f 3 ( ~ < 0 1 [ I S ( a )A 1(&,112,(Y)"- 11. The proof of Theorem 12.1 for n > 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 ,

ORDINAL RECURSION AND INDUCTIVE DEFINITIONS

257

{(u,u) : L2(u,u,A)}isa well-ordenngof type :6 Let ri+l(A) = {<(i,(u,uN:L2(u,u,A)}U A .

We can generalize Lemma 12.6.

Lemma 12.11. For all n > 0, there is an index s, such that for all p and all limit ordinals r N _ s(a).

As in Proposition 12.7 each A: is Xi, so T, check that for all n > 1, lTnl = p n . Parallel to Propositions 8.3 and 8.4, we have:

It is not difficult to

Proposition 12.13. (a) I [Xi,...,Xi] I
i

t]

The results regarding

inductive definitions are parallel to those in 59 on

E inductive definitions. Since all proofs are basically adaptations of the

techniques of 3 9, we omit them. An interesting open problem in this area is whether or not Ill;-mon I = In; 1. The fact t h a t n i - m o n =ni,whereas lli is muchlarger suggests that In;-monI
13. Ordinal recursion and the constructible hierarchy Our definition of wecursive is equivalent to the Kripke 151 - Platek [ 171 definition of Z1-definable (without parameters) over L a . In this section we explore the relationship between ordinal recursion and constructibility.

D. CENZER

258

Let 4, and \Ir denote formulas of the language of ZF (Zermelo-Fraenkel set theory - see Shoenfield [21] for details). We follow Levy in classifying formulas as A o, C , and so forth. Proposition 13.1. For all recursively regular ordinals a > w (or a!=w),all k and n > 0, and all R E a k ,R is a -C, iff R is C, definable over L , (without parameters).

Proof. (+) By induction on the class of p.0.r. functionals we see that all p.0.r. relations are A -definable over L,; the result follows by adjoining quantifiers on each side. (+) We first prove a lemma; let {F(o) : a€ORD} be Godel’s [9] enumeration of the constructible sets.

Lemma 13.2. The relations E, defined by E(o, T) i f f F(a) E F(T),and C, defined by C(CJ,T) i f f r = F(o),are p.0.r. Proof. E and Care defined by an induction similar to that which defines F. The result follows from this lemma by induction over formulas.

For structures SQ a n d q f o r the language of ZF, we write d < c M iff , , % iff I 4 and for all X, A i s an elementary submodel o f q a n d PQ < formulas 4, and all a E I 4 1, PQ 4, [a] implies d: /= @[a].

+

c?i3

Proposition 13.3. For all a!, 0, and n , (a) a! i s n-0-stable iff La< Lo; (b) S n ( 4 i f f La< L.

,,

(See Levy [ 161 for a definition of satisfaction for C, formulas in proper classes like L . ) (a) This follows from Proposition 13.1. (b) This follows from Theorem 1 1.7 (a) and Proposition 13.1. Combining Proposition 13.3 with Theorems 6.1 and 12.1, we obtain new characterizations for In; I and /Xi[.

ORDINAL RECURSION AND INDUCTIVE DEFINITIONS

259

Proposition 13.4. (a) In: I is the least a such thet La<,, La+; (b) IEiI is the least a such that La<==La,. There is no comparable characterization of I C i I or init,since for any a, La+ iff L, <*, L i + ,which is true for any regular ordinal a and La* iff La<,, La*, which is true for any stable ordinal a. There i s a characterization for each of the four closure.ordinals in terms of certain reflection principles. Definition 13.5. (a) (a!,a+)is X l ( l l , ) reflecting iff for any

La+i=@[a]-t3P
Xl(nl)formula @,

(b) (a,a*)is Ez(nz) reflecting iff for any Ez(nz)formula a,

La* l=a[a]+ 3 P < a . L p * +@[PI.

Combining Proposition 13.1 with Propositions 4.21 and 11.17 and Theorems 9.1 and 12.2 we obtain Proposition 13.6. (a) In! I is the least ordinal a such that (a,a') is C, reflecting; (b) 1 Ci I is the least ordinal a such that (a,a') is Ill reflecting; (c) 1 Cil is the least ordinal a such that (a,a*)is Z2 reflecting; (d) l l l ~isl the least ordinal a such that (a,a*) is 112 rejlecting; There is an obvious extension of this result to classes like (n;)"which we leave to the reader. In 5 14 we will obtain similar characterizations of ICA - L 1 and Ill: - L 1 for n > 2. We indicated in 3 11 that 6 s need not be countable; our next goal is to show that in fact it is a rather large constructible cardinal. The following result is very helpful. Proposition 13.7. There is a p.0.r. function FL such that for all ordinals a, L {Xfl .FL(p,a,a) : oEORD} = (hp .FL(P,U,(Y) : ZSZ]. Proof. Let FN (a, a) iff F(o) E aa iff 'do< a .3y < a! . 3<~a[C(7,(0,~))A E(T,u)].Define FL so thatFL@,a,a)=(F(a))(P) i f F ( o ) E a a : least y <(Y . 3 ~U<[ C ( T@,y)) , A E ( T ,a)], if FN(a, a)]; FL(p'a'a!) N- 0, otherwise.

260

D. CENZER

The second equality follows from a collapsing argument using Lemma 4.9.

+

Definition 13.8. (a) LC(a) iff L a is a cardinal; (b) is the a’th infinite constructible cardinal. Proposition 13.9. (a) LC is a - r l l ; (b) LC is m-recursive in S: (c> h a . N; is m-recursive in S.

Proof.(a)LC(a)iffVo--XP.FL(/3,0,a)mapsa (b) and (c) follow from Theorem 11.7 (d).

1-1 into some ~ < a .

Corollary 13.10. For any ordinal a, (b) S 2 ( a )+ a = Ni (a) S2(a) + LC(a);

Now if Y = L , then the least 2-stable is a rather large uncountable cardinal. Andreas Blass has pointed out in conversation that it can be proven by a simple forcing argument that CON(ZFC) -+ CON(ZFC+ 3 countable 2-stable). Since S n n Nf = @ for n > 1, rc4-recursion in Sn is the same as N,L recursion; in 3 14 we obtain results connecting the En+l- L relations over w with n - $-stability. It is interesting to note that although by Proposition 4.12 Sf” exists for each n , the two-place relation Sn(a)is not definable in Z F (being in fact “equivalent” to a satisfaction relation for L ) , so that the existence of an ordinal which is n-stable for all n is independent of ZF.

14. The constructible analytical hierarchy It follows from Proposition 13.1 that for any a and any n , all a - En relations are constructible. On the other hand, it is consistent with ZFC that there be a A$ non-constructible set of natural numbers (see JensenSoJovay [ 101). Therefore it need not be the case that every E$ relation over w be ~0 -En for any n. However, we are able to generalize Proposition 10.6 if we restrict the discussion to constructibly IlA relations.

Definition 14.1. A relation R is constructibly EA (R E EA - L ) iff R can be defined by a En formula with function quantifiers restricted t o L n “ w . Il; - L and other constructible definability classes are similarly defined.

ORDINAL RECURSION AND INDUCTIVE DEFINITIONS

Theorem 14.2. For all n, k and all R

26 1

5 wk,R is Zh+l - L iffR is HlL - En.

Proof. (+) Begin with Proposition 10.1 and replace additional function quantifiers by ordinal quantifiers using Proposition 13.7. (+) This is proven from Proposition 5.10 as was Proposition 10.2. We need some terminology for n - Hfi -stability.

Definition 14.3. (a) Sy(a) iff RSn(a, Nf) iff a is n - Hi stable; stable in A ; (b) 6, - A is the least ordinal n (c) is the a'th n - $t stable ordinal; (d) scn(a)is the least n - Nf-stable greater than a. We generalize Theorem 14.2(-+) by adding a set parameter.

Proposition 14.4. For any CA+2 - L relation Q on o X P(w),there is an Nf - X relation P such that for all rn and A, Q(m,A ) i f f 3 a < Nf,P(a,rn,A)iff 3a<6n+1-A.P(a,rn,A).

,

We obtain further results parallel to those in 5 10 from the following corollary to Addison's [5] uniformization theorem for - L.

IIi

Proposition 14.5. For any n > 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<6,-A .F(m,a,P,xA)- 1.DefiningIby bounding the quantifiers we have as in Lemma 12.5:

Lemma 14.10.(a) Forall m a n d a , X,,(m)----((y,m,62,a,62,(u); (b) foralla,62 -r" L62,a+l; (c) for all a = 6,,,, m E r " + l iff3o<sc2(a) . v T < ( ( u , ~ ) ) * . F ( ~ , u , T , x ~ . I ( o I , ~ , ( Y , ~1) ) " It is now easy to check that for any m, m E I'az+l + m E FUz. (2) As ir. the proof of Theorem 12.1, we can define a - L operator 'Y and such that for all CY <_ u2, 6, -Ta = 6, T" = U o < 6 2 , a { ( a , ~ ,:n{a},(m,S,) ) 2 %n}. Combining Propositions 13.3(a) and 14.8(b), we have

A+l

Proposition 14.11. For all n 2 1, 12: L

L"-%z+l

-

L 1 is the least ordinal a such that

scfl(a)'

We can prove directly an extension of Theorem 12.2. Theorem 14.12.For all n 2 1, - L I is the least ordinal a such that for all a , if V r < a . { a } s c n ( T { ~ , S r )then 4 , {a}sen(")(a,Sy)J. Definition 14.13.For any n >_ 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.

+.

.

ORDINAL RECURSION AND INDUCTIVE DEFINITIONS

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
Example (A): e = Z t (i.e., first-order existential positive operators), S = the set of hereditarily finite members of VA .

7

21 0

R.O. GANDY

Example (B): C = Ah+ (i.e., first-order positive operators), S = the smallest subset of VAsuch that (a ,S,E) is a model for the axioms KPU' of Kripke-PIatek set theory over a set of urelements. (For the description of KpU+ see Barwise's paper in this volume; also 2.4.1. below.) In both these examples some conditions must be placed on % in order to ensure that RHS ( 1) C LHS (1). A sufficient condition is that \u should have only finitely many functions and relations, and be such that C('") contains a pairing function for %.Barwise gives beautiful, short proofs of both these results, and example (B) is his theorem. (These proofs will appear in some lecture notes which Barwise is preparing, see the introduction to Barwise [ 3 ].) Barwise-Candy-Moschovakis had previously proved it when % itself is the structure of an admissible set. I think example (A) should be credited to Grilliot [ I ] ;but the class had been investigated from various points of view in Moschovakis [ 11, Montague [ 11 and Gordon [ 11. What is so fine aboutsBarwise's proof is that one does not need to prove first any intricate results (e.g. the stage comparison and pre-well-ordering theorems) about ccm); but these theorems are easily proved once (1) has been established. In Moschovakis [4] the reader will find statements and proofs of these theorems, valid even when (1) fails, and some discussion of this case. So far as I know the following problems are open for both examples. Problem I . Find minimal conditions on % for (1) to hold. Problem 2. Are there any examples where (1) fails, but ecm) is an interesting class in its own right. (In the simplest failure of (1) for example (B), e(")is merely the class of %-definable relations.) Problem 3. If the answer to problem 2 is 'yes', is it possible to find a restriction of the RHS which makes (1) true? Example (C). Here % is '37 and we identify N with w E V A . For C we take E:+~,II:, E: of non-monotonic operators conone of the classes nf+l, sidered by Aczel and Richter in [ 11. Then S = L, e ,. Of course this does not solve the problem (as they do) of characterising IC 1. It would be good to have a Banvise type proof of their results. As an interim measure we mention

INDUCTIVE DEFINITIONS

27 1

that the structure of L, (>, can be coded in C(”“); this does simplify the AczelRichter proofs.

To sum up: the wearisome feature of so much earlier work on generalised (and even ordinary) recursion theory has been the excessive use of notations and codings. The detads of these were frequently irrelevant to the results; bu without keeping track of the details one could not prove the results. This criticism also holds, I think, for those more elegant versions (e.g. Richter 111 where, rather than some particular system of coding, one deals with a type o system. The method described in this section avoids these Zongeurs. One passes as rapidly as possible to an equation like (1); thereafter one can use the actual objects (ordinals, cumulative sets) instead of their codes. 1.4.5. Invariant definability. For the most studied cases, C(”) has received elegant characterisations in terms of invariant definability. The classic references are Grzegorczyk, Mostowski and Ryll-Nardzewski [ 11, Mostowski [ 11 and Kreisel [ I ] . More recent work is in Kunen [ 11, Moschovakis [ 31, Grilliot [2], Barwise, Gandy and Moschovakis [ 11, and the papers by Barwise and Ville in this volume. Just as the direct methods appeal to those who like to think in terms of constructions (Pascal’s ‘espiritgeomtrique’) so do these to those people who prefer to think in terms of structures for a language or theory (‘espirit analytique’).But, so far as I know, there is no general approach to the problem of characterising a given e(”)in terms of invariant definability. Two particular problems might serve as first steps.

Problem 4. Give an invariant definability characterisation of (Il:+)(-) for any arbitrary structure, or for an admissible set. (Grilliot in [ I ] has shown that for acceptable % , this is the class of relations which are semi-prime-computable in E.) Problem 5. Characterise by means of inductive definitions the classes defined by Grilliot in his paper in this volume. I t is also relevant to seek invariant definability characterisations of the classes C of inductive operators. The only work known to me along this line is Feferman [ I ] . He showed that I::+ is the class of operators which are monotonic both with respect to the relation argument and with respect to

R.O. GANDY

21 2

end-extensions of the structure %. This suggests:

Problem 6. Develop a theory of the connection between invariant definability characterisations of C and those of d").

52. The method of semantic tableaux The results we prove here are not essentially new. The method is an obvious one, but I have not seen it used elsewhere; it is described in the hope that it may prove useful in other contexts. And we take this opportunity of showing how easy it is to work in the system of set theory with a structural collection of urelements (as introduced by Banvise in his paper in this volume). Given a structure (% ,S, E) of the kind described in 1.4.4., we shall use a, b , c to range over A , r , s, t to range over S , and x , y , z , u , u , to range over A US.

2.1. Definition. The class €-Prim (a) (or 3 for short) of €-primitive recursive functions over ,ZI is defined to be the last class 3 of functions over A U VA satisfying the following conditions. (i) If F is a function of gI, and if ; € " A ,

F'x=F?

0

=

otherwise,

then F' E 3. (ii) If R is a relation of \u and if x E n A and R x

R*x=O =

{O} otherwise,

then R* E 3. (iii) The following functions E 3. ( x ; y ) =Df {z : z E x v z =Y}.

ux

=Df

( y : (3u Ex)(y Eu)}.

C X ~ U= VD ~ xif u € u , =Df

y otherwise.

27 3

INDUCTIVE DEFINITIONS

(iv) P i s closed under explicit definition. (v) P i s closed under €-recursion; i.e. if G ,H E 9 and

Fxy = Gx?

if x € A ,

r

= H(F x,y)xy

otherwise

then F E 3 ,(where F r x , p {
A

z = Fzfy}).

Mutatis mutandis all the results proved in Jensen & Karp [ 11 hold also for 3. In particular the characteristic function of any A. relation (without parameters) lies in 3, and the graph of any function in 3 is A1. Also, U E 3 and (v) can be dropped in favour of (v’) which is obtained from (v) by substituting the equations: if x € A ,

Fx? = Gx? = H(U{FuU : u Ex})xy

otherwise.

c

Note. With the definition of given above, identity on A becomes a primitive recursive relation. I d o not know if a satisfactory theory of 3 can be developed which avoids this. 2.2.Notationsand hypotheses. Let 3 =(%,S,€)be such that D ( = A U S ) is closed under 3. Since D is closed under pairing we need only consider inductively defined subclasses of D (we reserve ‘set’ for members of S ) ; X ranges over P(D), and we introduce a 1-place predicate symbol ‘EX’. Let L = Ls,u(x) be the infinitary language for (3, X ) . We use a, b, ...,x, y, _.. ..., r, s, t, to denote variables of L of the appropriate sorts. We suppose that 3, V,M, W, are all primitives of L, so that we may suppose without loss of generality that negation is applied only to atomic formulae. A basic formula is an atomic formula or the negation of one. We suppose that L contains constants for all elements of D.We suppose that a 1:1 coding function g : L+D has been defined, and we identify formulae with their codes. We further suppose that g has been chosen so that all the functions of elementary syntax, and the truth definition for atomic sentences not containingx, belong to 9. (This imposes some limitation on ‘u; it is certainly possible if ‘u has only finitely many functions and relations.) The only realisations of L

214

R.O. GANDY

we consider are (3, X ) , so the truth of a sentence varies only with X . We denote the class of predicates definable by formulae in which V (resp. 3) does not occur by C , (resp. ll,). We use A o , E l , ..., to refer to predicates definable using conjunctions and disjunctions over finite sets. We introduce V, for the closure of El under (finitary) boolean operations. Finally, as in 5 1 , a ‘+’ indicates that ‘EX’ only occurs unnegated.

2.3.1. Definition. We define the semantic tableau for a sentence cp by induction on the level n. (1) At level 0 there is a single point P , the vertex and the sentence at P is cp. (2) L e t P be a point of level n , and let the sentence at P be $. (i) If $ is basic, P is a tip and there are no points below it. (ii) If $ is Ws, or (3x)O(x) then P is disjunctive. (iii) If $ is M s , or (Vx)O(x) then P is conjunctive. (iv) If $ is Ws or M s , then for each O E s there is a point Q, of level n + 1 immediately below P at which the sentence is 0 . (v) If Ic/ is (3x)O(x) or (Vx)O(x), then for each y E D there is a point of level n + 1 immediately below P at which the sentence is O(y). (vi) ‘below’ is the transitive closure of ‘immediately below’. Evidently a semantic tableau is well-founded and every ‘branch’ through it comes to a tip. 2.3.2. Definition. Let an assignment X to 2 be given; we define the grounded points of a tableau T for cp and their ordinals as follows. (i) If P is a tip, it is grounded iff the sentence at P is true, and in t h s case IPI = 0. (ii) If P is a disjunctive point it is grounded iff some point immediately below it is grounded, and in this case

IPI = Min { lQ 1 : Q immediately below P} + 1. (iii) If P is a conjunctive point it is grounded iff all points immediately below it are grounded and in this case

IP I = Supf { I Q I : Q immediately below P) (where Sup+Y = Sup { a + 1 : a E Y } ) .

INDUCTIVE DEFINITIONS

(iv) The tableau T is grounded iff its vertex V is, and then 1 TI

215 =

I Vl.

From this there follows straightforwardly: 2.3.3. Lemma. Under the assignment X for X , cp is true iff the tableau T for cp is grounded.

2.3.4. Now let CP E L+ be a positive inductive operator. The complete tableau T ( x , @ )for x E @("I is defined by modifying the definition of the tableau for @ ( x , X ) . Clause (2) (i) of 2.3.1. is altered to:(i) (a) If $ is basic and does not contain k,then P i s a tip (i) (b) If $ i s y EX, then P is (conventionally) disjunctive; there is just one point immediately below P , and the sentence there is cp(y,k). The definition of grounded for points of the complete tableau for x E CP" is again 2.3.2. Notice that 2 does not occur in the formula at a tip, so that t h s definition does not depend on an assignment for 2. 2.3.5. Lemma. x E a" i f f its complete tableau is grounded. This is readily proved, by lemma 2.3.3., using transfinite induction on

lylQ and PI.

2.3.6. Remarks. (1). I believe the notion of trees with both conjunctive and disjunctive points was first introduced by Beth in [ 11. (2) As Moschovakis first pointed out, and has greatly exploited, the notion of 'grounded' can be given a very intuitive explication in terms of game theory. Players R and v choose in succession a sequence of points on a branch through the complete tableau, starting at the vertex. Suppose P I , ..., P,, have been played and P,,is not a tip; ifP,, is conjunctive, A must play next, otherwise V. In either case the appropriate player must choose for Pn+la point immediately belowP,,. IfP, is a tip then the game is finished; it is a win for V if the sentence there is true, for A if it is false. Then player V has a winning strategy just in case the tableau is grounded. From this it is plain that x E CP" can be expressed using an o-sequence of quantifiers. If we code consecutive turns by the same player into a single turn, then the osequence becomes an alternating sequence of V's and 3's.

21 6

R.O. GANDY

(3) It is almost obvious that 'grounded' has been given a V 1 + inductive definition. This is verified in the proof of the following: 2.3.7. Theorem. 7'here is a J E ( V, +)("I such that for any 4, E L+, any x E D ,

Thus J is universal for @+)("I. The theorem generalises to the infinitary language theorem 6 of Moschovakis [2], and its proof derives from his.

Proof. We code a point P on a complete tableau by a finite sequence u ( P ) = ( u o , . . . , u ~ - ~the ) ;u i are just the sentences ( O , O ( y ) or cp(y)) of 2.3.1. (iv), (v) or 2.3.4. (ib) chosen to lead one from the vertex (coded by ( )) to the point P. Let R p u z be the relation which holds just in case either (i) u is a sentence of the form Ws or f h and z E s, or (ii) u is a sentence of the form y € 2and z is cp(y),or (iii) u is a sentence of the form (3y) O(y) and z is of the form O(y).Then R E 3 by the stipulation of 2.2. To see that this is true in case (iii), observe that either O(y) does not contain the variable y free, or the constant y belongs to the transitive closure of O(y). Using R it is easy to construct a function G E 3 such that G p x u = 1, 2 or 3 if u codes a tip, a conjunctive point or a disjunctive point on T ( x , @ ) ,and = 0 otherwise (this shall include the case that p is not a formula of L+ with at most one free variable). Let F be defined by (1) F p x ( ) = p(x ) if Gpx( ) # 0, F p x u = ( u ) ~ ~if -G~p x u # O and l h u > O , = @ €fl otherwise. Then if u codes a point on T ( x ,p), F p x u is the sentence at that point. Now we give an inductive definition of a class K . (2) G p x u = 1 A F p x u is a true basic sentence not containingx (cp,x,u) E K , = 0 V (p,X, u * z ) E K ) (p,x , u ) E K , # 0 A (p,x,u*z)EK)

-+

C p x u = 2 A (Vz)(Gpx(u * Z )

-+

G p x u = 3 A(2z)(Gpx(u*z)

+(p,x,u)EK, where u * z codes the sequence got by adjoining z at the end of the sequence coded by u .

211

INDUCTIVE DEFINITIONS

-

Note that if G q x u = 0, then ( q x u ) @ K . I t is a straightforward matter to verify that u is a grounded point of T ( x , p). (3) ( p , x , u )E K Evidently wo = ($3 4 @,9, ( ) ) EK a n d w l = ( @ E @ , @ , () ) $ K . Then there are H , ,H , E '? which satisfy (4) H l ( p , x , u ) z = ( p , x , u * z ) if G ~ p x u= 2 and G p x ( u * z ) f 0, = wo otherwise, H 2 ( q , x , u ) z= ( p , x , u * z ) if Gpxu = 3 and G p x ( u * z ) f 0, = w1 otherwise. So we can rewrite (2) in the form: (5) Q(w) v ( V z ) ( H l w z E K ) v ( 3 z ) ( H 2 w z E K ) + w E K where, since the truth function for basic sentences E 3, Q E 3'. But 3'5 A1 ; thus we see that K E (V';>". Finally we set

By (3) and 2.5.3., J satisfies the conditions of the theorem.

QED 2.3.8. Remarks (a) Besides providing a universal set for (I.,+)("), K allows us to compare ordinals using (v l+)(m) relations. More precisely the relations

are both (V;)(-).

Q(w)

For, from 2.3.3. and (5) we have +

l W l = ~

0,

G(W)O(W)l(W)2 =

.

'1 3

sup+ IH, w z I = (;in

IH2wzI+1.

And from these it is a straightforward matter to write down a V1+ simultaneous inductive definition for < K , G K .(For details see Moschovakis [2] .)

27 8

R.O. GANDY

Further, if one of wl, w2 E K , then either w1 < w2 or w2 < w l .Hence there is a partial function (or selection operator) with graph in (V,)@) which is defined if at least one of its two arguments is in K , and which wdl then select one of its arguments which is in K . For a general discussion of the principles involved see Grilliot [ 3 ] . For more refined theorems of the same kind see Moschovakis [4]. (b). The arguments in remark (a) depended on the fact that w E K if it is an initial member, or if all or some of its 'predecessors' Hiwz belong to K . Recently Aczel [ 11 has shown that the introduction of a special inductive class with this property is not essential. If (2 is sufficiently closed, then <@, <@are (?-inductive for any @ E C . And, of course, for the case we are actually discussing, Barwise's method (cf. 1.4.4.) can be applied. (c). The inductive set K has a similar role to 0 in hyperarithmetic theory. The realisation that in general one needs (V:+)@) Of course 0 E (II:+)("). to get a universal set for (Ah+)@) is due to Moschovakis. We discuss the problem of when V y + can be replaced by IIy+ in 52.5 below. 2.4. For our next application of tableaux we need: 2.4.1. Definition. (Barwise). 9= (2i,S,E) isadmissible iff i t satisfies the axioms of (KPU). It is an admissible beyond iff it satisfies the axioms of (KPV).We give an alternative characterisation: 2 is admissible iff (a) it is closed under 3 , and (i) (b) it satisfies the axiom of A,-collection:(VX E s) ( 3 Y 1cp (ii)

+

(3t) (VX E (3 Y E t) cp

where cp is any A, formula. %if it is admissible andA ES.

3 is admissible beyond

It is easily shown that an admissible 3 satisfies the axioms of A , separation and Z,-collection. We write o ( % ) f o r S n On. 2.4.2. Theorem. If

2 is admissible then

IE,I=o(a)

and ( E D + p = ED

.

This theorem makes plain why admissible sets carry a recursion theory

INDUCTIVE DEFINITIONS

27 9

similar to ordinary recursion theory. It is inherent in the original development of admissibility theory by Kripke and Platek, but I believe it was first stated (for C in lectures which I gave in Manchester and UCLA in 1968.

Proof. By a subtableau with vertex V of a complete tableap T = T ( x ,p) we mean a subclass Y of the class of points of T such that: (i) V E Y and all other points of Y lie below V ; (ii) if Q E Y is conjunctive, then all the points of T immediately below Q belong to Y ; (iii) if Q E Y is disjunctive then at least one of the points of T immedialy below Q belongs to Y . A subtableau Y is well-founded if: (iv) the sentence at every tip of Y is true (v) the relation 'below' on Y is well-founded. 2.4.3. Lemma. A subtableau is well-founded i f f all its points are grounded points of T. For 'if use induction on IPI; for only if use induction on 'below'.

Corollary 1. 7he union of a collection of well-foundedsubtableaux with vertex V is itself a well-founded tableau with vertex K For the union will be a class of grounded points of T which satisfies (i)-(iii).

Corollary 2. The sentence at the vertex of a well-founded subtableau is true under the assignment of CP" to X . 2.4.4. Lemma. For EL+ the relation 'Y is a well-founded subtableau with vertex uo of the complete tableau for x E am' is Z in Y , uo, x , cp. We refer, of course, to the coding of points introduced in the proof of theorem 2.3.7. Observe that the relation 'u is below u' is simply u C u ++Df lh(u) < lh(u) A (Vi < Ih(u)) ( ( u ) =~ ( u ) ~ )Using . the functions G and F it is easy to construct a primitive recursive predicate L ( Y ,uo,x,p) which expresses conditions (i) -(iv), for the quantifiers in those conditions

R.O. GANDY

280

are all restricted to Y . And (v) can be expressed by:

( 3 f f )(ff is a function with domain Y and

M ( Y ) -Df range

c On A ( v u,u EY )(u C u

+

H u
This is obviously I;, , and so

gives the required relation. Now let Q, E E D + . The proof of the theorem rests on the crucial.

2.4.5. Lemma. I f uo is a grounded point of the complete tableau for x E Q,= then there exists a well-founded subtableau Y with vertex u0 such that Y is a set (i.e., Y E D or Y is 'D-finite'). The proof is by induction on ( U ~ I , , ~ . If u0 is a tip of T the required subtableau is {uO}. If u0 is a disjunctive point, then by IH we have a well-founded subtableauy for some point immediately below u0 and ( j ; u O )is the required subtableau. If uo is conjunctive, then the sentence at uo is M s , say, and, by IH,

Hence, since D satisfies El -collection, there is a t E D for which

(veE

~( g) Z E t ) N ( p , x , u0 * e, z ) .

Now set

Y=

u,,,

U { ( z ; u O )z: E t A N ( p , x , u o * e , Z ) } .

For any 0 E s Y, =U(zEtAN(~,x,ug*8,z)} is a well-founded subtableau with vertex uo* 0 , by Corollary 1 to 2.4.3. But then Y is evidently a well-founded subtableau with vertex uO. Finally since D satisfies the A 1 -separation axiom and is closed under 3, Y E D as required.

' See footnote on page 299.

'

INDUCTIVE DEFINITIONS

28 1

This completes the proof of the lemma. Finally, to prove the theorem, we claim:

( ) is grounded on T ( x ,@) and the RHS follows by 2.4.4. ConFor if x E am, versely if the RHS holds, then q(x,@)") by corollary 2 of 2.4.3. QED

2.4.6.Corollary.WithD,@asin2.4.2.,ifweput

then \Im= am. For, since the subtableau Y of 2.4.4. belongs to D , so does

2.4.7. Remarks. (1) A corollary of this theorem is lemma 2.5. of Barwise [ 1J , which may be stated thus: the class of derivable sequents of LD,w, where D is admissible is C1/D. A comparison of Banvise's proof with ours shows, I think, the advantages both of working on forms of inductive definition, and of using the complete tableau. We have only 4 cases to consider(tips, 3,W,m) against Barwise's 10. And because the complete tableau contains all the different possible justifications of x E amwe have avoided having to deal with derivations which contain, hereditarily, sets of derivations as subderivations. ( 2 ) P.W. Grant has given (unpublished) a rather neat proof of the theorem for the case (C /D)(") based on the second recursion theorem for D-recursive functions. + (3) Suppose we consider a relativised inductive operator q ( B ) where B is a given subset o f D and atomsz E B occur only positively in &I In )general . lemma 2.4.4. fails: we can only assert that Y belongs to some extension D' of D in which the A&) axiom of collection holds. [This is in contrast to the particular case when D is the hereditarily finite structure overA.1 However, if B E Z D ,then 2.4.4. still holds, and so, if x E (@(B))" b = { y : y E B occurs

282

R.O. GANDY

(a($)"

on the subtableau for x E am}belongs to D. 1.e. if x E then x E (@(b))" for some 'D-finite' b ED. The Barwise compactness theorem is just a specid case of this fact. so, in this case, i f 1 E then is

weakly D-metarecursive in B. This suggests

(+(h))('"),

x

Problem 7. Are there easily characterised subclasses C(B) of EA+(B)such that

(e

= ( X : X is weakly (strongly) D-metarecursive in B}?

2.5. In this section we consider (no+)-. First we note that by using the proof of 2.3.7. it is easy to show: 2.5.1. Theorem. I j D satisfies the stipulations of 2.2., then

Now we prove our main result, which concerns those D all of whose members are countable. 2.5.1. Theorem. Let Y be a function such that for all s ED, { v s n : O
x E @"

-

( ( y , z ): R ' q x y z } is well-founded.

Proof. We use the coding for points on the complete tableau T for x E am used in 2.3. Since x, remain futed throughout the argument, we omit all further mention of them. We use p , q to range over the set Seq of codes for finite sequences from w . For definiteness we suppose 1 is the code for the empty sequence, and we suppose p C q impliesp < 4 . Let p : Seq -+ D ; we establish a mapping up from Seq into T as follows: (14

uo( 1) = ( )

(the vertex of T ) ;

(lb) if u , ( p ) is a conjunctive point of T

UP(P*O) = q P ) * P(P*O) ;

28 3

INDUCTIVE DEFINITIONS

( I c ) if u,(p) is a disjunctive point of T , at which the sentence has the form Ws, and if n > 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
secured)} ,

be the tree of non-past securedpoints of I'.I" is well-founded if every branch is secured at some point. And this is so iff the relation

restricted to

I", is well-founded.

2.5.2. Lemma. I f T is grounded,

I" is well-founded.

Let T be grounded. For a given p consider the sequence

R.O. GANnY

284

Pj+l = Pi * 0 = pi

if u p ( p j )is conjunctive,

* ( p m > 0) ( u p ( p j* m ) is grounded) if u,(pi) is disjunctive,

- Pi

if u,(pi) is a tip.

Since T is grounded, p i is always defined, and u,(po), u p ( p l ) ,... is a sequence of grounded points of T , each one immediately below the one before. But this sequence must terminate at a tip, u p ( p k )say, and then p is secured at p k .

2.5.3. Lemma. I f

r’is well-founded, then T is grounded.

Suppose r’is well-founded; we prove the following by induction up I”:(3) If p ( p ) E r’,there is a 4 < p such that up(4)is a grounded point of T . If p ( p ) is immediately secured, then u p ( p ) is a tip and so is grounded. If not, let the points of I” immediately below p ( p ) be p ( p ) * p ( p ’ * m), where p’ < p , and p ( p ‘ * m ) ranges over D.By IH, for each of these points there is a q < p ’ * m , with u,(4) grounded. If for any of them this 4 is < p , then (3) holds for p ( p ) . If not, u p ( p ’ *m ) is grounded, where p takes the values assigned by p ( ~ )at arguments < p , and can take any value at argument p ‘ * m. By (lb), (lc) or (Id) this means that all the points of T immediately below u p ( p ) are grounded points of T . Hence u p ( p ) is a grounded point of T ; so (3) is proved. But then, since p( 1) E T , up( 1) (= ( )) is a grounded point of T , and so T is grounded.

To prove the theorem we have to show that R (as given by (2)) restricted to r” is primitive recursive in v. Since obviously, R 9 3 , we have to show that ‘E r”is primitive recursive in v. But, given p ( p ) , equations (1) determine u p ( 4 )for q < p by an w-recursion which uses v and the G and F of 2.3. This concludes the proof of 2.5.1. Remark. An alternative proof could be constructed using semantic tableau rules similar to (but simpler than) those used by Lopez-Escobar in [ 11. To see the interest of this theorem, let

INDUCTIVE DEFINITIONS

285

w y = Sup+ {R : R is a $-recursive well-ordering whose field E D } .

We consider only admissible D of the form L, (so we are takingA = @).Let q be the least admissible ordinal such that for some x EL, ,x is not w enumerated by any L,-recursive function. Certainly q is quite a large ordinal. For example 1) is much greater than Po the closure ordinal for ramified analysis.

2.5.4. Theorem. If a is an admissible ordinal less than q then

Proof. LHS >, RHS follows from considering the inductive definition for the initial well-ordered segment of an L,-recursive ordering. For FWS 2 LHS we apply 2.5.1. Since an La-recursive w-enumeration of any s E L , is L, bounded and so belongs to L , , we can find it by search; thus there is an La-recursive v as in 2.5,1. And since there is an La-recursive well-ordering of L,, we can pass from the well-founded relation R’ to a well-ordering relation Q with

1Q12 IR’I.

Further, it is straightforward to verify the following by induction on the ordinals concerned. (1) If p ( p ) and q are related as in (3), then

Thus l Q l 2 Ix (recall that R ’ , Q depend on x,p). This suffices to prove RHS 2 LHS in 2.5.4. It is known that there are admissible ordinals <

o0 for which

R.O. GANDY

286

(see Gostanian [ I ] ; h e calls them bad ordinals). Thus there are certainly ordinals a for which I V + /L,I > III, + /La/. I have found it hard to get much insight into @ID)(-!. Here are two problems which may well be easy, but which I could not solve.

Problem 8. Are there admissible D (preferably with A = @) for which

Problem 9. Is it true that

53. Inductive definitions with type 2 parameters 3.1. The natural way of relativising a class C? of inductive operators overA to a given total or partial function { : A + A is well-known and well understood:

one enlarges C by allowing atoms f x y to occur positively in the defining fomiulae for 2 , where is the graph of {. If C satisfies some simple closure conditions all goes very smoothly, and the theorems about C(") are easily relativised to theorems about C?({). We begin an investigation here of relativisation to a type 2 functional F : A A + A ; in particular we wish to connect this relativisation with the recursion theories of Kleene and Platek. We consider only the case where C? is A:+ (first-order, positive) or A!+ (quantifier free, positive), or Xy+. Further we assume thatA contains a pairing function ( ) with projection functions ( ),,,( and a copy of the natural numbers (so that it is an expansion of an 'acceptable' structure).

3-

3.1 . l .Notations. Capital italics stand for subsets of " A . To avoid a plethora

of qualifying marks we shall not always distinguish between a relation over A and its coding as a subset o f A . ThusR, {(x,y) : xRy}, {(x,y) : x R y } may all be denoted by R . The variables {, 77, range over ( A +p A ) , the set ofpartial functions; f , denote their graphs in the above ambiguous sense. For Y C A , (x, Y ) denotes { ( x , y ) : y E Y } ,and conversely, Y,, the x-th component of Y is { y : ( x , y ) 6 Y } . Letters F,G stand for partial functionals from (A to A . In order to code their graphs we define, for a binary relation R ,

4

INDUCTIVE DEFINITIONS

[R : y ]= ( O , R ) U { ( I , y ) } ,

287

and then set

F = {[f : FC] : { E D o m F } .

F is consistent iff

3.1.2. Let z E P(P(A)), and let 2 be a formal variable for it. Let the class of formulas C be given by initial and closure conditions; we enlarge i t to C(2) as follows: (i) c p c~ c p qi); ~ (ii) if cp E C (Z), {x : cp} is an abstract of C(Z); (iii) if T is an abstract of C (Z), c E 2 is a formula of C (2); (iv) other closure conditions as for C . Now let be included in a language for positive inductive definitions, and consider a progressive operator @ given by @X= {u : cp(x,X, 2)) where cp(x,X, 2) E C (Z), and has no free variables other than x. Aczel (in [ 11) has pointed out that if z is monotonic (i.e. X 5 Y A X E 2. -+ Y E z) then @ is a monotonic operator. Further, for C = AA+ much of the theory of can be lifted up to (e(Z))(-);in particular Aczel's proof of the stage comparison theorems goes through. Aczel makes use of the fact that Z can be treated as a quantifier: (zx)cp holds just in case {x : cp} E 2. We add the remark that provided z is monotonic we may introduce 2-rules into the calculus of sequen ts: 1 : from { r k cp(x), A : x E Y } for some Y E infer r k (Zx) cp(x),A 2 I : from {I-, cp(x) k A : x E Y } for some Y which meets every X E Z infer r,(Zx) ~ ( x I) A. These rules permit cut-elimination. This should make it possible to extend Barwise's method (cf. 1.4.4) to this case. -+

3.1.3. Provided F is consistent, it makes sense to introduce the monotonic set of sets F" defined by:

F" = { Y :( 3 X E F ) ( X C Y ) } .

zag

R.O. GANDY

And then

F{ = y +-+ [f : y ]

(1)

E F O

From now on we always assume that F is consistent. We note that a consequence of the definition of F” is: y E R g e F + + [ A XA : y ] E F o .

(2)

3.2. We use Kleene’s schemata SIbS8 to define.the class %(F) ofA-theoretic functions partial-recursive in F. We take 9( = %, and for simplicity ignore relativisation to a function - that is we drop S.7. We do not insist that F be defined only for total functions, so that S.8. reads:

...,u,, z = (8, (m, 0, 1 ), 2, z

where U = u

-

A t {zl) (L, t , F) E Dom (F)

);

both sides are defined iff

.

For the case considered this is, in effect, Platek’s definition of ‘recursive in F ’ and we shall refer to it as ‘Kleene-Platek’, or ‘K-P’ recursion. If the domain of F consists exactly of the total functions we say that F is clean, and write

F€%:

We write ‘{z} (a)? ’ for ‘{z} (a) is defined’; (Moschovakis introduced this purpose, but a thumbs up sign seems more appropriate). We set D = D ( F ) = {(z,U): { z }(L, F)?} ,

V = V ( F ) = {((z,U),y): {z}(Z,F)=y} The class 92+(F)of sets semi-recursive in F is defined by

%+(F)={ Y :( 3 z ) ( Y = D , ) } . Similarly

%*(F)= { Y : Y , N - Y €9‘(F)}

‘4’for

INDUCTIVE DEFINITIONS

289

(This would be pronounced as 'bi-semi-recursive in F'.) I t is known that not every non-empty set of 'wz+(F)is the range of a total function recursive in F; that is why, following a warning from Kreisel in [ 2 ] ,we avoid using 'r.e. in F' for W(F). Also, in general, 72'(F) # %?(F).

3.3. We now seek connections between the definitions in 3.1. and those in 3.2. 3.3.1. Theorem.For any consistent F over % (i)

72+(F)C(XY+(Fo))(m),

(ii)

(E:+(F"))(") = (A;+(F"))(") .

Remark. We separate out the two parts because (i) can be generalised to other structures, but (ii) depends essentially on the fact that 9 is finitely generated. The proof of (i) consists of writing out formally an inductive definition CP which follows Kleene's definition of D(F) (for (ah+)). An existential quantifier is essential in the clause corresponding to substitution (S.4.). Further details will be found in Grilliot [ 11, which also introduced the 'counting down' technique necessary for the proof of (ii). We illustrate this by a simple example. Suppose an inductive definition for X has just one existential clause:

We replace X by the component X , in the other clauses, and replace the above clause by:

3.3.2. Theorem. If F satisfies the condition (*) below, in particular if F €% , then

290

R.O. GANDY

(*) There is a total function 77 E (Z:(F"))(-) in the domain of F , such thai none of the partial functions 7) { i : i < n } (n = 0, 1 , 2 , ...)are in the domain of F. This theorem is implicit in GriUiot [4]. We illustrate the method of proof by a particular example. Let @En!+( F " ) be associated with the formula ( V t ) p l ( t , x , k , 2 ) , a n d l e t q = ernwithOEZy+(Fo);(ingeneral oneshould consider q = 0,").Consider the inductive operator E Ly+ (F") given by:

*

(whereX3$ denotes {z : (3,(x,z))EX}). We claim that q: =@.".To see that this is so it is sufficient to observe the following facts. (a) For some0 < Is/,@= q . (b) I f x E @ " , then for some?< l * l , ~ ~ , x = q . (c) If n = ( p t ) pl(t,x, Grn,F"),then 7

For the general case, one applies the above trick repeatedly, starting from the inside, so as to get rid of all the V's from p. The 3's can be got rid of by 3.3.2.( ii). QED 3.4. Now we would like to be able to establish a converse to 3.3.1. Under any reasonable definition of 'function recursive in F' however, we would not expect Rge F to be recursive in F. But by 3.1.3.(2) R g e F is definable from F " ! Because we required F" to be monotonic, we have allowed [K : y ] E F" even when the relation R is not functional. But in the proof of 3.3.1. and 3.3.2. we never needed this extension from 6 to Fa.In fact those

291

INDUCTIVE DEFINITIONS

theorems remain true if 6 be substituted for F" throughout. The disadvantage is, of course, that the operators in F are not monotonic, and we cannot apply Aczel's results to them. SOwe make the following definition: 3.4.1. Definition. The operator CP(F")determined by substituting F" for 2 in the formula q ( x , x , 2) is said to be functional (at F) iff

(NF"))" = (@(F))". As the example of V(F) and D(F) shows, it may be obvious from the form of cp that @(F") is functional at all F. We write CP E F n - e ( F o ) to mean that CP E C(F") and is functional at F. We recall that E E% and satisfies E{ = 0 if (3x) ({x = 0), E{ = 1 otherwise. 3.4.2. Theorem.If F E 'X , then (i)

(ii)

(Fn - Ai+(F")>(") = %+(F,E) = (Fn - A;+(F"))(") (Fn

-

= %(F,E) = (Fn-

A;+(F"))('")

A;+(F"))('") .

The equality of the extreme members is given by 3.3.2. For (i), %+(F, E) C RHS is proved by a simple extension of the proof of 3.3.1. To prove LHS (i) Cc>;! +(F,E) we apply the first recursion theorem. We first show that for cp E A;+ (2) there is a functional Jv,with indexj,+,, defined by Sl-S9 such that if Rget C ( 0 ) then (1) (V$(cp(&Dom{,F) +J,+,GC,F,E) = 01, and (2) (V$(JvG {, F , E) = 0 cpG Dom t, F")). We proceed by induction on the construction of p; we only exhibit relevant occurrences of variables and constants. Case (a). cp(i;) is basic, not containingk nor 2. Jv($ = 0 if cp($ and is undefined-otherwise. Then Case (b). is T ( $ EX. Then J , O ={(TO). Case (c). q o is {z : ~(i;,z)} E Then JJG) - J ~ ( u , ( I , F ( h t . 0 1 u ) ( J ~ ( u , ( O , ( t , u )=) )0)))). Suppose {z : $(u,z)} E F. Then +

c p o

Z.

(Vt) ( 3 !u ) J/ (z, (0,( t ,u))) .

29 2

R.O. GANDY

So, assuming (1) holds with $ for cp, the h term in the definition o f J 9 is a total function, 77 say, and $@,( 1, Fq)) is true. But thenJ9(Z) = 0, so (1) holds. Conversely, suppose J,J$ = 0. Then the h term must define a total funcSimilarly tion 7, and so by (2) with $ for cp, to,<> E ( z : $6,~)). ( 1 , F q ) E {z : $(U,z)).Hence {z : $(U,z)}EFo;i.e.(2)holds. Case(d).rpis$l A $ 2 . Then J,(ii) =Jil(ii) *J$&ii). Case (e). cp is v $2. Then J90{h}O'$L,ji2jU,t,F,E) where h is an index such that the RHS is defined with value 0 just in case

Such an index h exists by Gandy's selection operator theorem. (For a proof see Moschovqkis [ 51 .) Now we can Fpply the first recursion theorem (for a proof see Platek [ 11) to obtain a partial function { E 32 (F, E) which is the minimal solution of

But then, by ( l ) , (@(F))-C D o m t , and by (2) D o m t L(@(F0))-.So, if @ is functional at F, (@(F"))"= Dom {, which suffices to prove (i). To prove (ii) we note that we can strengthen the result about h (which shows that *(F, E) is closed under the 'strong or') to get a k (corresponding to 'strong definition by cases') satisfying

This completes the proof of the theorem.

3.5. Discussion. I believe I was the first to realize that %(F,E) (or 32(F) if E E 9?(F)) had a 'nice' recursion theory. Since then Sacks and his colleagues at MIT have carried it to stratospheric levels of soplustication (see, e.g. Sacks [ 13 and MacQueen [ 11). Theorem 3.4.2. represents an attempt to explain why this was possible. It also suggests a different way of developing the theory. Namely one starts from (Fn - A + (I?")(") and then follows the lines sug-

A

INDUCTIVE DEFINITIONS

29 3

gested in 3. l., checking functionality where necessary. The theorem is not easily weakened or generalised. If we drop the requirement F € cK* then we cannot any longer use E to compare ordinals of computations. However, the use of Hinman's E# (E'S = 0 if (3x)({x = 0), = 1 if (Vx)(fx > 0), undefined otherwise) will yield an h as in (c) above, so that (i) holds with E# replacing E. But, as Platek pointed out, a functional K which satisfies Kcgx = 0 if {(x) = 0, = 1 if g ( x ) = 0 and S(x) undefined, is not consistent. So it seems that to restore (ii) in this case one would need to introduce a multi-valued search operator. One would then be operating with relations rather than functions and it would seem more plausible to take the inductive definitions as primitive rather than to base them on an artificial notion of computability or recursiveness cooked up to make (ii) true. Let us consider now extending the theorem to structures other than 9. Theorem 3.3.1. (ii) will fail, so the inductive definitions to consider will be Fn - Ey + (F").As Crilliot has shown in [ 11 the existential quantifier in the inductive definitions calls for a search operator in the recursion theory. One might expect to restore the theorem then, by replacing A: by Ey, and interpreting %(F) as 'search-computable in F'. But t h s means that F must act on many-valued functions - i.e. relations, and the last sentence of the previous paragraph may be re-applie d . Suppose now we take 92 as given, and seek for an appropriate class of inductive definitions. In particular we consider trying to find C so that for

FE3C (1)

(C(F"))(-)

= 92+(F).

Grilliot [4] has shown that %+(F)is not closed under union, so C? cannot be closed under v. At first sight this seems to make the case hopeless, as one needs v to connect separate clauses. However, examination of the clauses in the inductive definitions o f D ( F ) and V ( F ) show that they are deterministic in the sense that if e.g., (z,?) E D then the relevant clause is determined by z . Indeed Kleene has chosen the definition of 'index' so that this shall be so. But, alas, there is still a non-deterministic feature. The clause for S4 (substitution) has the form

(2)

( ~ , v ) ( ( ( z ~ , ; ) , , v ) € V A (z,,U,,V)ED)+(Z,G)ED

where z1 and z 2 are determined by projection functions from z . We cannot eliminate the 3 by counting down, since this uses a non-deterministic V .

R.O. GANDY

294

Compare (1) with

(3i< 2 ) ( ( z ,i,;) E D ) -,( z ,uj E x .

(3)

It seems almost impossible to imagine a criterion which would permit (2) but reject ( 3 ) . On the other hand (3) appears to lead one outside 92 +(F).To be more precise, what Grilliot shows is that there is no mthod uniform in F for presenting the union of two members of 92+(F) as a member of %?+(F).So one poses: Problem 10. Is there an F such that

%?+(F) is not closed under union?

To sum up. For an arbitrary structure the classes (Fn A; + (Fo))(")(Fn - Zy +(I?"))(=) promise to have nice closure properties, and be amenable to a variety of techniques. For !Jl when E is recursive in F E3c , the classes coincide with%?+(F). When E is not recursive in F the techniques cannot be applied and %?+(F) appear to behave very differently. (I think that all that is known is contained in Grilliot [ 6 ] ) .I t is not yet clear whether in this case %?+(F) should be regarded as merely pathological. ~

$4. Discussion A number of important or interesting questions received scant or zero attention in 5 1-3. We indicate some of these by listing further problems. Then we discuss the possibility of applying the theory of inductive definitions to proof theory. Problem 11. Find conditions which ensure that (em)(-)= (C+)("). Spector showed that for case (H) of 1.3. the equation holds for all relevant

e. I had hoped that semantic tableaux would provide a proof for the case

considered there; maybe they can be forced to do this. An entirely different approach would be along the lines suggested by Feferman (cf. 1.4.5.). One considers C and ecm) not just for a, but for a class of models for the theory of 21. For this class one can apply Lyndon's theorem to get Cm = C? +. One then has to work one's way back - via semi-invariant definability - to +I(") for itself.

(e

INDUCTIVE DEFINITIONS

2 95

The next problem is concerned with generalising the boundeness theorems of hyperarithmetic and hyperprojective theory. Let

Problem 12. Under what conditions is C?(<-) = C('")?

Moschovakis [4] shows, for arbitrary %,that the equation holds for C = Ah+. It is obviously not true (in general) if C= Cyt. The problem is important because when the equation holds one has a natural definition for 'finiteness' in &). Problem 13. Give a direct proof that, for countable admissible A , (XI+/A)(") = (s - a;+/A)(").

Here 's - l l i 7is 'strict niyas introduced by Banvise in [2] (i.e., formulae which can be put in the form (VXCA)cpwith cp E Zl). Present proofs use the and then apply the completeness fact that the validity predicate is s theorem for Lwl,w.I t is even possible that the above equation holds for uncountable A ; although then it is known that in general 1s - n;+l is much greater than I El+ I. If the method of semantic tableaux could be extended to s - n: it should provide answers to this and other problems. In 9 1-3 we used languages which contained constants for all members of A . There are many cases in which this is much too crude and will prevent interesting distinctions from being made. (For some examples, see Hinman & Moschovakis [ I ] .) In [ 11 and [2] Moschovakis systematically allowed for the use of parameters only from a given B G A . If one uses the method of embedding (1.4.4.) then one would also restrict the use of parameters in the axioms of KPU. There is not exactly a problem here; just a line to pursue. Another very obvious line is to investigate what happens to the results and problems of 9 3 if one relativises to an 'F' of type higher than 2. One should then consider not only inductively defined subsets o f A ('I-sections'), but inductively defined subsets of pm(A).(See Grilliot [4] and [5] .) There is one part of mathematical logic which is wholly concerned with inductively defined sets; namely that part of proof theory which is concerned

296

R.O. GANDY

with the study of formal proofs. At a first encounter it may look as if no kind of general study of the sorts of inductive definition used will be of much use here: one studies a particular formal system, and evolves methods which may be specific to it. And it could be that a theory of inductive definitions has little or nothing t o contribute. One does not, after all,expect such a theory to tell one anything interesting about a particular inductively defined subset of N such as the prime numbers. But I think recent work does suggest that there are certain patterns or forms of inductive definition waiting to be discovered. The intuitive definitions I have in mind are: (a) of cut-free (infinite) proof trees (see, for examples, Schutte [ I ] ) ; (b) of reducible or computable proofs (ranging from Gentzen’s original consistency proof to e.g., Martin-Lof [ 11); (c) of ordinal notations and their ordering relations. It should be observed that even when the end product of the definition is in fact a recursively enumerable or recursive set, the inductive definitions used are usually nf (not necessarily monotonic - see, for example, Martin-Lofs definition of ‘computable’). What one might hope from a theory of forms is that inductive definitions of the same form would have the same closure ordinal, and would endow it with the same structure (e.g., by exhibiting the same pattern of principal sequences or by defining the same functions on it). The evidence for optimism is as follows. First, it is often easy to spot the irrelevant elements in a given definition, and also to discover which are the clauses which really ‘do the work’. Second, the same ordinals with, at least approximately, the same structure turn up in numerous different contexts (e.g. eO, r,,,F ( E ~ ~ +One ~ ) has ) . the feeling, often, that this is no accident, without quite being able to discern the cause. Third, the method originated by Bachmann [ I ] and since extensively elaborated (for example, in Pfeiffer [ I ] and Isles [ 11) establishes natural-seeming connections between various inductively defined sets of notations and sets (in particular the finite number classes) for which there is an obvious classification. (A simplification of Bachmann’s method, due to Feferman and Aczel, has been explored, described and extended in Bridge [ 11 .) A further, and less arbitrary connection with a known classification is provided by Martin-Lofs conjecture (in [ I ] ) that the provable well-orderings of his formal system for n-fold iterated @+ inductive definitions have precisely those ordinals for which Pfeiffer et al. provided notations using the n + I-th number class. Now the closure ordinal of n-fold iterated KIP+ inductive definitions is precisely the n + I-st admissible ordinal, which is the ‘recursive’ analogue of H,.

INDUCTIVE DEFINITIONS

297

Martin-Lofs system and his normalisation procedure have a naturalness and transparency (as compared, say, with work on ni-analysis) which suggest that it should be possible to find the reason for the connection. Lastly, I quote an example where a ‘form’ of inductive definition can actually be given, rather than merely hinted at. Richter [ 11 introduced the operation [al,G 2 ]; this acts like a1on sets which are not closed under a 1 , and like a2on those which are. The idea can readily be extended to finite sequences: [ a l , a 2 , ...,@.,I. If the aiare ZT+,then the closure ordinal of the resulting @+ operation is < on. Schmidt, in her dissertation [ 11, has shown how cO can be characterised using transfinite sequences of Ey+ operations, labelled by previously introduced notations. The resulting ‘form’ is not ideal, since it does not cover other rIy+ inductive definitions which are known to close off at eO.But it suggests a promising direction for further investigation.

References P. Aczel [ 11, Stage comparison theorems and game playing with inductive definitions (to appear). P. Aczel and W. Richter [ 11, Inductive definitions and analogues of large cardinals, in: Conference in Mathematical Logic, London ’70, Springer Lecture Notes in Maths. 255 (1972) 1-9. H. Bachmann [ 11, Die Normalfunktionen und das Problem der ausgezeichneten Folgen von Ordnungzahlen, Vierteljschr. Naturforsch. Ges. Zurich 95 (1950) 115- 147. K.J. Barwise [ 11, Infinitary Logic and admissible sets, J. Symbolic Logic 34 (1969) 226- 25 2. K.J. Barwise 121, Implicit definability and compactness in infinitary languages, in: The Syntax and Semantics of Infinitary Languages, Springer Lecture Notes in Maths. 7 2 (1968) 1-34. K.J. Barwise [ 31, this volume. K.J. Barwise, R.O. Gandy and Y.N. Moschovakis [ 11, The next admissible set, J . Symbolic Logic 36 (1971) 108-120. E.W. Beth [ 11, Semantic construction of intuitionistic logic, Mededelingen der Kon. Ned. Akad. v. Wet., new series 19 (1950) no. 11. J.E. Bridge [ 11, Some problems in mathematical logic (systems of ordinal functions and ordinal notations) D. Phil thesis, Oxford, 1972. S. Feferman [ 11, Uniform inductive definitions and generalized recursion theory, ASL meeting, Cleveland, Ohio, April 30, 1969. R.O. Gandy [ 11, General recursive functions of finite type and hierarchies of functions, Annales de la Facult6 des Sciences de l’Universit6 de Clermont, Maths. 4 Fascicule (1967) 5-24. C.E. Gordon [ 11, A comparison of abstract computability theories, Ph.D. dissertation, UCLA, 1968.

298

R.O. GANDY

R. Gostanian [ 11, The next admissible ordinal, Ph.D. dissertation, N.Y. University, 1971. T.J. Grilliot [ 11, Inductive definitions and computability, Trans. Amer. Math. Soc. 1 5 8 (1971) 309-317. T.J. Grilliot [ 21, Implicit definability and hyperprojectivity, Scripta Mathematica, to appear. T.J. Grilliot [ 3 ] , Selection functions for recursive functionals, Notre Dame Journal of Formal Logic 1 0 (1969) 225-234. T.J. Grilliot [ 4 j , Recursive Functions of Finite Higher Types, Ph.D. Dissertation, Duke University, 1967. T.J. Grilliot [ 5 ] , Hierarchies based on objects of finite type, J. Symbolic Logic 34 (1969) 177 - 182. T.J. Grilliot [ 6 ] , On effectively discontinuous type-2 objects, J. Symbolic Logic 36 (1971) 245-248. T.J. Grilliot [ 7 ] , Abstract recursion theory: a summary, this volume. A. Grzegorczyk, A. Mostowski and C. Ryll-Nardzewski [ 11, The classical and the w complete arithmetic, J. Symbolic Logic 2 3 (1958) 188-206. D. Isles [ 11, Regular ordinals and normal forms, in: Intuitionism and Proof Theory, A. Kino e t al. (eds) (North-Holland, Amsterdam, 1970) 339-361. R.B. Jensen and C. Karp [ 1 1, Primitive recursive set functions, Proc. Symposium Pure Math. Amer. Math. Soc. 13, Part I ( 1 9 7 1 ) 143-176. P.G. Hinman and Y.N. Moschovakis 111, Computability over the continuum, in: Logic Colloquium '69, R.O. Gandy, C.E.M. Yates (eds.) (North-Holland, Amsterdam, 1971) 77-105. S.C. Kleene [ 11, Recursive functionals and quantifiers of finite types 1, Trans. Amer. Math. SOC.91 (1959) 1-52. G. Kreisel 11, Model theoretic invariants, in: The Theory of Models, J. Addison e t al. (eds.) (North-Holland, Amsterdam, 1965) 190-205. G. Kreisel [ 21, Some reasons for generalising recursion theory, in: Logic Colloquium '69, R.O. Gandy, C.E.M. Yates (eds.) (North-Holland, Amsterdam, 1971) 139-198. K. Kunen [ 11, Implicit definability and infinitary languages, J . Symbolic Logic 33 (1968) 446-451. E.G.K. Lopez-Escobar [ 11 , An interpolation theorem for denumerably long formulas, Fundamenta Math. 5 8 (1965) 254-272. D.B. MacQueen [ 11, Post's problem for recursion in higher types, Ph.D. dissertation, M.I.T. 1972. P. Martin-Lof [ 11, Hauptsatz for the intuitionistic theory of iterated inductive definitions, in: Proc. 2nd Scandinavian Logic Symposium, J.-E. Fenstad (ed.) (North-Holland, Amsterdam, 1971) 179-216. R. Montague [ 11, Recursion theory as a branch of model theory, in: Logic, Methodology and Philosophy of Science 111, B. van Rootselaar, J. Staal (eds.) (North-Holland, Amsterdam, 1968) 63-86. Y.N. Moschovakis [ 11, Abstract first order computability I , Trans. Amer. Math. SOC.138 (1969) 427-464. Y.N. Moschovakis [ 21, Abstract first order computability 11, Trans. Amer. Math. Soc. 138 (1969) 465-504. Y.N. Moschovakis [ 31, Abstract computability and invariant definability, J. Symbolic Logic 34 (1969) 605-633. Y.N. Moschovakis 141, Elementary induction o n abstract structures (North-Holland, Amsterdam, 1973).

INDUCTIVE DEFINITIONS

299

Y.N. Moschovakis [5],Hyperanalytic predicates, Trans. Amer. Math. Soc. 129 (1967) 249-282. A. Mostowski [ 11, Representability of sets in formal systems, Proc. Symposium Pure Math. Amer. Math. Soc. 5 (1962) 29-48. H. Pfeiffer 111, Ausgezeichnete Folgen fur gewisse Abschnitte der zweiten und weiterer Zahlklassen, Dissertation, Hannover, 1964. R. Platek 111, Foundations of Recursion Theory, Ph.D. Dissertation, Stanford University, 1966. W. Richter [ I ] , Recursively Mahlo ordinals and inductive definitions, in: Logic Colloquium '69, R.O. Gandy, C.E.M. Yates (eds.) (North-Holland, Amsterdam, 1971) 27 3 -288. G.E. Sacks [ 11, this volume. D. Schmidt [ 11, Topics in mathematical logic (characterisations of small constructuve ordinals; constructive finite number classes) D. Phil. thesis, Oxford, 197 2. K. Schiitte 111, Beweistheorie (Springer, Berlin, 1960). R.M. Smullyan [ 11, Theory of formal systems (Princeton University Press, 1961). C. Spector [ 11, Inductively defined sets of natural numbers, in: Infinitistic Methods (Pergamon, Oxford, 1961) 97-102.

Note added in proof. Moschovakis has recently made a very considerable extension of the method of embedding. He has shown that ifA C Y is a 'nice' transitive set, and if C is any class of inductive operators overA satisfying certain rather weak closure conditions, then there is an admissible set A* such that

1

0 E s)

Note that the existential quantifer in N should also be exposed, and a bound (for assigned to it.

J.E.Fenstad, P.G.Xinmnn (eds.}, Generalized Recursion Theory @ North-Holland Publ. Comp., 1974

INDUCTIVE DEFINITIONS AND REFLECTING PROPERTIES OF ADMISSIBLE ORDINALS Wayne RICHTER The University of Minnesota

and Peter ACZEL University of Oslo and Manchester University

Contents Introduction

301

Part I. Reflecting properties 0 1. Summary of definitions and results 3 2. Elementary facts 8 3. Ordinal theoretic characterisations 0 4. The relative sizes of the first order reflecting ordinals 8 5. First oIder reflecting ordinals and the indescribable cardinals 0 6. Stability

306 313 3 17 322 329 333

Part 11. Inductive definitions p 7. First order inductive definitions, I p 8. Closed classes of inductive definitions p 9. First order inductive definitions, I1 010. Higher order inductive definitions, I 01 1. Higher order inductive definitions, I1

3 37 341 347 35 1 35 6

Appendix. Proof of the coding lemma

361

References

38 1

Introduction

An operator or inductive definition (i.d.) r : P(w) +P(o) determines a transfinite sequence (I'g : E ON) of subsets of o,where I'h = U {r(rE) : < A}. The closure ordinal I r (of r is the least ordinal h such 301

302

that

W. RICHTER and P. ACZEL =

rA. The set defined by r is r" = rlrl.r is monotone if C Y w . For monotone r we have

r(X)C r(Y ) whenever X

Monotone inductive definitions have long been used in logic and in particular in recursion theory. For example the definitions of the terms, formulas and theorems of predicate logic may be naturally formulated as monotone inductive definitions. More generally Post's production systems give a wide class of monotone inductive definitions, for defining sets of strings of symbols in a finite alphabet. These lead to a natural characterisation of the class of recursively enumerable sets of integers. All these inductive definitions have closure ordinal 5 w . But inductive definitions with larger closure ordinals may also be considered, and they determine notation systems for ordinals in the following way. For each x E r" let lxlr be the least ordinal X such that x E I-.*+'. Then (r",I I 1')is a notation system for the ordinal lrl. Note that because I I I' maps r" onto I rl, I I71 must be a countable ordinal. For example let A be the i.d.:

A(X) = { l} U { Z x : x EX} U { 3 . S e : V n [ e ] ( n €X} ) , where [el is the e'th primitive recursive function, in a standard recursive enumeration of them. Then (Am,I ),.I is a slightly modified version of Kleene's system of notations for the recursive ordinals, i.e. A" is a complete II! set such that I A1 = wl, the first non-recursive ordinal. Note that A is monotone. Certain monotone i.d.'s are basic to Kleene's definition of recursion in higher type objects, [91. Also monotone i.d.'s are extensively investigated in [ 131. As w1 is a constructive analogue of the first uncountable ordinal, it was natural to try to f o r k l a t e constructive analogues for larger initial ordinals by constructing systems of notations for them. This led to the use of non-monotone i.d.'s. (See [ 141 and [ 151.) An independent development led to the Kripke-Platek theory of recursion on admissible ordinals. These ordinals are a constructive analogue of the regular ordinals, the first two admissible ordinals being w and w The main aim of this paper is to formulate constructive analogues for large regular ordinals, and to obtain notation systems for them using non-monotone inductive definitions.

ADMISSIBLE ORDINALS

303

Our results will be concerned with classes C of those i.d.'s that are definable in a certain way. Thus we say that the i.d. r is Jlz if { ( x , X ) E w XP(w) : x E r(X)}is definable by a n ; formula in the language of finite types over arithmetic. Similarly we define the classes of X k and A; i.d.'s. For example the i.d.'s involved in Post's production systems are all Zy when coded on o. The operator A, above, is an example of a lIy monotone i.d. We shall write n",mon for the class of m o n o t o n e n k i.d.'s. Similarly for C h - m o n and A;-mon. Given a class C of i.d.'s we will be interested in I C? I = Sup { I rl : I?€ C} a n d I n d ( e ) = { X C w : X L m I?'= for some E C } .H e r e x < , Y means that X is many-one reducible to Y . In many cases 1 C 1 can be compared with o(32) for a suitably chosen class 72 of relations on w . a(%) is defined to be the sup. of the order types of the well-ordering relations in72 . Thus it is well-known that o1= o ( A Y ) = o ( A j). We next list some of the earlier results on the ordinals of i.d.'s. Proposition. (i) = I C ~= Iw ; (ii) (Spector [201) In(i)-mont= I n ,1- m o n l = wl; (iii) (Candy, unpublished) I @ l = wl; (iv) (Richter, [ 161) In![ is a large admissible ordinal; e.g. much larger than the first recursively Mahlo ordinal; (v) (Putnam, [I411 =4Ai); (vi) (Gandy,unpublished) IEi-rnonI = w(A;).

IAiI

We now summarise our results. 7r; is the least ordinal h such that (LA,€) sentence. u h is defined using X k sentences. For a precise reflects every definition see 5 1 .

In general the characterisations of the closure ordinals of i.d.'s must be more complicated. I f A is a relation on ordinals let n k ( A ) be the least ordinal h such that (L,[A],E,A) reflects e v e r y n k sentence. Similarly for o z ( A ) . Let rk(r)= nk(A,) where A , = {(n,ol) : a E ON & IZ E FLY}.Also let

$(r) = o",(A,).

304

W. RICHTER and P. ACZEL

Theorem C. For m, n > 0 (i) /IlkI = r$(F) where F is complete H k (ii) IE$l = flm(r) where FiscompleteZk The proofs of the above characterisations actually give much more information. In each case, as well as characterising I I we may also characterise Ind (C). In the next result we use the notion of a “closed” class C . Each IlL and X k is a closed class f o r m > 0 and every closed class C has a “-complete” element. See $8 for a definition of this notion.

e

Theorem D. If e is a closed class 2 Ily, r is C-complete and (i) X is admissible relative to A , ; (ii) X is projectible to w relative to A , ; i.e. there is a A-recursive in A , injection f : h -+ w ; (iii) Ind (C) = {XC w : X is X-r.e. relative to Ar}.

X = I C I then

When C? is “sufficiently absolute” then A , 1 X is X-recursive, so that the relativisation to A , may be omitted in the statement of Theorem D. This is the case when C i s l I i + l , X m0 + 2 , 1 1 i or Xi. Results along the lines of Theorems C and D have been recently obtained, independently, by Moschovakis. Moreover he has generalised them to classes of inductive definitions on arbitrary abstract structures. In the next result we locate the ordinals of inductive definitions in relation to the ordinals of certain wellorderings. Part (i) has been independently obtained by Cenzer (see I S ] ) .

Note t h a t m + n > 2 i s e s s e n t i a l i n ( i ) a s I A : ( > _ l I l ~ l > w 1 = w ( A : ) . W h e n m+n > 2 Sacks has shown in [18] that w(A$) is a stable ordinal, so that I A: 1 is stable. It might be conjectured from this that I A: 1 is at least admissible. But we have:

ADMISSIBLE ORDINALS

305

Theorem F. I A: 1 is not admissible. The diagram in (ii) of Theorem E leaves open the order relationship between several pairs of ordinals of the diagram. The next result, obtained independently by Aanderaa in [ I ] , gives us some more information.

Theorem G . If m, n > 0 then

IJIkI # IZ;

I.

When m = n = 1 this result was first proved by showing directly that ni # 0:. But we do not know if n; # u k for n + m > 2. The proof of Theorem G is symmetric b e t w e e n n k and Z k and hence gives n o information on the relative magnitudes of the two ordinals. This is explained by the following result of Aanderaa, which we state here for completeness. (See [ I ] .) PW(C) denotes that C has the pre-wellordering property. See [ 1] for a precise definition.

The following summarises what is known about when the pre-wellordering property holds.

i)

i),

Proposition. (i) PW (n and PW (Z: (ii) v = L implies PW ( x k ) f o r m > 2, (iii) PD implies PW(n:,+,)and PW(Z:i,+2)for m > 0.

Here PD denotes the axiom of projective determinacy. It follows that l f l i l < l Z : i l andlZ:il a ; (ii) w ( n i ) = u ( A ; ) a n d w(A;)<w(Z:i) 2.

~(AL)

3 06

W. RICHTER and P. ACZEL

These results lead to an improvement of Theorem E(ii) in certain cases. For example we have 0: = 1 X f 1 > In: 1 = 7 r i >: 1 A l l > w ( n i ) > w ( T i ) = ~ ( b ; ) , Inil> I X i I >_ u i > o ( C i ) > o(ni)= w ( A l ) = IA :I, and in;I 2 n; > ~ ( 2 ; ; ) . = 7 r i remains open. Also, the relationship Whether I C i l = C J ~or between 7ri and I C i l or u i is not known. The paper is divided into two parts. In Part I we give alternative characterisations of some of the reflecting properties, compare them with the reflecting properties for the indescribable cardinals, and investigate their relative magnitudes. See $ 1 for a survey of the definitions and results of Part I. This part makes no use of inductive definitions and may be read without reference to Part 11. In Part I1 we prove the results stated in this introduction. Most of t h s part depends only on 8 1 of Part I, so that the reader mainly interested in inductive definitions can probably omit the other sections of Part I on a first reading. In $7 we examine first order inductive definitions. In particular we give upper bounds to their ordinals, proving half of Theorem A. For the other half we need the construction introduced in $8. In this section we formulate the notion of a closed class of operators. The construction of the notation systems Toand the associated coding lemma are the key to getting lower bounds for the ordinals of inductive definitions, and to proving Theorem D. The coding lemma is proved in the appendix. The proof of Theorem A is completed in $9. Theorems B and C are proved in § 10, while 3 1 1 has proofs of Theorems E, F and G. Many of the results in this paper were first announced in [ 21.

PART I. REFLECTING PROPERTIES $1. Summary of definitions and results In this part we shall study some of the classes of ordinals that will be used to characterise the ordinals associated with inductive definitions. These classes of ordinals will be defined in terms of certain “reflecting properties” closely analogous to those used in defining the indescribable cardinals of lianf and Scott. (See 171 and also LCvy’s [ 111 for a detailed discussion.) In order to bring out this analogy we shall start by considering the indescribable cardinals.

ADMISSIBLE ORDINALS

3 07

We shall use a perhaps excessively large language d: within which we can conveniently formulate all our reflection properties. d: has the usual propositional connectives, and has variables and quantifiers for all finite types (variables of type 0 range over individuals, those of type 1 range over sets of individuals, etc.). d: also has a name (individual constant) for each set and a name (relation symbol) for each relation on sets. (We will use the same symbol for the object and its name.) In particular E will denote the membership relation between sets. The restricted quantifiers (Vx Ey), (3x E y ) are defined in the usual way. Formulae of d: may be classified according to their prenex form. When doing this we shall follow LCvy in ignoring restricted quantifiers that do not bound unrestricted quantifiers. A formula i s n ; (Xh) if it is logically equivalent to a formula in prenex form which first has m alternating blocks of type n universal and existential quantifiers starting with a block of universal (existential) quantifiers and then has quantifiers of types
...)a,).

We can now define the (weak) indescribables.

1.1. Definition. Let X C On and a E On. If cp is a sentence of d: then (Y reflects cponXif

(Note that On is the class of all ordinals and that we indentify an ordinal with its set of predecessors.) a reflects cp if a reflects cp on On. a is Kl; (E;)-indescribable [on X ] if a reflects [on X ] every (Xh) sentence of 2. Some properties of the indescribable cardinals are summarised in the following theorems. Proofs of most of these m y be found in [ 1 I ] . 1.2. Theorem. a is H!-indescrfbable

(Y

> o is regular.

W. RICHTER and P.ACZEL

308

Let Rg = { a > o : a is regular}. The ordinal a is Mahlo on X if for evely f : a -+ a there is a /3 > 0 closed underf such that E X n a.

1.3. Theorem. (i) the following are equivalent a ) a is n:-indescnbable on X , b) a is L :-indescribable on X , c)a=sup(Xna). (ii) the following are equivalent a ) a is n! -indescribable on X b) a is -indescribable on X c ) a is Mahlo on X . (iii) a isn,!,-indescribabZe on X e a is C~+l-indescribable on X . (iv) Zjn>Oorm>2(n>Oorm>3)tken aisnn,(Ln,)-indescribable on X a is II; (Z il)-indescribableon X 0 Rg.

-

Hierarchies of classes of large cardinals have been obtained by iterating such operators as L and M where for X 5 On:

L(X)=

EX : 01 = s u p ( x n a ) }

M ( X ) = { a E X : a is Mahlo on X } . Iterations of an operator F are defined by transfinite induction on A:

F ~ ( x= ) Xn

n

v<

F(F@(x).

The elements ofLh(Rg) are the (weak) A-kypen'naccessibles,while the elements of M h ( R g ) are the (weak) A-kyperMakZo ordinals. Let H 1 ( X ) = { a E X : a is ny-indescribable on X } and let Hn+2(X)= { a E X : a is 11;indescribable on X } . Then by Theorem 1.3 H , = L and H 2 = M. The relative magnitudes of the ordinals in H,(Rg) may be indicated by using the following diagonalisation of iterations:

FA(X)={a>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<,P(R(/3)) for all a E On. (P(x) is the power set of x).

1.5. Definition. R(a)reflects cp on X C: On if

(X;)-indescribable R(a)reflects cp ifR(a) reflects cp on On. a is strongly [on XI ifR(cr) reflects [on x] every nk ( x:) sentence of L. The properties of the notions of Definition 1.5 closely resemble those of Definition 1.1. The strong @-indescribables coincide with the strongly inaccessible ordinals. For n > 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.

W. RICHTER and P. ACZEL

> 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

:

W. RICHTER and P. ACZEL

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 < p < a . Hencea is a limit number. So L, b Vx 3y(x E y ) , which implies that there is a 0< a such that Lp I= Vx 3y(x E y ) . Hence a is a limit number > 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

W. RICHTER and P. ACZEL

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<,VxVy 3zcpi(x,y, z). By Lemma 9 of [ 6 ]we may prove: 2.5. Lemma. There is a X y - formula Sat(x,y) of L, such that ifO(x) is a Z formula of L, with x as only free variable and a = r O(x)l then for all rud closed M , i f the constants of O(x) are in M then a E M and

M k Vx(O(x)

+-+

Sat (a,x))

.

Using this lemma we see that the transitive rud closed classM is admissible if and only i f M O 1 where O 1 is then!sentence VuVu[Vx E u Sat(u,x) 3zVx E u Sat (U,X)']. The theorem now follows if we let (5, be Oo A 8,.

--f

Proof of Theorem 1.9. (i) b) * a) is trivial. c) =j b). Let a = Sup (X n a) and let cp be a I l y sentence such that La I= cp. If a l , ..., a, are the individual constants occurring in cp then a l , ...,a, E L, so that there is a /3 E X n a such that a l , ..., a, € L and hence Lp t= cp, as cp is rIy. So a is rI; -reflecting on X and hence X 2-reflecting

6

on X , by (iii). a) * c). Let a be nt-reflecting o n X and let /3 < a. Let cp be the sen€ 0). Then La k cp, so that as cp is rI: there is a y € X f? a such tence 1(/3 that L7 k cp and hence /3 < y. Hence a = Sup ( X n a). (ii) *. Let a ben!-reflecting on X. Then a is @-reflecting and hence by Theorem 1.8 a E Ad. Now let f : a+ a be a-recursive. Let O(x,y) be a Ey formula of L, that defines the graph off on L,. Then L, Vx 3y(O(x,y) v (1On(x) A y = 0)). Hence there is a 0 E X n a such that Lp k Vx 3y(O(x,y) v (1 On (x) h y = 0)). Hence /3 > 0 is closed underf. So a is recursively Mahlo on X . (ii) +=. Let a be recursively Mahlo on X and let cp be a @ sentence of LE

ADMISSIBLE ORDINALS

317

such that L, k cp. As in the proof of Theorem 1.8 we will suppose that cp has the form Vxl 3yO where O is C: and define the a-recursive function f : a -+ a, and the ordinalPo 0 is closed under g. From the definition o f g it follows that 0> P o is a limit ordinal and is closed under f so that Lp k Vxl 3yO. Thus a reflects cp as required. (iii) is trivial. So for the converse let a be n;-reflecting on X and let cp be such that L, I= cp. cp has the form 3x1 ... 3x,0(x1, ..., x,) where O(xl, ..., xn) isn,.0 So there areal, ..., a, E L, such that L, k O(al, ..., a,). Hence there is a p E X fl a such that Lp B(al, ..., an). But then on X . Lp I= 3x1 ... 3x,t7(xl, ..., x,). So a is C,+l-reflecting 0 (iv) + is trivial a s X fl Ad CX.For the converse we use the I'I!- sentence oo given by Theorem 2.4. Let Oo be a Zy- sentence expressing the existence uo h d o , and uo A O o is a n $ - senof an infinite set. Then a E Ad L, tence of & . Now let a be n; @;)-reflecting on X and let be a n ; (E;) sentence of -C, such that L, t= cp. Then because of the restrictions on n and m cp A uo A O o is a n; (Z;) sentence of L, such that La I= cp A uo A Oo. Hence there is a / ? E X fl a such that Lp t= cp A uo A 8,. Hence 0 E Ad so that a reflects cp on X n Ad showing that a is II; (C;)-reflecting on X fl Ad.

5 3. Ordinal theoretic characterisations Let us call-an; (z;) *-reflecting [on XI if a reflects [ o n x ] every Ilk (E;) sentence of Lp. Our proof of Theorem 1-10will be a little indirect, in that we first prove Theorems 1.8* and 1.9*, obtained from Theorems 1.8 and 1.9 by replacing 'reflecting' everywhere by '*-reflecting'. But first we need the following lemma.

3.1. Lemma. There is a bijective primitive recursive function N : ON + L such that if(Vx < a) zx < a then L, = N"a. Proof. In Lemma 3.2 of [8] a primitive recursive bijection N : ON + L is obtained from Godel's primitive recursive surjection F : ON + L by successively removing repetions in the values of F. Examining their definition of N it is not hard to see that N'Ia = F'Ia for all limit ordinals a. If (Vx < a ) 2x < a then either a = 0, a = w or a has the form a = ep. Clearly Lo = (3 = "'0. If

W. RICHTER and P. ACZEL

318

a = w or a = E~ then in [ 121 it is shown that L, = F'Ia and hence it follows that L, = N " a as a is a limit ordinal. This result relativises to give a bijection NA : ON + L[A] which is primitive recursive in A , such that L, [ A ] = NA"a if ( V x < a) 2x < a. 3.2. Lemma. If a is n! *-reflectingthen (i) a is a limit ordinal > w . (ii) a, b < a *a+b < a. (iii) b < a =$ 2b w . (ii) Given a < a we prove that a+b < a by induction on b < a. If b = 0 this is trivial, If 0 < b < a and V x < b a+x < a then, as a is a limit ordinal, for some a Vx 3 y [ x < b + a + x + l = y ] .So by reflection V x < b a+x+l
+

+

3.3. Theorem 1.8*. a isn! *-reflecting-& E Ad.

Proof. Let a be KI! *-reflecting. Then by (i) of Lemma 3.2 a > w and a is a limit ordinal so that as we have already observed L, is rud closed. Hence it suffices to show that L, satisfies Xy-collection. So let L, l= V x E u 3y\k(x,y,b) where q ( x , y , z ) is Z:-. (We can assume without loss that there is only one existential quantifier 3y and only one constant b.) We must find c E L, such that (*>

L,

V x e a3yEcq(x,y,b).

Let R C 3 ON be the primitive recursive relation given by

ADMISSIBLE ORDINALS

319

Let a = N(ao), b = N(Po). By the previous lemmas La = N"a so that

where RE(a,P)-N(cw) E N ( @ ) .Hence, as a is Il! *-reflecting, there is a

P < a such that P I= Vx(RE(x, ao) + 3yR(x,y,Po))A Vx 3y (2* = y ) .As Vx < 0 ( F < P), L@= N"Pso that

(*) follows if we let c = Lp E L,.

3.4. Theorem 1.9*.

Proof. This follows the same pattern as the proof of Theorem 1.9 and so will be omitted. In the proof of (iv) we need the next lemma, which replaces Theorem 2.4. 3.5. Lemma. mere is a I$only ifa b ul.

sentence u1 of .Cp such that a is admissible ifand

-

Proof. Let us assume that then!- sentence uo of L, given in Theorem 2.4 is in Prenex form with Z matrix \ k ( x l , ...,x k ) . Now let R ( a l , ..., ak) /= \k(N(al), ...,N(ak))for al,...,ak E ON. Then R is a primitive recursive relation. Let B o be obtained from uo by replacing \k(xl, ...,x k ) by R ( x l , ..., x k ) . Then if L, ="'a

8-

a /= 0 0

-

L, I=0 0 .

Hence by Lemma 3.1 we can let ul be B o

A V x 3 y 2x =y ) .

We can now turn to the proof of Theorem 1.1C (i)-(iii) of Theorems 1.9 and 1.9* yield the theorem in the cases Ilk ( m 5 2 ) and Ek (rn 5 3). By Theorems 1.8 and 1.8* and (iv) of Theorems 1.9 and 1.9* the remaining cases need only be proved when a E Ad and X E Ad. With these restrictions the remaining cases will follow from:

320

W. RICHTER and P. ACZEL

3.6. Lemma. Let n + rn > 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

W. RICHTER and P. ACZEL

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,<& (Y

k Ol(al, ...,a,)-a

l=62(a1, .-.,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

W. RICHTER and P. ACZEL

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

W. RICHTER and P. ACZEL

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

W. RICHTER and P. ACZEL

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
r(fF A) 0,

if

f"XE 2 ,

otherwise,

for f < K .F is K -bounded since F ( f ) (f)is determined by values off on h < K . Let a be a witness for F. It is clear that a can be chosen 2 2. Let g : X + 2 such that r(g) > a and letf : K -+ K so thatf X = g andf(t) = 0 for f 2 A. Thenf"a 2 C a. Since a is a witness for F , F ( f ) ( O )
c

r

330

W. RICHTER and P. ACZEL

which is a contradiction. To show (a) * (c) let p be a Il: sentence of d: such that K k p. We must find a 0 < (Y < K such that (Y p. Let P be one of the standard bijective mappings of On X On onto On, and K , L be the associated pairing functions (cf. LCvy [ 1 I]). We first switch from set quantifiers to quantifiers of binary relations which are characteristic functions, and then switch to the language with unary function quantifiers instead of set quantifiers. In this language with the aid of P, K , L we can put formulas in a normal form (cf. Rogers [ 171, where this is done for formulas of second-order arithmetic). Thus there is a quantifier-free formula Q such that K k Vf3EQ(f, E) and for every a 5 K which is closed under P,

Furthermore, Q can be chosen so that in Q there is no nesting off(i.e. no terms of the formf(f( ...))). For a givenfand E the truth or falsity of Q ( f , t ) is determined by the values off for finitely many arguments and the answers to finitely many questions about membership in the relations appearing in Q. Since there is no nesting off the finite set of arguments off needed depends only on { and the relations in Q but not onfitself. Thus,

where

Hence, since K is regular,

where

Let h(P) = p q . C(P,q). Then h : K

+K

and for allf,g : K

+K

,

-

ADMISSIBLE ORDINALS

r

331

a.

( 1) f r h (0)= g h (0)3 [vt I P. QV, t ) ~ ( g , Let G : K~ ' K so that (2) C ( f )= pu [P"uX u< u & 35 5 U . Q ( f , E) & h ( t ) 5 u ] , and let F : K~ -+ K~ so that F ( f ) ( P ) = G(f) for all 0< K . F is K-bounded since -+

Let a be a witness for F and letf : a + a . We show 3 t < a. Q ( f ,E ) . Let g : K + K so that g Fa =f.Theng"a C a and F(g)"a E a since a is a witness. Let & = pc. Q ( g ,t).Then 6 , h ( 6 ) 5 G ( g ) IF ( g ) ( 0 )< a by definition of G. Thus from (l),

and hence p t . Q(f,E) = 6 < a. Thus a Vf 3 [ Q ( f , t ) and since a is closed under P (by (2)), a cp. A proof that (b) & (c) *(d) appears in L6vy [ 111 p. 217. It remains to show (d) *(a). Let K be strongly ni-indescribable and F : K~ + K~ be K bounded. We show that F has a witness. Let

+

Then X C R ( K ) .Note that (f1 y, t

, ~E X) * F ( f ) ( t ) = 7). Since F is K-bounded,

Since K is strongly n -indescribable there is an a < K such that

i.e. 0 < a < K and " a v t < a 37,q < a 7,t , ~E X ) n~ ( 4 . i (3) We show a is a witness for F. Let f : K -+ K such that f "aC: a. Since f 1a E (ya a n d f l a 1y y for y < a,we have from (3)

wr

=fr

V[

< a 377 < a F ( f ) ( $ )= 7 ,

i.e. F ( f ) " a E a

.

332

W . RICHTER and P. ACZEL

5.3. Remark. In the proof of (d) * (a), the assumption that F is K-bounded cannot be eliminated. For each K it is easy to define an F : K~ + K~ which is not K-bounded and has no witness. 5.4. Proof of Theorem 1.16. K is 2-admissible iff K is n!-reflecting. Suppose K is @-reflecting. Let {$IK map K-recursive functions t o K-recursive functions. We show t has a witness. By hypothesis,

By using the T predicate this is equivalent t o

The sentence on the right is equivalent t o a @ sentence cp(t),Since K i s n 30 reflecting (and hcnce II!-reflecting on Ad) there is some a E K n A d such that a /= cp($). But by the definition of p($) this implies maps a-recursive functions to a-recursive functions and hence a is a witness for $. Now suppose K is 2-admissible and let cp be a @ sentence of X p such that K /= cp. We show that K reflects cp. For simplicity we assume that cp is of the form Vx 3 y V z $(x,y,z) where $ is a Z;: formula with constants less than K . Let a be admissible so that all constants in $ are less than a. We introduce certain Gijdel numbers of a-partial recursive functions which, by the uniformity of the Normal Form and S- m - n theorems, can be chosen t o be independent of the particular choice of a. First choose a < a so that

Let g be a primitive recursive ordinal function such that

ADMISSIBLE ORDINALS

333

and let t < w be a Godel number (independent of a ) ofg. Then from (4),

{t}, maps a-recursive functions to a-recursive functions.

+

Since K cp, by ( 5 ) {tIKmaps K-recursive functions to K-recursive functions. Since K is 2-admissible there is an a E K n Ad which is a witness for E . But then by (9,a I= cp.

5.5. Remark. The definition of 2-admissible given here is equivalent to the definition which appears in [2]. The full definition of n-admissible is given in [21.

8 6. Stability In this section we prove Theorems 1.18 and 1.19. Note that if A <x y B and A C 5 B then A
s

6.1. Lemma. If a is a+l-stable then a is admissible.

Proof. Let L, k Vx €alp where cp is a Xy formula 1,. Then La+] VxEacpb whereb=L,EL,+l.HenceL,+l k 3zVxEacpZ.Ifaisa+lstable then L, k 3zVx Eacp'. Hence L, satisfies Zr-collection. The lemma now follows, as a is clearly a limit ordinal so that La is rud closed.

Proof of 1.18. a is HA-reflecting if and only if a is a+]-stable. Let a beIIA-reflecting. Let cp be a Z: sentence o f & , with constants only I= cp. We may assume that cp has the form for sets in L,, such that

334

W. RICHTER and P. ACZEL

3x1 ... 3x,,8(xl, ..., x,) where 8(xl, ..., x,) is@. So let a l , ..., a, E such that t= 8 ( a l , ..., an). As L,+1 = Def(L,) there are HA formulae 8 l(uo), ..., 8,(v0) of LE, with constants in L,, such that ai = { b E La : L, i= 8,(b)} for i = I , ...,n. Let 8’ be obtained from @ ( a l ..., , a,) by first replacing every occurrence of ai E s by 3 y E s Vz (z E y ++ z E a j ) and then replacing every occurrence of s € a j by e,(s) for i = 1, ..., n. Clearly 0’ is a HA sentence such that L, i= 8’. As a is HA-reflecting there is a 0 < a such that Lp i= 8’. Now ifai = { b E Lp : Lp k 8,(b)} then a; E as we may assume that x1 actually occurs free in 8(xl, ..., xn) so that the constants of Oi(uo) are also constants of 8’ and hence are constants for sets in Lp. It follows that O(a;, ...,a;) and hence L, k cp. L, Conversely let a be a+l-stable. Let uo be the H!- sentence given in Theorem 2.4. Let \k(x) be a Ey formula of ZE that defines L inside each admissible class A. Hence Vx\k(x) expresses V = L and the transitive models of oo A Vx\k(x) all have the form Lp for some admissible p. Now let cp be a HA sentence of L, such that L, k cp. Then L, cpl where cpl is cp A uo v VxJl(x) is also an; sentence of L,. Hence k 3x(trans(x) A where is obtained from cpl by restricting all quantifiers to x , and trans(x) is Vy E x V z E y ( z E x ) . A s a i s a + 1 - s t a b l e i t f o l l o w s that La/= 3x(trans(x)Acp(iY)). Hence there is a transitive set in L, that satisfies cpl. But this must have the form Lp for some admissible 0 and Lp k cp. It follows that a reflectscp, so that cy is nh-reflecting. We now turn to the proof of 1.19. In fact we shall prove a generalisation of that result in 6.4. Some of the ideas in [4] will be basic to our proof. For a transitive setA letA+ be the smallest admissible set such thatA € A + . I f S C A we say that S i s H k (E.”,) overA if S = {a € A : A k cp(a)} for some II; (E;) formula cp(x) of XE. Theorem 3.l(a) of [4] states that ifA is a countable transitive set closed under unordered pairs then for S CA, S is H i overA if and only if S is Zy over A’. The proof of this result in [4] may be made to yield the following formulation which gives us the extra information we shall need.

+

+or’)

cpy’

6.2. Theorem. (i) Ifp(ul, ..., un) is a formula of& then there is a Ly formula cp+(uo,u l , ..., v n ) of LE having the same constants as cp(ul, ..., u,) such that for every non-empty countable transitive set A and every admissible setBsuch t h a t A E B , i f a l , ..., a n E A t h e n

3 35

ADMISSIBLE ORDINALS

(ii) If ip(ul, ..., u,) is a Zy formula of L, then there is a FIf formula

cp-(ul, ..., u,) having the same constants as cp(ul, ..., u,) such that i f A is an infinite transitive set containing the sets whose constants occur in cp(ul, ..., u,) then f o r a I , ..., a,EA

Proof. We shall require some familiarity with the infinitary languages LB for admissible B.See for example [3]. (i) Let ip(ul, ..., u,) be a n t formula of L,. We may assume that it has the form V X , ... VXmO(ul,..., u,) where B ( q , ..., u,) is a FIh formula of L, with extra relation symbolsX1, ...,X,. Given a non-empty transitive setA we may define the infinitary sentences q 0 ( A ) and \kl(A) as follows: q o ( A )is AuEAVy(y€ a V b E u ( y = b )and ) \kl(A) is V x V a E A ( a = x )Then . the Hence if models of *o(A) A \kl(A) are all isomorphic to ( A, E r A a l , ..., a, € A then (1) A cp(al, ..., a,) iff \ko(A)A \kl(A) @(a,, ..., a,) is logically valid. Note that ifA E B where B is admissible then (*,,(A) A \kl(A) + e(al, ...,a,)) E B i.e. it is a sentence of L,. Now it is a routine matter, using [3] to find Z: formulae of LE q ( u o , ul, ..., u , + ~ )and x(uo), such that i f A , B,a l , ..., a, are as above then if

-

+

-+

b€B (2) B k \k(A,al, ..., a,, b ) iff b = (\ko(A)A q l ( A ) @(al,...,a,)), and if b is countable then (3) B x(B) iff b is a logically valid sentence of LB. (3) follows from the completeness theorem for countable infinitary sentences (see Theorem 2.7 -+

of ~ 3 1 ) .

\k(uo, ..., u,+,) may be chosen to have the same constants as cp(ul, ..., u,), while x(uo) may be chosen to have no constants. Now let cp+(uo, ..., u,) be 3u,+l(\Ir(uo, ..., u , + ~ )A x(u,+l)). The result follows from (1)-(3) using the fact that (*&I) A *,(A) e ( a l , ...,a,)) is countable i f k is countable. (ii) Let cp(ul, ..., u,) be a Zy formula of L,. Let KP be the theory of admissible sets, as formulated in [3]. Then by 3.3 of [4], ifA is a transitive set -+

W. RICHTER and P. ACZEL

336

and Y3 is an end extension o f A U { A } that is a model of KP then % is an end extension o f A + . Hence if a l , ..., a, € A then A+ k O(al, ..., a,) iff % O(al, ...,a,) for every A-model % of KP where an A-model of KP is a model of Kp that is an end-extension o f A U { A } . Now by the downward Lowenheim-Skolem theorem every A-model % of KP has an elementary subsystem 8'4 8 that is an A-model of KP of the same cardinality as A , assuming that A is infinite. Every such A-model 23' is isomorphic t o ( A ,E ) for some E C A X A . Then there is an f : A + A and a € A such that (a) ( A , E ) is a model of KP (b) f : ( A ,€ A ) E (a,, E r a,) where aE = {b € A : hEa) ( c ) b, 5 aE for all b € a E . It follows from the above thatA+ k O(al, ...,a,) iff ( A , E )k 6 ( f ( a l ) , ... ...,f( a,)) for all E C A X A , f : A + A a n d a € A such that (a) &(b) & (c). I t is now a routine matter to find the required H i formula obtained by forinalizing the right-hand side of the above equivalence. 6.3. Definition. An admissible set A is ni-reflecting i f A and a is transitive) for all n],sentences cp of L,.

cp * 3a € A (a

cp

The following is a generalisation of 1.19.

6.4. Theorem. The counkzble admissible set A is n ;-reflecting if and only if A < ~ AO+ . Proof. Let A be a countable admissible set that is H:-reflecting. Let cp be a C y sentence of L,, with constants only for sets in A , such that A + k cp. Let T be Vx 3y(x E y ) . Then, by (6.2)(ii) with IZ = 0, A k=cp- A T . Hence a k cp- A T for some transitive a € A . It follows that a is an infinite transitive set such that a /= cp ~. By (6.2) (ii) a+ cp. But as a+ C A and cp is E it follows that A k cp. Hence A < z y A + . Conversely, let A < q A + and let cp be a ni sentence of LE such that A /= y . Then by (6.2) (i) with n = 0, A' cp+(A). Hence A+ k cp, where cpl is the Cl) sentence 3x(trans(x) ~cp'(x)). But cp, only has constants for sets in A , so thatA cpl 1.e. A k y'(a) for some transitive set a € A . AsA is countable so is a, so that by (6.2)(i) a cp. ThusA is ni-reflecting. In order t o obtain 1.19 we need:

7

+

+

ADMISSIBLE ORDINALS

337

6.5. Lemma. L, is ni-reflecting iff a is n:-reflecfing.

Proof. Let a be Hi-reflecting. If cp is a rI! sentence of L, such that L, i=cp then Lp k cp for some 0 < a.But now a = Lp is a transitive element ofA such that a k q. Hence L, is Hi-reflecting. Conversely, let L, be nf-reflecting. Let u be the fl! sentence uo A Vx\k(x) occurring in the proof of 1.18. If cp is a r1: sentence such that L, cp then L, =i cp A u. Hence there is a transitive set a E L, such that a /= q A u. But a = Lp for some 0 < a.So Lp k cp for some p < a.Hence a is nf-reflecting. Now 1.19 follows from 6.4 and 6.5 when we observe that (La)+ = La+for every ordinal a.

PART 11. INDUCTIVE DEF€NITIONS

57. First order inductive definitions, I We begin by considering inductive definitions which are either recursive or closely related to recursive inductive definitions. These very simple cases illustrate some of the principles used in characterizing the closure ordinals of more complicated inductive definitions. 7.1. Definition. For any inductive definitions ro,rl let

Let r = [r0, In constructing the transfinite sequence ( F a : a E On) one repeatedly applies Po until closure under r0 is reached, (i.e. until a h is reached such that r0(rh) C r’); then rl is applied once; then rois repeatedly applied until closure under Po is reached, etc. rl is applied only when closure under Po is reached. Note that if ro(I”)C r’ then r(r’) = rl(r’). I [Po, rl]I is the least X such that both ro(r’)E I” and r,(r’) 5 F A .

rl].

For any recursive relation R and inductive definition r, the truth or falsity ofR(n, r’) is determined by the answers to a finite number of questions about membership in r‘. For limit h, the answers to these questions are the same as the answers to the same questions about membership in I’E for suitable large $ < X. Hence for recursive R and limit h,

W. RICHTER and P. ACZEL

338

7.2. Proposition. (i) IKI:l = w , (ii) I [n:,n",i= w 2 , I [II!,HO,,~~,II

Proof. (i) Let Hence, ~1

r En:.

= w 3 ,etc.

Then for some recursive R , n E F ( X ) --R(n,

X).

E r(rw) - ~ ( n rw) , * R ( n , rE) for some l < w , by (1)

c

c rw.

+ n E r(rt) rt+I

Thus r ( Y WC ) r Wand hence I rl I w . To show In! I 2 w, let r o ( X ) = { O} U {( 1,x) : x E X } . Then roE @. 0 E r " and I 01 = 0; if n E rr and In I = $, then ( 1 , n ) E rg and I( l , n ) l = f + 1. Thus I Po I 2 w. (ii) Letn E r O ( X ) - R o ( n , X ) a n d n E r l ( X ) - R l ( n , X ) whereRo and R are recursive. Then as in the proof of (i) we have: ( 3 ) Iflimit A then ro(rh)r hand hence r(rh)= rl(rh). Now let n E I ' ( P 2 ) . We show n E P2. Since limit 02,n f rl(rW2) by (3 ), i.e.R,(n, rW2). Then by (2) there is some [ < w2 such that V6 < w2. f 5 6 * R l ( n , r'). Since the limit ordinals are cofinal in w2 there is some limit 6, $ 56 < w 2 ,and henceRl(n, I").Thus n E I',(r').,BUt since = r(r6) and hence n E r(r6) C E I'w . Let r0be as limit 6, r1(r6) above and let rl(X) = ( ( 2 , ~ ): x EX}. Let r = [r0, I?,]. It is easy to show that i r 1 > 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 ) z o such that t < f ( x ) < a and hence1 R ( n , z O ,F f ( x ) ) . But then by definition off, V z I x R ( n , z , l ? f ( x ) ) In . particular R ( n , z o , F f ( x ) ) which is a contradiction. Case 2. Not limit a.Then since f is non-decreasing there is some y such that for all x > _ 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
< ol,y < w,

- n ~ r ( r cr w ) ~l .

Hence r ( r w l C ) r W 1so that IFIS ol. A different proof appeared in [2]. The present proof is due to Grilliot. As in the definitions of [@, @], [@, @, @] etc. we may define [C,, C,], [C,, C,, C z ] etc. for any classes C, ... of i.d.'s.

e,,

Proof. The first inequalities in (i) and (ii) are trivial, so we turn to the last inequalities. (i) Let r = [ I-,, r,] where roE E; and rl € E:+l. Then r' E nh and : E < h ) is Xy on L, for h E Ad, so that by the proof of Lemma 7.7 hence If h E Ad then r0(rh) C r h and hence r(rh)= rl(rh).Suppose (4) first that n is even. Then for some recursive R a EQ(X)

IJ

3 x , v x , ... 3x,R(a,x,, ... x , , X ) .

Hence by (2) and (4), if A E Ad then

)

ADMISSIBLE ORDINALS

34 1

for some X z + 2 formula Q(u) of E, that is independent of h E Ad. Now let K be the least element ofM,+,(Ad). Suppose (I E r(rK). Then as Q(u). As K is Z:+2-reflecting on Ad K E Ad it follows from (5) that L, there is a h < K such that h E Ad and L, P Q(u)- Hence by (5) a E r(rh)E rK. Thus r(rK) C r Kand hence lrl IK. If n is odd then for some recursive R (I

E

-

r,(x)

3x,,Vxl ... VX,R(U,X,,,..., x,, X )

Hence using (2) and (4) again, if h E Ad then

for some X : + 2 formula Q(u)of .& that is independent of h E Ad. The rest of the proof is as before. (ii) This follows the same pattern as the proof of (i). Let r = [ro, rl,r2] where Po E Z!, rl E X:+l and r2E X k + l . The proof of (i) shows that (4') If h EM,+l(Ad) then [ro, rl](YX) C rxand hence r(rh)= r2(rh). Then as in (5) if h EM,+l(Ad) ( 5 ' ) (I E r(r9 L, k= e ( a ) for some formula Q(u) of L, that is independent of h EM,+,(Ad). The rest of the proof is as in (i).

xi+2

-

In the next section we will prove results which will enable us to reverse the inequalities in this lemma. !j 8. Closed classes of inductive definitions

In this section we formulate the notion of a closed class C. The results in this section will enable us to give characterisations of IC I and Ind (C) for many of these classes. 8.1. Definition. f : A

5, r if

342,

W. RICHTER and P. ACZEL

(a) f is a recursive function and { f ( e ) }is total for all e ; (b) if {el : X I , Y then { f ( e ) }: -L ( X ) I, r ( Y ) . A 5 , r iff : A for some f . A IIr is defined similarly.

5,

8.2. Theorem. If A 5 , 5, replaced by I

r then A m 5,

I'" and I A I _< I PI. Similarly, with

This is an immediate consequence of the following: 8.3. Lemma. If A 5 ,

g:A

~S,

ra.

r there is a recursive function g such that for all a ,

Proof. Let f : A 5 , r. By the recursion theorem there is an e such that {e}= {f ( e ) }= g , say.g is total since {f ( e ) } is. We show by induction on Q thatg : ha I , P.Suppose

{ e } = g : A @5, I'p and hence g = { f ( e ) }: A(A p,

5,

r(I'p)

for all fl
8.4. Definition.

r is C-complete if r E C and e = { A : A 5 , r}.

8.5. Theorem. I f

r is C-complete then I e I = I I'l and Ind (C) = { X

:X

5 , rm}.

Proof. Let r be C-complete. As E C, IC I > 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}.

<,

8.6. Theorem. There is a l l ~ + l - c o m p l e toperator. e Similarly for X z + l Proof. We shall need the following folklore result, which is well-known when

ADMISSIBLE ORDINALS

343

n = 0 or n = 1, but is equally true for larger n. 8.7. Proposition. There is a universal IT&+l operator. Similarly for L&+l. By this we mean a n & + l operator such that every n&+, operator A has the form A(X) = r,(X) = {XI ( a , x )E r ( X ) } for some a E W . We will show that r islI&+l-complete. Let Al(X) = { ( e , x ) : x E A({e}-'X)}. When n > 0, A 1 is easily seen to hen",,, and hence has the form I?, for some u E W . Now let f be a recursive function such that { f ( e ) } ( x )= ( u , ( e , x D .Then f : A 2, I'. When n = 0 we must be more careful as A1 may not be rIi+l.We will define a rI:+l operator A 2 such that if { e } is total then ( e , x )E A1(X; ( e , x ) E A2(X). T h e n f : A 5, r if we let A 2 = r,. Let rp(X,x) be formula defining r. By separating out positive and negative occurrences of X in q ( X , x > 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

W. RICHTER and P. ACZEL

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
EON^^.

Note that the definition ofM has the appearance of a set inductively defined by an i.d. But the situation is complicated by the fact that the definition ofM,+, depends not only on the previously definedM,, but also on (Mi,, : a EM,). Given any sequence Mt : [ E O N ) we use the following notation.

M=U{ME:[EON} l ~ l = p [ ( a E M ~ + ~for ) a€M

IMI = Sup{lxl : x E M } M,* = {(x,y) : x , y EM, & 1x1 I ly I}

for (YEON

M * = u{M,* : ( Y E O N } . I f X c o l e t 9 ( X ) = { x : (x,x)EX}andX<,={y: (y,x)EX&(x,y)$X}. I fora E M a . Then clearly Ma = 9(M,*) a n d M I U=(Mi),, Hence the definition ofM,+l above may be written

M,+1 = M a u @(M,*) where

ADMISSIBLE ORDINALS

345

Notice that 0 is Hy. We will see below that M * is inductively defined by a l l y i.d. We now generalize the above procedure to an arbitrary 0. 8.10. Definition. For any i.d. 0,%@ = (M',

Mo

=

11) is defined by:

b

Ma+l =Ma U 0(M,*)

M,=U{M,:a
does not have an obvious inductive definition we show that

8.1 1. Definition. For any i.d. 0,0, is defined by 0,(X) = u { ( x , y ) : x,y E 0 ( X ) \ S(X)}.

{ ( x , y ) : x E T(X) & y E 0 ( X ) \

qx)}

8.12.Remark. For closed C? note that 0 E C * 0,- E C . 8.13. Lemma. For all a 0:- =M,* and hence 101 ,= -

lM'l

and @:=M*. -

Proof. Note thatM,*+l = M i U 0, ( M i ) . The result now follows by induction on a.

8.1 4. Equivalence Theorem. If C is closed and 0 is C -complete then I C I = IM @I and Ind ( C ) = { X C w : X I , M'}. Proof. If C is closed and 0 E C it follows from 8.12 and 8.13 that IC121M'I a n d I n d ( C ) > { X C w : X < , M @ } a s O , EC and x E M @ -(X,X)€@:. -

For the converse it is sufficient to prove the following. 8.15. Lemma. I f I? 5, 0 then there is a recursivefunction f such that for all a f : r" 5, Ma;hence r" Me and I rls IM' I.

<,

W. RICHTER and P. ACZEL

346

Proof. Letfo be a recursive function such that { f o ( e ) } ( x )= ( { e } ( x ) , { e } ( x ) ) . S o {e}--' 7={ f o ( e ) } - l . Let f l : r Srn0. Then for total {a}, r{a}-' = {f,(a)}--' 0. Then

Now choose e such that {e} = { . f l f o ( e ) } and l e t f = { e } . Then rf-'7=f--'0.Hence F f - ' ( M , ) = r f - ' S ( M * ) = f - ' O ( M , * ) , so thatf-lM, U r ( f - l M 0 ) = f - ' M , Uf-'@(M,*) = f-7( M , U @ ( M z ) )= f-'M,+,

. It follows by induction on Q that F a = f - l M , ; i.e. f : I"y 5, M,,

e

In 57 we have seen how to prove, for certain C , that 1 I IK where K is the least reflecting ordinal of a certain kind. In order to. show that I C I = K we will choose a 'good' notation system% = (M,II) such that IMI I I CI and show thathf has the required reflection property. A s M is a set of notations for the ordinals < / M I , statements about ordinals< / M I can be rewritten as statements about M . The reflection property for [MI will then follow from closure properties of M . The Coding Lemma below gives a formulation of this rewriting process for X: statements.

8.16. Definition. A notation system 311 = ( M , 11) isgood i f % = %' where O ( X ) = Z ( X ) u Q ( X ) and E(X) = {0}u { ( ] , a b, ) : a E 7 ( X j & V x [ b ]( x , X , , ) E 7(X)}U { ( 2 , a , b ) : a E 7 ( X ) orb E 7(X)},and @ ( X ) is alwaysdisjoint f r o r n ( O } ~ { ( I , a , b: )a , b E w } U { ( 2 , a , b ) : a , b E o } . I f LXis good then an ordinal X 5 /MI is W-good If Z ( M l ) ClzJ,. Thus [ M I i s m - g o o d , but usually there will be%-good ordinals< IMI. 8.17. Coding Lemma. Let 311 = (M, 11) b e a good notation system and let T , = {(x,(u) : a: E On & x EM,}. Then (i) b'very W-good ordinal is in Ad ( T , 1. (ii) For evely Ly- forinula q ( u l , ..., u n ) of .Cp(Tm)there is aprimitive recursive function h such that for every %-good ordinal X

a , , ..., a n E M h & X t = p ( l a l l , ..., IanI)-h(al, (iii) I f

X is %-good then for X E w X is h-r.e. in Tm 1 X

-

..., a n ) E M A X

<,, M A .

347

ADMISSIBLE ORDINALS

This lemma will be proved in the appendix.

8.18. Corollary. L e t W , T,, be as in the lemma, and let f (a)= pn [n E M & In 1 = a] for a < IMI. Then for each W-good ordinal X f I’ h : X + o is a X-recursive in T , I’ X injection. Proof. f ( a ) = p n [ ( n ,a + l )E TT & (n,a) 4 T,]. Hence f 1 1is A-recursive in T,, 1 X f o r m - g o o d A. I t is clearly an injection.

8.19 Theorem. Let C 2 @ be closed and let r be C-complete. Let h = I C I andAr={(n,a):a
,

Proof. Let 0 ( X ) = X ( X ) U @(X) where @ ( X )= ( ( 3 , a ) : u E F(X)}. Then as E En: E C and e is closed it follows that 0 E C . Also Xx(3,x) : r ( X ) < , 0 ( X ) for all X and hence f ’ : F 5,0 where f : r F and f ’ is a recursive function such that {f’(e)}(x) ( 3 , {f(e)}(x)). Hence 0 is Ccomplete. Now 7?2= 311’ is a good notation system and by the Equivalence Theorem I C I = IMI and Ind (C) = { X E o : X 5 , M}. Hence by the Coding Lemma and its corollary the theorem follows as long as we replace A , by T , I\ h. I t only remains to show that A and T , 1 h are X-recursive in each other. But by 8.15 there is a recursive function h such that h : F a 5 , M& for all a. Hence ( n ,a ) E A (h (n),a) E T,, 1 , so that A is 1-recursive in T , I A. For the converse note that as F is C-complete and 0, E C it follows that g : OF 5 , r for some g. Hence g : M,* Srnr” for all a,b y 8.3 and 8.13. so

<,

N_

-

,

,

showing that T , 1 X is A-recursive in A , .

3 9. First order inductive definitions,

II

We are now ready to characterise the ordinals of first order inductive definitions.

348

W. RICHTER and P. ACZEL

9.1. Theorem. (i) I Hy I is the least element of Ad; (ii) I [@JI,"]I is the least element ofM,,+l(Ad); (iii) I [ny,Il~,rl~]l is the least element ofM,,+l(M,+l(Ad)),

etc.

9.2. Remark. By 7.8 this theorem is also true when each n: is replaced by 0 zk+l.

Before proving the theorem we derive some immediate consequences.

in,"i= J~;+,I = n'n+l 0 = o hoof. In:l=Icyl=w=n(: by 1.9(i),7.2and7.3.1nyl=IZ!l=wl=n2 0 by 1.8 and 7.4. Forn > 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 '
Proof. (i) This folIows from Theorem 7.4, whose proof assumed the result In(i)-monI >_ 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)<], -

-

ADMISSIBLE ORDINALS

349

as (a;)<Ell:. 7?2 is a good notation system so that by the Coding Lemma E Ad. Let cp be all:,, sentence of L, such that a k cp. We may assume that cp has the form

a = IMI

where \k is a ET- formula of L, and cl, ...,ck E M . By the Coding Lemma there is a primitive recursive function h such that for all%-good ordinals h

Now choose e E w such that

Then it follows that f o r m - g o o d h

Hence as LY k cp and a i s m -good

Now if h = I(3, e )I then h < a , X ism-good, and hence admissible, and (3,e)E@A(Mi)so that h i=cp.ThusaEM,+I(Ad) a n d @ =lO, l<_l [lly,ll:]I, as required. When n > 0 is even the proof is as above except that

and the ll:+, sentence cp now has the form VX13X*

... 3 X , l * ( X ]

with 9 , c l , ..., ck as before.

)...)X,,ICll ,..., I C k l )

W. RICHTER and P. ACZEL

35 0

In case n = 0 let @.,(X)= X.Then as before a = IMI 5 I [fly, n,"] I and a € Ad. In order to show that (Y EM,(Ad) we must show that a is a limit of admissibles. So let f l < a . Then fl = la1 for some a E M . Then ( 3 , a )E M as Z ( M * ) C M . Let A = I(3,a)l
aEO(X)-aEE(X)v

[ Z ( X > S 9 ( X ) & a€cpA,(X)]

v [ Z ( X ) u cpL(X)

c 9(X)& a E @AI(X)]

are as in (ii). Then as in where @L1(X)=( ( 5 , e ) : e E Q n ( 7 ( X ) ) } and a,, (ii) 0, = [Z<,(@&,(cpA1)5] E [ny,lli,lT,"] so t h a t & = IMI =

10,17 I [ncn;,n;li.

As in the proof of (ii) we may show that a EM,+,(Ad). Moreover we may show that l(5,e)l EM,+l(Ad) whenever (5,e) E M . Hence using once more the argument in the proof of($ we can show that aEMn+,(Mm+l(Ad)). The next result characterises I n d ( C ) for certain classes of first order inductive definitions.

9.6.Theorem.Ife isany ufthecZassesn(i', [lly,n,"], [Il~,fl~,fl,"],etc. and h = IC I then there isa T'E C such that h = lrl and Ind(C) = {X C w : X I, r-} = { X C w : X i s A-r.e.}. 9.7. Remark. This result also applies to the classes n,"+, and to the classes obtained from the ones considered by replacing each KI: by ZE+l.

Proof. The proof has the same form in each case. We illustrate with C = [@, n,"]. In the proof of 9.5 (ii) a good notation system W = W @is defined such that !MI 5 h and \MI EMn+l(Ad), But h is the first element of M n + , ( A d ) b y 9 . 1 . H e n c e I M I = X . S o i f r = 0 < E C t h e n h = I r I . By t h e c o d i n g k m m {X 5 w : x is A-r.e.} S {X w : X%, rm}, asM I , M * = r". . Hence { X C o : X is A-r.e.} 5 Ind (C) as r E C. A E C implies that A" = Ah is A-r.e. by 7.5. Hence I n d ( C ) 5 {X 5 w : X is h-r.e.}, proving the theorem.

c

For completeness we conclude this section with the following easily proved result.

ADMISSIBLE ORDINALS

35 1

9.8, Theorem. (i) Ind(rI!)={XEo: Xisr.e.1. (ii) Ind(X7) = { X E w : Xis a “recursive”unionof arithmetical sets}.

5 10. Higher order inductive definitions, I In the previous section it was shown that the closure ordinals of certain classes of first-order inductive definitions are reflecting ordinals of a prescribed form. In this section we obtain corresponding results for higher type inductive definitions. The techniques are similar to those of $87 and 9. In Lemma 7.5 it was shown that for r (rt : t; < A) is uniformly Zy on L, for h E Ad. For other r this need not be the case and this makes the characterisation of In; 1 for m , n > 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) >

-

W. RICHTER and P. ACZEL

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-

W. RICHTER and P. ACZEL

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 ) :<,1, I1(X) for all X . Let a = g ( ( a l ,...,a&). Then for ?X-giiOd

x

I n gciicral wl: c.inirot expect (ha1 II1,Tl = n,"' or In,"'[= u r for m,n > 0. ortlcr t o use 1,eiiima 10.1 to show that I n : , for example, we would w,mt to h o w that for I'EH!, X E L f \ P ( o ) * r(X) E L I . But there is no "3 n3 gu,i~ariteethat I" = I'(@)belongs to 1, 1 or even to L. hi the case of 11; and n3 Z Iiowever, we c a n do better by making use of a result due to Barwise, L a t i d y m d Moschowkis, formulated here in Theorem 6.2. 1ct I n be the class of recursively inaccessible ordinals.

Inil

Iii

1,

ADMISSIBLE ORDINALS

355

i

10.6.Lemma.If'I' is n or Z then (Y6 : E < h )is uniformly E: on I,, for h E In. Proof. It is sufficient to show that if h E In then (i) x E L, =$ r ( x n w ) E L,, and (ii) { ( x , y )E L, x L, : r(x n u ) = y ) is E: on L, uniformly for A E In, as we may then carry through the proof of Lemma 7.5. Let r be n:. Then by 6.2(i) there is a Ef- formula cp+ of %, such that if A , B are admissible sets and x E A E B then

But if h E In and x E L, then x E Lp for some admissible p p+ so

<x.

12

Er(xnW)

-

L,,+

t=

< h and dso

P+(L~,~,X).

Hence r ( x n w ) is Zy on Iir+ so that r(x n w ) E Lp++lC L,, proving (i). To prove (ii) let uo be the n2- sentence of 2.4. Then if x , y E L, and h E In, r(x nu) = y if and only if there are transitive sets A , B E L, such that [A,Bare transitive&xEAEB&A + u o & B ~ u o & y ~ w & V i z E o (B i= cp+(A,n,x) n Ey)]. The expression [...I can be defined by a X:formula \E(A,B,x,y) of l,, independent of h, so that r ( x n u ) = y L, l= 3a 3b\E(a,b,x,y), proving (ii).

-

If

r is Zt

then the proof is as above except that p+ is now n

-

r.

Proof. In{l>n;and I E ! l 2 a i followsfrom 10.5.Note thatbyTheorem 1.9 n:, a: E In, T ; is nf-reflecting on In and a: is Xi-reflecting on In. Hence by Lemma 10.6 we may use Lemma 10.1 with n = rn = 1, X = In and A = 0 to get In:I< ni and I E:l I u i .

356

W. RICHTER and P. ACZEL

8 11. Higher order inductive definitions, II Recall from the introduction that a(%?) = Sup{l 0. We will prove Theorems E, F and G of the introduction.

Proof. (i) We first show that [A; o f A C w . I t suffices to find r E A;

Let < be a A; such that 1 rl = I<[. Let

12a(A",.

well-ordering

As n > 0 r is clearly A;. By induction on a, = {x € A : Ix I< < a} where (XI< is the order type o f x in the well-ordering<. Hence l r l = [
;

Then Q is a univalent system of notations for the ordinals less than IF/. Let

-

The < is a well-ordering and = 1 rl. It suffices to show that < i s A;. For X C w X w let X , = { y : y X k } . I f S 2 w X w and Y C w let @ ( S , X ,Y) S is a (strict) well-ordering of Y & Vk @ Y(Xk= $9) & vk E Y ( X , = {r(xl): / S I C ) t~ ) v k , i Y~ ( ~ # Z = . X#, X J &

u

w,< X,) c u,< w

X, .

ADMISSIBLE ORDINALS

35 7

Then as m , n > 0 and m + n > 2 @ is A;. Clearly @ ( S , X ,Y) if and only if S is a well-ordering of Y of order type Irl such that X , = rIk'sfor k E Y , and x k = f o r k 4 Y . Hence if

so that Q is A;.

As

it follows that < is A .; be complete Z; such that a 4 I'T. Let r,(X) = {a} for all (ii) (a) Let X and let = [rl,r2]. Then E A;+l and I rl = lr,l t 1 = I X;l f 1. Hence 1 A$+lI > I X k I ; 1 A",,1( > ln&lis proved in the same way. (b) is just 10.5. For (c) let
(1)

3f Vk VZ [k < 2!=' f ( k ) < f(Z)] .

A 2 1-4and hence 1<1 is not X; -reflecting The proof that is similar to the above, interchangingll; and X; throughout and replacing (1) by

Then A k cp

r z 2 w(E;)

(d) is trivial. Remark. We do not know of any cases where equality holds in (c). Note that

W. RICHTER and P. ACZEL

358 T;

5 Ink I < ?rk+l.Thus ?rkand Ink I are not too far apart. Similarly for

uk and IXLI.

i

1 1.2. Theorem. I A I is not admissible. Proof. We shall use the following fact extracted from 57.10 of [19].

Proposition. There is a I l i relationJC o X w X ?(o)such that I' is A: if and only if r ( X ) = F,(X) = iy : J ( n , y , X ) ) for some n < w.

11.3. lamma. I f h is recursively inaccessible then

:n

< o & 5 < h ) is

Z: on L ~ .

Proof.By 10.6,aseachI', i s l l : , ( r i : . $ < h ) i s Z y o n L k f o r a l l n < o . B u t as r, is IIi uniformly in n , and the proof of 10.6 is uniform, :5
(I'i

11.4. Lemma. I A

i I is a limit of admissibles.

Proof.Leta
@<(a U fx : O,(X)

A(X) = -

-

2 X & x E a}.

T h e n A i s A; a n d l A 1 = l O < l + l . H e n c e I M I < I A l 5 l A ~ I , s o t h a t a <_ 1 M 1 < I A 1 and 1 MI isadmissible proving the lemma. Note that we have shownthatl~l
4

We can now prove the theorem. Suppose that h = 1 A: I is admissible. Then by the previous lemma it is recursively inaccessible and hence by Lemma 11.3 : n < o & g < h ) i s X y onL,.Letf(n)=Ir,Iforn<w.Thenf :o+h is A-recursive, as

(I'i

359

ADMISSIBLE ORDINALS

But A = 1 A i l = Sup,<,)r,) of A.

= Sup,<,f(n),

contradicting the admissibility

We conclude this section by showing that under very general conditions

I el # 1 1e I. We also obtain a related spectrum result.

11.5. Definition. If C is a class of inductive definitions then the spectrum sp(c) of is {iri: r a?).

e

11.6. Definition. A closed class I'l E C, where

e is V-closed if Zy 5 C? and 1'E

C implies

e)

11.7. Theorem.If e is V-closed then I C I 4 sp (1 a n d in particular

iei#iiej.

Note that the last inequality fotlows'because IlCI E s p ( 1

e).If

IeI
m, n > 0. In particular we get

11 -8. Corollary.If m, n > 0 then

Ink 1 # I X k 1.

We turn to the proof of the theorem. Let C be V-closed. Let I' be C complete and A E 1 It is sufficient to show that I A I f 11'1 as 11'1 = I CI. Let O(X) =

e.

W. RICHTER and P. ACZEL

360

As is first order closed 0 E C . As r and hence by Theorem 8.14

e

Lemma 1. IF1 = (MI. Lemma 2. For a I IMI,

<-,,,0 it follows that 0 ise -complete

ADMISSIBLE ORDINALS

Lemma 3. For a A Q I,M4a.

< IMI and i < 4 , ( 4 ,a ) € M4a+i - a

361 E A&. In particular,

Proof. We use induction on a. ( 4 , ~ ) E M ~ ~ + ~ - 3 v3 <[ 4v at + i & d ( a , v ) ] -3p
3 j < 4 q a , 4 p ti)

3P
-a€

AQ.

Since I I'l= IMI it suffices to prove: Lemma 4. I A I f IMI. Proof. Suppose I A I = IMI = a. We get a contradiction by showing I A I
V x [ x E A ( { a :( 4 , a ) E M a } ) * ( 4 , x ) € M Q ]

By Lemma 3 , V X [ XE A(Ap) * x € A @ ] , andhence I A l < p < a .

Appendix. Proof of the Coding Lemma SA.1. Acceptable ordinal systems. We begin by discussing certain closure properties on systemsw and show that if % is a good notation system, then 9R has these closure properties.

362

W. RICHTER and P. ACZEL

A . l . Definition. Let 'h?= ( M ,11) be any ordinal system and B C w . W is Brestricted productive if there is a primitive recursive function p such that for every n E w , (i) V x [ { n ) ( x , B ) E M l~p(n)EM$Ip(n)12sup{I(n}(x,B)I + 1 : x E w } ; (ii) p ( n ) E M ~ V x [ { n } ( x , B ) E ~ . p is called a B-restricted productive function for%'. The closure condition (i) is analogous to the closure of infinite regular cardinals with respect to mappings from smaller ordinals. (ii) is a technical requirement which ensures that there are not extraneous notations inM.

m

A.2. Definition. is acceptable if there are recursive functions j , 0 and a primitive recursive function p such that (i) j : M 5 , M and for a E M , if lj(a) I 5 a then J(MIQI)is recursive in M, uniforrnly in a, where J is the complete i.d. J ( X ) = { x : 3 y T X ( x , x , y ) } ; (ii) if a E M then An. p ( a , n ) is an Mla,-restrictedproductive function for %. (iii) a E M v b E M * a &3 b E M & inf { l a I, I b I) 5 la v b I, where Ix I = I MI if x 4fig. We say that 3tI is acceptable in terms of j , p , 0 . We next show that there are functions j , p , @ such that if CLT is a good notation system and h ism-good then W Ais acceptable in terms o f j , p , 8

A.3. Lemma. Let 311= ( M , 11) be a good notation system. If a E M then J ( M l a , )is recursive in M , for all a 2 la I + 2 , uniformly in a. Proof. Let Q! >_ ( a (+ 2 and let e be a recursive function such that

Note that 1 $ M . It suffices to show there is a recursive functionf such that for all x, x 4J(Mlal) f (a,x)E M a . Now

--

x~J(~lul)-Vt7~'~'(~,X,t)

vt[e(a>X)l(t,Ml,,) = a EMl,l+I

( 1 , a , e (a,x ))E Ma

.

ADMISSIBLE ORDINALS

363

Thusletf(a,x)= (l,a,e(a,x)). To show that 9Z!satisfies A.2 (i) it remains to find a recursive function j so that for all%-good X,j : Mh 5, M , and for a EM,, [@)I = la1 + 2. Let el be a recursive function such that for all a, t [el(a)](t,MIO,) = a. Then for %-good A, aEM,

+ Vt[eI(n)](t,Mlol)E =M a

,

~ ( l , O , e l ( a ) ) E M h & l ( l , O , e l ( a ) ) Jlal = t 1.

Also a 4M A 4 ( 1, 0, el@))$ M A . Thus let j ( a ) = ( 1 , 0,e (( 1,0, el (a)))).

A.4. Lemma. There are functions j , p , 0 such that i f % is a good notation system and X is %-good, then 3n, is acceptable in terms of j , p , @ . Proof. I t remains to find v andp. It is easy to see that a 0b = (2,a, b ) has the desired property. To find p 3 let e be a primitive recursive function such that for any n and X E w , the range of ht[e(n)](t,X)equals (0) U the range of A t . { n } ( t , X ) .Then for a EM,,

and if Vx{n}(x,MIuI) EM, then

Thus it is easy to see that we can choosep(a,n)

= (l,a,e(n)).

In view of Lemma A.4, to prove the Coding Lemma it suffices to prove the following: A S . Theorem. Let W = ( M ,11) be acceptable in terms of j , p , 8 . (9 IMI E Ad(Tm ); (fi) For e v e v Xi- formula cp(ul, ..., un) of LP(Tm), there is a primitive recursive function h such that

364

W. RICHTER and

P. ACZEL

Furthermore, h is completely determined by the functions j , p , 8 ,a member u o E M , and the formula q. (iii) Let F be an ordinal function which is I M 1-partial recursive in T, . Then there is a recursive function k such that for al, ..., a, E M , if F(lall, ..., la,l) isdefined then k(al, ..., a,)EMandF(lall ,..., la,/)< Ik(al, ..., a,)l. (iv) X E w is I MI -r.e. in T, i f f X 5 M.

,

The remainder of the appendix is devoted to the proof of Theorem A S . Suppose % = ( M ,11) is acceptable in terms of j , p , 8. Let u o E M and luol = 0. In Lemmas A.6-16 below the reader should observe that the functions described are either independent of the particular acceptable system or are completely determined by j , p , @and uo. (In Lemma A.8 the choice of e is independent of %; in Lemma A.9 an index of h can be found as the value of a recursive function of the indices of the f i , giwhich is independent of%.) A.6. Lemma. There is a recursive function +M such that: (i) a E M & b E M 3 a tMb E M & la tMb I > max { l a (ii) a tMb E M * a E M & b E M .

b I};

Proof. Let e be a recursive function such that

andleta tMb=p(u0,e(a,b)).

3A.2. ‘MI-recursion. We next define a class of partial number-theoretic functions based on%. These functions behave very much like the functions partial recursive in the type 2 functional E of Kleene [9], where for f E w w , 0 if 3 r [ f ( t ) = O ] , 1 otherwise.

ADMISSIBLE ORDINALS

365

Using these functions we are able to carry out computations involving 5% which are needed to show that IMI is admissible. The following definition by schemata of the predicate {z}%(x) = y parallels the corresponding definition by Kleene [9] of the partial recursive functionals of finite types. As described in [9] this definition by schemata may be viewed as a transfinite inductive definition. The essential difference here is that there are an infinite number of starting functions in the case SO. Thus the characteristic function of each Ma for a < IMI is given outright. In the following, x a n d y are abbreviations for xl, ..., x, andy ...,y m , respectively.

0 if x EMla,,

S0.a ( ( 0 ,~ , a ) } ~ ( x=)

SI.

i

1 otherwise,

for each a E M ;

{(l,n)}%(x)=xl + 1 ;

~ 2 .{ ( 2 , n , q ) j C " [ ( x ) =;q ~ 3 . {(3,n)}* S4.

(XI

= x1 ;

{ ~ 4 , n , a , b ) } 3 " ( ~ ) ~ { a } " ( { b } 3 " ( ;x ) , x )

s5.

(

S6.

{(6,n,k,a)}- (x) = {a}* (xl) ;

{(5,n+l,a,b)}m(0,x)- {u}*(x) {( 5, n + 1,a, b )Im ( y + I , x) = {bjM( y ,{ ( 5 ,n + I , a, b)Iw ( y ,x), X) ;

where x1 is obtained from x by movingxk+l to the front.

s8. {(g,n,a)jx (x) = E ( h t . {a}w(t,x)) ; where both sides are undefined if for the given x, A t . {a>im( t , ~is)not total.

S9.

{(9,n+rn+l,rn)}*(z,~,y)={ Z } ~ ( X ;)

Since we are defining only partial number-theoretic functions there is no S7 clause. {z}" is called them-partial recursive function with index z. is%-recursive if it is everywhere defined. Note that if z is the index of an %-partial recursive function, then ( z ) ~is the number of variables of the function. It is easy to prove the standard theorems of recursive function

{zp

366

W. RICHTER and P. ACZEL

theory with the exception of the normal form theorem. In particular the Kleene S--m-n theorem, the Kleene second recursion theorem and the theorem on definition by cases are proved exactly as in 191. Thus we have:

A.7. Lemma. For each m 2 1 : there is a primitive recursive function S m ( z , y I ,...,y,) such that if f ( y l , ...,y , , ~ ) isan %-partial recursive function with index z then for each fixed y ...,y , , Sm(z,y , ...,y , ) is an index of f ( y l ,...,y m , x ) a s a f u n c t i o n o f x .

,,

,

A.8. Lemma (Second recursion theorem). Given any %-partial recursive fiinction f ( z , x ) an integer e can be foutid such that { e } m ( x )= f ( e , x ) . A.9. Lemma. I f f n , J ~g a, r e m -partial recursive then the function

is Cm -partial recursive. A.10. Lemma. I f f is %-partial recursive, then so is p y [f ( x ,y ) = 01.

A.1 I . Remark. I t is clear from scheme SO and the fact that the %-partial recursive functions are closed under composition, primitive recursion and the p-scheme, that each function partial recursive in some M,, where (Y < [ M I , is -partial recursive. 'M I t is convenient to deal with functions ofjust one variable. Let { z } [ a ] = { Z ) ~ ( ( U ~ ..., ) , ( a ) ( z ) l A land ) let D = { ( z , a ) : { z y m[a] is defined}. The inductive definition described by schemata S&S9 associates with each ( z , a )E D an ordinal as follows. I(z,a)l'm= 0 if ( z ) is ~ 0, 1, 2, or 3, that is if {z}w((a)o, ..., (a)tz),-,) is defined by one of SO-S3. In case S4, letting a = (a)()> (a)(,)l 1> -.'I

L

{z}" [a]= { z j m ( a )= {by'({cfm(a),a)= for some b , c. Thus let

{ h f Mi({cYrn~ a l , a ) ]

ADMISSIBLE ORDINALS

367

The cases S5, S6, and S9 are similar. For example in case S9,

Thuslet I(u,(z,a,y))lq = I(z,a)l'm {z)" [a] = {z)*

+ 1 . In case S8 we have

(a)= E(ht. {b}M(t,a)) = E(ht. {b)* [(t,a)])

for some b. In this case we define

The ordinal function 1 1% makes possible proofs by induction. The following lemma and corollary are a generalization of the fact that a function recursive in E is actually recursive in0, for some a < w1 (where Ois from Kleene P I ).

A.12. Lemma. There are recursive functions f and g such that (i) ( z , a ) E D o g ( z , a ) E M ; (ii) If (z,a) E D then for all x,

Proof. The recursive functions f and g are defined siniultaneously by the Kleene second recursion theorem of ordinary recursive function theory. (ii) and (i) in the direction * are then proven by induction on l(z,a)Icm. (i) in the direction is proven by induction on I g(z,a)l. A rigorous proof would require an elaborate definition of f a n d g involving a number of auxiliary functions arising from applications of the S-m-n theorem of ordinary recursive function theory. Instead of this we give an informal description suppressing explicit reference to most of the auxiliary functions. We begin by assuming (z,a) E D and show in case Si how f (z,a) andg(z,a) must be defined in this case so that they satisfy (ii) and (i) in the direction *. Then we show in S i that iff (z,a) and g(z,a) are defined as in S i theng(z,a) E M implies < z , a )E D . The reader familiar with the Second Recursion Theorem will have

W. RICHTER and P. ACZEL

368

no trouble in showing, if desired, that there actually exist such recursive functionsfandg. Case SO. (z,a) E D and

=i 0

if (a10 -fl(z),l

1

otherwise.

7

Thus let g(z,a) = ( z ) ~and choosef(z,a) so that for all x,

Case S’O. z = ( 0 , l , ( ~ ) andf(z,a) ~) = ( z ) ~ E MSince . (z)2 E M , {z}“ [a] N {(O, l , ( ~ ) ~ ) } ~ ( ( is a )defined ~) by clause SO in the definition of { }” .

The definition of f a n d g is trivial in cases Sl-S3, and easy in cases S5, S9. We consider in detail cases S8 and S4. Case S8. (z,a) E D and {z}” [a] = E(Xt.{b}%

o

if 3t.{bYm [ ( t , a ) ]= 0 ,

[ ( t , P ) ] )=

1 otherwise.

where b = ( z ) ~ By . the Induction Hypothesis,

and for all t , x,

Since 3n is @restricted productive we can find u E M such that for all t , Ijg(b,(t,a))l< IuI. ( m r e precisely u = p(uo,e) where for all 1, { e } ( t , M , u o l ) = jg(b,(t,a)). e of course depends ong, a and b.) Then by A.2.(i),

ADMISSIBLE ORDINALS

369

where these reducibilities are uniform in a, b, t. Then from (2), (3) we can ] Then find a u (depending on a , b ) so tfiat for all t , {b}%[ ( t , ~=) {u}(t,Mlul). choose w so that

Let g ( z , a ) = j ( u ) . Then choose f ( z , a ) so that for all x,

It follows from (4) that f(z,a)satisfies the desired equation. Case S'8. z = ( 8 , n , b )andg(z,a)EM. S i n c e g ( z , a ) = j ( u ) , j ( u ) E M ;since j : M 5 , M , u EM; since u = p(uo,e) E M we have by A.2 (ii), for all t, {e}(t,MIUol) = jg(b,(t,a)) E M and hence g ( b , ( t , a ) )EM. Also by A.2(i),(ii), for all t.

Hence by the Induction Hypothesis, for all t , ( b , ( t , a )E) D,i.e. {b}% [(t,a>] is defined. Hence {z}~[a] --E(Xt. {byx [ ( t ,a ) ] )is defined, i.e. ( z , a ) ED. Case S4. (x,a>E D and {z>" [a] = {b}% [ ( { c } ~[a] , a ) ] Let . d = ({c>" [a] , a ) . T h e n ( c , a ) E D a n d ( b , d ) E D , and I(c,a)lT, l(b,d)lrK< l(z,a)l". By the induction hypothesis, g(c,a) E M

and

g(b,d)E M ,

W. RICHTER and P. ACZEL

37 0

and for all x,

We begin by showing how to choose g(z,a) so that g(z,a) E M and

By (5) we can find a u (depending on a, b,c) such that for every x,

By A.2(ii), p(g(c,a),u) E M and

Letg(z,a) = p(g(c,a),u) tMjg(c,a). It is clear from (8) thatg(z,a) satisfies (6). To find f(z,a) observe that by (6) and A.2(i),

uniformly in b. d ; and

(10)

Mlg(c,u)l S J ( % c , a ) l )

itMig(z,a)l

'

uniformly in c,a. From (5), (9) we can find u, w ,y so that for all x,

37 1

ADMISSIBLE ORDINALS

where w is obtained by eliminating d in the previous equation by referring back to the definition of d and then using (lo), andy is obtained by using (10). Thuslet f(z,a) = y . C a s e S Y 4 . z = ( 4 , n , a , b ) a n d g ( z , a ) E M Since .

and henceg(b,d)EMand Ig(b,d)l
A.13. Corollary. Let X C w . (i) X i s 92-r.e. iff X & M ; (ii) X is%-recursive iff X 5,Ma for some a < IMI; (iii) If h is %-partial recursive there is a recursive function k such that for all a, h ( u ) E M * k ( u ) E M & Ih(u)IS Ik(u)l ;

-

(iv) If h : w + M and h is %-recursive, then Sup {I h(x)I : x E w } < I MI

{ z ) (~n ) is Proof. (i) If X is %-r.e. then there is a z E w such that n E X defined. So i f g is as in Lemma A.12 then n E X O g ( z , ( n ) )EM. Thus X 5, M . Now suppose h is a recursive function such that h : X 2, M and Then, choose e so that {e}%(n)- ( ( 0 , I,h(n))}%(O).

n EX

--

h( n ) E M ( ( 0 , l,h(n))}%(O)

(elTK( n )

is defined

is defined.

W. RICHTER and P. ACZEL

37 2

Thus X is %-r.e. (ii) i t follows from Remark A . l l that i f X is recursive inMa for some (Y < / M I , then X is 9Zrecursive. For the other direction it suffices to show that each total function { z } ~is recursive in M a for some a < IMI. By A.12, for all n,

Choose e so that for all n , {e}(n,MlUo,)= jg(z,n) and let c = p ( u o , e ) . Then c E M a n d I c l > Ijg(z,n)l for alln.Hence for alln,Mlg(z,n)l ( n ) ) m M , g ( z , ( n ) )' l ) Then from (1 l), {z}" is recursive in MICI. (iii) L e t h = {z}" , a = ( a ) , andk(a)=p(g(z,a), f(2,a)). F o r h ( a ) E M , { ~ } ' ~ [= ah (]a ) E M ; h e n c e b y A . l 2 , g ( z , a ) E M a n d f o r allx,

Thusk(a) E M and

(iv) Let k be as in (iii). Then for allx, k ( x ) E M and Ih(x)l < Ik(x)l. Choose e so that for all x, k(x) = {e}(x,MlUol).Then p(uo,e) E M and for A x , Ih(x)l
5A.3. Selection. Using @and the fact that by SO every Ma is W-recursive uniformly in a notation for a , given a E M v b E M it is possible to decide %-recursively whetherlal
ADMISSIBLE ORDINALS

37 3

Using A.9 it is easy to see that d is 772-partial recursive. If a E M & la I 2 Ib I thenh(a,b) E M a n d la1 2 la v bl < Ih(a,b)l;Hencea EMlh(a,b)l& b $Mlal and so{kh(a,b)}'lr(a)=O& { I ~ ( a ) } ~ ( b 1. ) =Thusd(a,b)= 0. Similarly, if Ibl < la1 t h e n h ( ~ , b ) E M a n d e i t h e r a E M ~ ~ (b~ E , ~M) l~a& Io r a $ M l h ( a , b ) l and hence by the definition of d, d(a, b) = 1. The following argument is similar to Gandy's unpublished proof of the existence of selection functions associated with functionals of type 2.

A.15. Lemma. ljrzere is an 5%-partial recursive function u such that if A t . {z}(t) is total then 3t { z } ( t ) E M * u(z) is defined & {z} (u(z)) E M .

Proof. L e t g be the recursive function of A.12 and let e be obtained from the second recursion theorem (Lemma A.8) SO that

e is found by using the recursion theorem in a manner similar to the proof of Kleene [9, XVI]. L e t y be the least t such that {z}(t) EM; equivalently,y is the least t such that {e}m(t,z) = 0. We show by induction on x that {e)'M(y-x,z)=x for O < x < y . T h i s i s true i f x = Osincein thiscase {e}' (y -x, z) = {elm( y ,z) = 0 = x. Suppose x > 0. Then by the induction m hypothesis {e)cm(y-(x-l),z) =x-1. In particular, since {e} (y-(x-l),z) is defined, g ( e , ( y - (x-l),z)) EM. Since also {z}(y-x) $ M , we have d ( { z } ( y - x ) , g ( e , ( y - ( x - l ) , z ) ) ) = 1. This implies by (12),

374

W. RICHTER and P. ACZEL

Settingx = y this gives {e}m(O,z)= y . Thus it suffices to let u ( z ) = {e}m ( 0 , z ) .

A.16. Corollary.Let Q Cn+' w be%-r.e. Then there is an W-partial recursive function X x v y Q ( x , y )of n variables such that for all x ,

Proof. Let Q ( ~ , y ) - { z } ~ ( x , y is) defined. Then Q(x,y)

{ S m ( z ,x)]" ( y )

(j

-g(Styz,x),

is defined

(y,) E M .

Let e be a recursive function such that { e ( x ) } ( y )= g(Sm(z,x), ( y ) ) .Then

Thus let vyQ(x,y) N ue(x). The following lemma summarises some of the properties of %-r.e. relations.

A. 17. Lemma. (i) I f f is m-partial recursive, then the relation f ( x )= z is %-r.e. (ii) I j Q ism-r.e., then the function

f

(XI

=

z if

QW),

undefined otherwise ,

is %-partial recursive. (iii) The relations y E M & 1x1< l y l ; y E M & 1x1 I l y l , E~ M & ly I 51x1 are W-1.e. (iv) The W-r.e. relations are closed under conjunction, disjunction, universal and existential number quantification, and inverse images by %-recursive functions.

Proof. Suppose j is W-partial recursive. Let

ADMISSIBLE ORDINALS

375

(2,1,0) if x = z ,

e ( x , z )= 0 otherwise, andg(x,z)

= {e(f(x),z)}*(O). Theng is %-partial recursive and g(x,z) is defined e e ( f ( x ) , z ) )= (2,1,0)

-f(x) =z . To prove (ii) let Q(x) -g(x) is defined, and letf(x) = O - g ( x ) + z . To prove(iii)wehaveyEM& Ixl
ism-partial recursive. Then

-

T o treat existential quantification, let Q(x,y) by A.16,

-

{z}”(x,y) is defined. Then

3 ~ Q ( x , -Q(x,vQ(x,v)> v) {z}“ (x, v y Q ( x , y ) ) is defined.

$ A.4. Proof of Theorem A5. A.18. Lemma. There is an 9?i-partial recursive function g such that if z E M and {e}M( u , x ) is defined for each u E MI,, then

W. RICHTER and P. ACZEL

316

Proof. L e t g ( e , z , x )= v y Q ( e , z , x , y ) where

A.19. Lemma. For each ordinal function Fprimitive recursive in T, there is an %-partial recursive function h, such that if a l , ..., a , E M then h,(a,, ...,a,) EM and

Proof. We shall use the characterisation of the ordinal functions primitive recursive in K K given by the following schemata (see [S]): 0 if x E M , (i) F ( x , y ) =

1 otherwise

(ii) F ( x ) = x i (iii) F ( x ) = 0

(iv) F ( x ) = x + 1

x ifu
y

otherwise.

(vi) F(x,y)= G(x,H(x),y) (Vii)

W , y )= G(H(x),y)

(viii) F ( z , x ) = G(sup,,,F(u,x),z,x)

.

For (i) we first need an %-recursive function k such that I k ( n )1 = n for all n. Let k ( 0 ) = u o and

ADMISSIBLE ORDINALS

317

Then k has the desired property. Now in case (i) let lull = 1 and

A.20. Lemma. (i) I MI is closed under functions primitive recursive in Tw . (ii) Let R C On be primitive recursive in Ten. Then

is 371-r.e.

Proof. (i) is an immediate consequence of A.19. Let A = {(al, ...,.a,). a l , ...,u, E M & R ( l a l J , ..., la,l)}andFbe therepresentingfunctionofR. Then using A. 19,

(al ,...,~ , ) E A-al -al

,...,a, EM&F(Iall ,..., la,l)= 0 ,...,a, E M & h F ( a l ,...,a,)EMI .

ThusA = nM n h$M1 which is%-r.e. using A.17.

W. RICHTER and P. ACZEL

378

A.21. Lemma(proof of A.5.1). IMI EAd(T%).

Proof. By Theorem 3.7 it remains to show that if R sive in T, , a < [MI and

3 0 n is primitive recur-

then there is a < A < ]MI such that (14)

Vx < h 3 y < hR(a,x,y) .

Suppose ( 13) holds. Let Ic I = a and f (x) = vy [ y E M & R( Ic I, Ix I, I y I)]. Then f ism-partial recursive by A.20. L e t g be the %-recursive function defined by: g(0) = c and

Then Ig(n)l < Ig(n+1)1 and 1x1 < Ig(n)l * If(x)I 5 Ig(n+1)/.Let h = Sup,,, I g ( n ) I . a < h < IMI by A.l3(iv). We show that h satisfies(l4).

( 14) then follows from ( 15) and the definition off.

A.22. Lemma(proofofA.S(ii)).IfR c n l M I is ]MI-r.e. in T, then thereisa primitive recursive function h such that a l , ...,a, E M & R( I all, ...,I a,

I)

-

h(al, ...,a,) E M .

Proof. Let R be MI -r.e. in Tm . Then there is a primitive recursive relation S such that

R(a

ADMISSIBLE ORDINALS

31 9

It follows that A is W-r.e. Hence by A.13 there is a recursive function h , such that A = hi1(,M). It remains to find a primitive recursive h. By the S-m-n theorem for ordinary recursive function theory there is a primitive recursive function such that, lettinga = a l , ..., a,, {g(a)}(x,Mluol)= h l ( a ) for all x.Let h(a) = p(u,,, g(a));then h is primitive recursive and

Remark. The above proof is the only place where we use the fact that p is primitive recursive instead of just recursive. A.23. Lemma (proof of A.5 (iii)). Let F be [ MI-partial recursive in T q . Then there i s a recursive function k such that feral, ...,an EM, ifF(lal1, ..., la,\) isdefined then k(al ,..., a , ) E M a n d F ( l a l I ,...,la,/)< Ik(al ,...,a,)l. Proof. By the normal form theorem relativised to Tm , the graph of F is I MIr.e. in Tq ; hence by A.22 there is a recursive function h such that

Let f ( a ) = v y [ y E M & h ( a l , ...,a,,y) EM].Thenfis %-partial recursive. Hence by A. 13 there is a recursive function k such that

Then for a EM, if F(lall, ...,la,/) is defined

380

W. RICHTER and P. ACZEL

A.24.Lemrna(proofofAS(iv).XCois [MI-r.e.in T% i f f X < , M .

Proof. M is IMI-r.e. in T% since n EM- 301 < / M I . Tq (n,.). Hence if X<,M, X is also ]MI-r.e. in 27%. Now suppose X is [MI-r.e. in T, ,and let k be an W-recursive function such that Ik(n) I = n for all n (see the proof of A. 19). Then using A.22 there is a recursive function h such that n EX-

Ik(n)I E X

-hk(n)EM

X is the inverse image of t h e W -r.e. set M under the W-recursive function h k and hence i s V - r . e . by A.l7(iv). HenceXS, M b y A.I3(i).

Remarks. (i) The definition of an acceptable ordinal system differs slightly from that Gven in [ 161. The requirement t h a t j (called there g ) be a many-one reduction of M to M is necessary for the proof of A. 12 and its omission was an oversight in [ 161. The other change is the requirement that p be primitive recursive instead of recursive, and as mentioned above this is only to ensure that the function h of A S is primitive recursive. (ii) AS(iii) is not used elsewhere but it appears to be of interest in its own right and its proof comes naturally from our construction. It was used in [ 161 in an earlier proof of some of our results but is not needed in our present formulation. (iii) The method we have used in proving the Coding Lemma is to utilize techniques from the well-developed theory of recursive functionals of type 2. In particular, the crucial results needed about W-recursion, namely the Boundeness Theorem (A.13 (iv)), and Theorem A. 15 on the existence of selection functions are proved by standard methods from the theory of recursive functionals of type 2. On the other hand the theory of recursive functionals may be regarded as a part of the theory of inductive definitions. This suggests that an ultimately simpler and more elegant proof of the Coding Lemma in a more general setting can be provided within the “pure” theory of inductive definitions.

ADMISSIBLE ORDINALS

38 1

References [ 11 S. Aanderaa, Inductive definitions and their closure ordinals, this volume. [ 21 P. Aczel and W. Richter, Inductive definitions and analogues of large cardinals, in:

Conference in Mathematical Logic, London '70 (Springer, Berlin, 197 1) 1-10. [ 3 ] J. Barwise, Infinitary logic and admissible sets, J. Symb. Logic 34 (1969) 226-251. [4] J. Barwise, R.O. Gandy and Y.N. Moschovakis, The next admissible set, J. Symb. Logic 36 (1971) 108-120. [5] D. Cenzer, Analytic inductive definitions, Abstract, Notices, A.M.S. 703-E2, Vol. 20 (1973) pA376. [6] K. Devlin, An introduction to the fine structure of the constructible hierarchy, this volume. [7] W.P. Hanf and D. Scott, Classifying inaccessible cardinals, Notices of the A.M.S. 8 (1961) 445. [8] R.B. Jensen and C. Karp, Primitive recursive set functions, in: D. Scott (ed.) Axiomatic Set Theory, Proceedings Pure Math. 13 (Amer. Math. SOC.Providence, R.I., 197 1) 143-176. [9] S.C. Kleene, Recursive functionals and quantifiers of finite types I, Trans Amer. Math. SOC.91 (1959) 1-52. [ 101 S. Kripke, Transfinite recursion, constructible sets and analogues of cardinals, in Lecture Notes prepared in connection with the Summer Institute on Axiomatic Set Theory, held at Los Angeles (1967). [ 111 A. Gvy, The sizes of the indescribable cardinals, in: D. Scott (ed.) Axiomatic Set Theory, Proceedings Pure Math. 13 (Amer. Math. SOC.,Providence, R.I., 197 1) 143-176. [ 121 Th.A. Linden, Equivalences between Godel's definitions of constructibility, in: J.N. Crossley (ed.) Sets, Models and Recursion Theory (North-Holland, Amsterdam, 1967) 33-43. [ 131 Y.N. Moschovakis, Elementary Induction on Abstract Structures (North-Holland, Amsterdam, 1973). [ 141 H. Putnam, On hierarchies and systems of notations, Proc. Amer. Math. SOC.15 (1964) 44-50. [ 151 W. Richter, Constructive transfinite number classes, Bull. Amer. Math. SOC.73 (1967) 261-265. [ 161 W. Richter, Recursively Mahlo ordinals and inductive definitions, in: R.O. Gandy and C.E.M. Yates (eds.) Logic Colloquium '69 (North-Holland, Amsterdam, 197 1) 273-288. (171 H. Rogers, Jr., Theory of recursive functions and effective computability (McGrawHill, 1967). [ 181 G . Sacks, The 1-section of a type n object, this volume. [ 191 J.R. Shoenfield, Mathematical logic (Addison Wesley, Reading, Mass., 1967). [ 201 C. Spector, Inductively defined sets of natural numbers, in: Infinitistic Methods (Pergamon Press, Oxford, 1961) 97-102. [21] H. Tanaka, On analytic well-orderings, J. Symb. Logic 35 (1970) 198-204. [ 22) S.C. Kleene, On the forms of the predicates in the theory of constructive ordinals, 11, Amer. J . of Math. 77 (1955) 405-428.

PART IV AXIOMATIC APPROACHES AND GENERAL DISCUSSION

J.E.Fenstad, P. G.Hinman (eds.), Generalized Recursion Theory 0 North-Holland Publ. Comp., 1974

ON AXIOMATIZING RECURSION THEORY Jens Erik FENSTAD University of Oslo

Generalized recursion theory can be many different things. Starting from ordinary recursion theory one may e.g. move up in types over w, or look to more general domains such as ordinals, admissible sets and acceptable structures. Alternatively, one may want t o study in a more general setting one particular approach to ordinary recursion theory, thus e.g. try to develop a general theory based on schemes or fured point operators, or work out a general theory of inductive defmability ,or develop in a suitable abstract setting the various model theoretic approaches such as representability in formal systems or invariant and implicit definability. The approach of this paper is axiomatic. This is nothing new. Of previous axiomatic studies of recursion theory we mention Strong [ 141, Wagner [IS], and Friedman [4]. Our interest in the axiomatics of generalized recursion theory was more directly inspired by Moschovakis [ 101, and any one familiar with his “Axioms for Computation Theories” will soon see our dependence upon his work. Our objective is two-fold: First to contribute to the discussion and choice of the “correct” primitives for axiomatic recursion theory. Second to indicate new results, partly proved, partly conjectural, within the (modified) Moschovakis framework. First one general remark on axiomatizing recursion theory. This may in itself be a worthy objective. Through an axiomatic analysis one may hope to get a satisfying classification and comparison of existing generalizations (technically through “representation” theorems and “imbedding” results). And one may,perhaps, also obtain a better insight into the“concrete” examples on which the axiomatization is based. But it is not clear - and some disagree that the field is at present ripe for axiomatization. Hence we are approaching our topic in a tentative manner. 385

386

J.E. FENSTAD

As Moschovakis in [ 101 we take as our basic relation

which asserts that the “computing device” named or coded by a acting on the input sequence u = ( x l , ..., x n ) gives z as output. Let 0 denote the set of all computation tuples (a, a,z) such that the relation { a } ( u ) = z obtains. It is possible to write down axioms for a computation set 0 which suffice to derive the most basic results of recursion theory, say up through the faed-point or second recursion theorem. However, many arguments seem to require an analysis not only of the computation tuples, but of the whole structure of “subcomputations” of a given computation tuple. Now computations, and hence subcomputations, can be many different things. And in an axiomatic analysis of the variety of approaches hinted at in the opening paragraph of the paper it would be rash to commit oneself at the outset to one specific idea of ‘computation’. In [ 101 Moschovakis emphasized the fact that whatever computations may be, they have a well-defined length, which always is an ordinal, finite or infinite. Thus he proposed to add as a further primitive a map from the set 0 of computation tuples to the ordinals, denoting by la, u,z lo the ordinal associated with the tuple (a, u, z ) E 0. In t h s paper we shall abstract another aspect of the notion of computation. We shall add as a further primitive a relation between computation tuples

which is intended to express that (a’, u ’ , z f ) is a subcomputation of (a, u,z), or, in other words, that the computation (a, u,z) depends upon (af,u’,zf). The basic axioms will state that the relation is transitive and wellfounded.

Remark. In our approach we have chosen functions and computations rather than sets and inductive definitions as basic notions. We have also chosen to exhibit the codes for the computations directly in the axioms rather than tried to develop the theory in a more “coordinate-free’’ or invariant manner. It is, perhaps, still an open question which will be the most “useful” way to organize generalized recursion theory into a theory.

ON AXIOMATIZING RECURSION THEORY

387

The rest of the paper wdl be divided in four sections. In Section 1 we give the basic definition of a computation theory. In this we follow Moschovakis closely, making the modifications necessary due to our use of the subcomputation relation < instead of the length concept. In Section 2 we list some basic facts about computation theories. This is in all essentials a repetition of material from [lo], but is included for the convenience of the reader. In this part of the theory there does not seem to be much difference between the length concept and the subcomputation relation. The important thing is that both allows us to carry over to the abstract setting certain results proved in the “concrete” examples by transfinite induction on associated ordinals, or, alternatively, by a course-of-value induction on “subcomputations”. The basic result here, due t o Moschovakis in the axiomatic setting, is the first recursion theorem. The section concludes with the definition of regular computation theories. Such theories have selection operators, hence we have a reasonable theory for the computable and semicomputable relations. And for this class of theories we can introduce an adequate notion of finiteness, a set being finite if we can computably quantify over it. In Section 3 we discuss the problem how to strengthen regularity. Of several possibilities we have chosen to emphasize two: One is the idea that a “computation” should be a finite object in the sense of the theory; the other is that the theory should satisfy the prewellordering property. The first we can express by requiring that for every (a, u, z) E 0 , the set

is finite in the theory. One formulation of the other says that the set

is computable in the theory (where [la, u,zll is the ordinal of S(a,u,z)). In Section 4 we move beyond the “normality” conditions discussed above, representation theorems and imbedding results being the main themes. Our exposition will be sketchy for several reasons: One is that a complete development would be too long, - another and more important one is that several of the results are still in a preliminary stage. In conclusion I would like to acknowledge my great debts to Peter Aczel and Peter Hinman, who patiently have explained many results and methods

J.E. FENSTAD

388

of general recursion theory to me, and to Johan Moldestad and Dag Normann, who have with great enthusiasm participated in the investigations reported on in this paper.

1. Computation theories: basic definitions In this section we give the basic definitions using the subcomputation relation as primitive notion.

Definition 1 . A computation domain is a structure

\u = ( A , C , N , s , M , K , L ) , where A is the universe, N CI C C A and ( N , s 1N ) is isomorphic to the nonnegative integers. Cis called the set of codes. M is a pairing function on C, i.e.

a,b E C

iff M(a,b ) E C ,

and

M(a,b) = M(a', b') E C

implies

a = a' A b = b'

K and L are inverses to M , i.e. they map C into C and

c = M(a,b) E C

iff a = K ( c ) A b = L (c) .

To facilitate the presentation we introduce some notational convention (following Moschovakis [lo]). We use x, y , z, ... for elements in A . a, b,c, ... for elements in C. i, j , k , ... for elements in N . for finite sequences from A . u, 7,... u, 7 or ( u , 7 ) denotes the concatenation of sequences. And as usual lh (u) = the length of the sequence u. A computation tuple is any sequence (a, u,z) such t h a t a E C a n d l h ( a , u , z ) 2 2. Definition 2. The system (O,<) is called a computation structure on the domain % if < is a transitive relation on the set of computation tuples and 0 is the wellfounded part of <.

389

ON AXIOMATIZING RECURSION THEORY

Thus (a, o,z) E 0 iff the set

is wellfounded with respect to the relation < (a, u,z), then (u’, u’,zf) E 0.

<. Note that if (a, u,z)

E 0 and

(a’, uf,z’)

Note: We have built into our definition the convention that something which looks like a computation, i.e. an arbitrary computation tuple (a, o,z), is not a computation if and only if its “subcomputation tree” contains an infinite descending path. In practice this may not always be so, but if an attempt at a computation stops after a finite number of steps without giving a bonafide computation, we can always start repeating ourselves in some suitable way so as to obtain an infinite descending path. As in

[lo]

we shall make use of the notions of partial multiple-valued (pmv) function and functional. We recall some notations: iff z Ef(o). f(4 z iff z is one value off at u. iff Vz [f(o) + z iff g(o) + z ] . f(o) = g(o) iff f(o) = {z}. f(4 = z iff ~ ( o ) + z A V U [ ~ ( U )* + u =Uz ] . fSg iff VaVz[f(o)+z * g(o)-+z]. A mapping is a total, single-valued function. Let (O,<) be a computation structure on %. T o every a E C and every natural number n we can associated a pmv function {a}: in the following way: -+

{a}:(u)-+z

iff I h ( o ) = n A ( a , o , z ) E O .

Definition 3. Let (O,<) be a computation structure on %.A pmv functionf on A is @-computableif for some .f€ C

i

We call a @-code o f f and write f = ments off.

ti}”, where n is the number of argu-

390

J.E. FENSTAD

A pmv functional on A

maps pmv functions onA and elements o f A into subsets ofA (including the empty subset, 8). cp is called continuous if

Definition 4. Let (@,<) be a computation structure on the domain '21. A pmv continuous functional cp on A is called @-computable if there exists a (p E C such that for all e l , ..., eQE C and all u = (xl, ..., x n ) from A , we have: a) cp({el};l, ..., {eQ>;Q,(I) + z if t + l g n ( e l , ..-,ep,0) z . b) If cp({el};', ..., {e&:Q,u) + z , then there exist pmv functionsgl, ...,g, such that ...,g Q CteQ>:Q andcp(g1, -..,g~,,u)-+z. i . g l L tell:', ii. For all i = 1, ..., Q , ifgi(tl, ..., t,i) -+u , then +

Note : This is the first essential use of the suncomputation relation. For a motivation of the definition of O-computable functional, see [ 10, p. 2091. Definition 5 . Let (@,<) be a computation structure on the domain 91. to,<) is called a computation theory on \21 if there exist @-computable mappings p l , ...,pI3 such that the following functions and functionals are @computable with @-codesas indicated and such that the iteration property holds:

11-VIII. Similar to I and state that the following functions are O-computable: Identity function, the successor function s, the characteristic functions of C and N , the pairing function M and the inverses K and L .

ON AXIOMATIZING RECURSION THEORY

391

X-XII. Similar to IX and state that the following functionals are @-computable: Primitive recursion, permutation of arguments, point evaluation. XIII. Iteration property: For all n,m ~ 1 3 ( n , m is ) a O-code for a mapping S z ( a , x l ,...,x,)suchthatfor alla,xl, ...,x , E C a n d a l l y l , ...,y , € A : (9 tal;+m(x,, ...,x,, Y l , ...,r,> = {S#,Xl, ...,X,>l(Jq, ...>Y,>. (ii) If {a};+”(x,, ...,x,, y ...,y,) + z , then

Remarks. 1 . The missing parts of the definition can be found in [ 10, pp. 205-2061. 2. If we drop the primitive <, the rest can be stated using 0 alone, and we arrive at Moschovakis’ notion of a pre-computation theory. This part seems to contain the basic core (i.e. “pre-Post” theory) of any systematization, including the fixed-point or second recursion theorem. 3. T o every tuple (a,u,z) E 0 there is associated an ordinal Ila, u,zll = the ordinal of the set S(o,o,zl. 0 with this ordinal assignment is a computation theory in the sense of Moschovakis.

2. Computation theories: basic facts 2.1. Inductive generation of theories and equivalence. Let ‘? be al computation domain and let

f = f i,...,f, and

cp=cp1,..-,cpk

be sequences of pmv functions and continuous pmv functionals on A . It is possible to construct a theory PR [f,cp], the prime computation theory generated by f and cp, which in the following precise sense is the least computation theory which makes all the functions f and functionals cp(uniformly) computable. Definition 6 . Let (@,<) and (Of,<’) be computation theories on the same domain ‘u. We say that 0’ extends 0 ,in symbols,

392

J.E. FENSTAD

(dropping, as we usual do, the explicit reference to < and < ’) if there exists a @’-computablemappingp(a,n) such that p(C,N) C N and such that for all n-tuples u, all a E C and z E A i. (a, u,z) E 0 iff (p(a,n), u,z) E 0’. ii. I f (a, u , z ) , (af,u’,z’) E 0 and (a, u,z) < (a‘, u’,z’), then (p(a,n), 0 , ~<’ ) (p(a’,n’), o f , z ’ ) . If 0’ 5 0 and 0 5 O r ,we say that 0 and 0’ are equivalent and write 0 0‘.

-

Remark. It seems that Moschovakis’ motivation for his version of the notion of equivalence (see [ 10, pp. 217-21 81) is even more appropriate for the present version.

As in [ 10, p. 2181 we have the following result which justifies the claim made above. (i) Let ( 0 , < )be a computation theory on V l , and let f and cp be sequences ofpmv functions and continuous functionals on ?I. Then

and if H is any other computation theory on 91 such that 0 < H and f are Hcomputable and cp are uniformly H-computable, then

Remark. There are some difficulties in carrying over (iii) [ 10,p. 2191 to the present frame. We mention this since this is the only example of a result in [ 10, 3 3 1-81 which has not had an immediate counterpart. (The difficulty is that if we pass from 0 to H via a map p and then back to 0 via a map q , the ordinal of (a, u, z ) is less than the ordinal of(q(p(a,n), n ) , u , z), but the former is not necessarily a subcomputation of the latter. 2.2. The first recursion theorem. The theorem was proved by Moschovakis in the axiomatic setting. The proof carries immediately over to the present set-up.

Theorem. Let te,<)be a computation theory on 8 . L e t cp( f,x) be a

ON AXIOMATIZING RECURSION THEORY

393

@computablecontinuous pmv functional over A . Let f ' be the least solution of V x € A : cp(f,x)=f ( x ) . Then f* is @-computable. The theorem is particularly important in discussing the relationship between recursion theory and inductive definability. It corresponds to the fact, and can sometimes be used to show, that I;, inductive definitions has a I;1minimal solution.

2.3. Selection operators. We first give the definition of O-semicomputable and @-computable relations. Definition 7 . The relation R ( o ) is @-semicomputableif there is a @-computable pmv function f such that

R(o) iff. f ( o ) + 0 . The relation R(o) is @-computableif there is a @computablemapping f such that

R(o) iff. f ( o ) = 0 . The existence of a selection operator seems to be necessary in order to prove some of the basic facts about O-semicomputable and @-computable relations, such as closure of O-semicomputable relations under 3-quantification and disjunction, and also to prove that a relation R is @-computableiff R and 1 R are O-semicomputable. Definition 8. Let (@,<) be a computation theory on 'ZI. An n-ary selection operator for (@,<) is an n t l - a r y @-computable pmv function q(a, o) with @-code4 such that (i) If there is an x such that {a},(x, a) + 0, then q(a, cr) is defined and vx [q(a, u) + x * {a} ( x ,0 ) + 01. (ii) If {a} ( x ,a) + 0 and q(a, a) + x , then

(a,x,o,O)<(G,a,o,x).

394

J.E. FENSTAD

Remarks. 1. For a motivation of (ii), equally valid in the present case, see [ 10, p. 2251. 2. By an oversight Moschovakis [ 10, p. 2551 only required {a}(q(a,a), a) + 0 rather than V x [q(a,u) + x * la} ( x ,u) + 01, which is necessary when working with pmv objects. 2.4. The notion of finiteness in general computation theories. Any good general approach to recursion theory must embody a suitable notion of “finiteness”. We repeat the basic definitions from [ 10, pp. 230-2331. Definition 9. A computation theory (@,<) on % is called regular if (i) C = A . (ii) Equality “x =y” is @computable. (iii) (@,<)has selection operators. Definition 10. Let (@,<)be a regular theory on ’u, and l e t B C A . By theBquantifier we understand the continuous pmv functional EB(f)defined by 0 if 3 x E B [ f ( x ) + O ] .

1 if V x E B [ f ( x ) + 11 . The set B is called @-finitewith @-canonical code e , if the B-quantifier Es is @-computable with @-codee. We refer the reader to [ 101 for a list of five properties of this particular notion of finiteness which may justify the claim that it is “natural”. But it would be premature to conclude from this that the present version gives all the properties of ordinary (i.e. true) finiteness necessary for the combinatorial arguments of e.g. degree theory.

3. Extending regularity In this section we will discuss the problem of how t o extend regularity. Of several possibilities we have chosen to emphasize two.

ON AXIOMATIZING RECURSION THEORY

395

A. The idea that a “computation” is a finite object in the sense of the theory seems to play an important role in many arguments of recursion theory. “Computation” is not a primitive of our system, but a perhaps satisfactory approximation consists in requiring that a computation tuple depends on only a finite number of other tuples, i.e. for every (a, u,z) E 0, the set

is finite (uniformly in (a, u, z)) in the sense for Section 2.4.

Note: A more complete technical statement would require that there is a 0 computable mappingp(n) such that for each (a, u,z) E 0 {p(n)},(a, u,z) is a @-canonical code for S(,,,,z). And since we are dealing with sequences of arbitrary finite length, we assume some suitable coding convention.

Remark. A computation theory in the sense of Moschovakis is a pair (0, )lo), where 11 is a map from 0 into the ordinals. Using a length-function it seems to be difficult to capture the idea that a “computation” should be a finite object in the sense of the theory. As a first approximation one could consider the set

But one cannot outright say that this set, which is the set of all computations with smaller length, is a finite object in the theory. Indeed, if this requirement ismade, the set of natural numbers necessarily will be finite in 0. Some restriction must therefore be added and Moschovakis proposed to compare computations of equal length, i.e. he required the finiteness of the set

It is possible to proceed on this basis, and it may have some advantage later on (see Section 4), but it is not in my opinion a satisfactory conceptual analysis of the motivation behind normality (see [ 10, p. 2331).

B. The prewellordering property is another important tool in general recursion theory. One formulation is as follows. Let ~la,u,zll denote the ordinal of the

396

J.E. FENSTAD

set S,,,,,,,, (a, u,z) E 0. The theory (@,<)has the prewellordering propery if there exists a @-computable function p ( x , y ) such that if either x E 0 or y E 0 then p(x,y) is defined and single-valued, and whenevery E 0, then

in other words, the set {x 1 x E 0 A ((x(1 <_ ( ( y11) is @-computable, uniformly iny. We shall make some remarks on the relationship between the finiteness of the sets S(a,u,zland the prewellordering property. It is convenient to introduce some terminology.

Definition 11. Let (0,<) be a computation theory on a domain ’ZI. 1. (El,<) is called p-normal if it is regular and has the prewellordering property (see B above). 2. ( 0 ,<) is called s-normal if it is regular and the sets S(a,o,zlare (uniformly) @-finite for ( a , u , z ) E 0 (see A above). 3 . ( 0 ,<)is called strongly normal if it is both p-normal and s-normal.

Remarks. 1. “Normality” in the sense of Moschovakis [ 101 requires that the sets in (*) above are uniformly @-finite. 2. I f the domain is @finite, normality in the sense of Moschovakis, p normality, and strong normality lead to essentially the same class of theories, viz. the so-called “Spector theories” (see Section 4.1). (i) S-normality implies a “weak” form of the prewellordering property: We can define a @-computable pmv function q(x,y) such that if x,y E 0,

then q(x,y)= 0 iff IIxII I llyll .

This is so since we can computably quantify over finite sets, hence have the following recursion equation

ON AXIOMATIZING RECURSION THEORY

397

< is O-semicomputable, then we have the prewellordering property. In this case we have the following recursion equations for the function p(x,y): (a) p(x,y) = 0 if V x ’E S, 3y’[y’< y A p(x’,y’)= 01. (b) p ( x , y ) = 1 if 3x’[x‘< y A V y ’ E Sy p(x’,y ’ ) = 11. (ii) If we in addition assume that the relation

Note : Since < is defined for all tuples of the form (a, u , z ) and O is the wellfounded part of <, the assumption that < is O-semicomputable is rather problematic. However, the following argument may add some plausibility. An arbitrary computation tuple ( a , u,z) may or may not represent a “true” computation. But as soon as we are given a set of instructions a and an input sequence u we should be able to start “generating” the “subcomputations”, and this is what the O-semicomputability of < is intended to express. This argument seems to carry force in the case of single-valued computations. It is in the case of multiple-valued computations that we would have difficulties in extending a relation
,I)

(v) P-normality does not imply s-normality. “Ordinary” recursion theory over w can be constructed in such a way as to provide a counterexample.

398

J.E. FENSTAD

4. Beyond normality Moving beyond normality there is one fundamental distinction to make: Either a theory (@,<) has a domain which is finite in the sense of the theory, or its domain is infinite. 4.1. Theories over finite domains. In analogy with Moschovakis [ lo] we call a p-normal (or, which amounts to essentially the same - see the remark following Definition 1I in Section 3 - strongly normal) theory (@, <) on a domain 8 a Spector theory if the domain A is @-finite. For such theories one can prove a great number of results which were originally established for hyperarithmetic and hyperprojective theory (see Moschovakis [9] and [ lo], - a detailed exposition within the axiomatic set up can be found in Vegem [ 161).

Hyperarithmetic theory over w is the theory of recursion in 2E,hyperprojective theory is a generalization of t h s to more general domains. How different is an arbitrary Spector theory from recursion in some functional over the domain, i.e. what kind of representation theorems d o we have for Spector theories? We state some results. But first a few terminological remarks. In many cases it is convenient to assume that the search operator v is computable in 0, where v ( f ) = { x If ( x ) -+ 0). It is known from [ 10, p. 2661 that if 0 has a selection operator, then 0 is weakly equivalent to @[v], hence, for reasons detailed there, we may as well work with @[v],which allows us greater freedom in defining pmv objects. For convenience we also introduce the following notations: sc(@) = the set of all @-computable relations on '21. en(@) = the set of all O-semicomputable relations on 9 ( . (The notation en (O), the envelope of 0, is taken from Moschovakis [ 1 11.) 4.1.1. For any s-normal (0,<) on a domain '21, there exists a continuous partial functional F on 91 such that 0 - PR [F] .

This result is jointly due to J. Moldestad and D. Normann, and the proof uses the uniform finiteness of the sets S(a,o,z).D. Normann has also adapted the

ON AXIOMATIZING RECURSION THEORY

399

main result of Sacks [ 121 to show 4.1.2. For any Spector theory over w there exists a total functional F such that (and

sc (0)= sc ( F )

en ( F )C en (0))

Remark. D. Normann actually proves a more general result: Let M be a countable admissible set which satisfies local countability and A O-dependentchoice, then: (i) M = L & ~ ) for , some generic class K E O(M) = M n On; (ii) M is Kadmissible, and (iii) O(M) is the least 0 such that L f is K-admissible. - 4.1.2 is an immediate corollary. (Note that the somewhat complicated hierarchy for type-2 recursion used by Sacks [ 121 is avoided in this approach. I t may be essential for the k-section result.) We expect that Normann’s result can be adapted to admissible sets with urelements, yielding a generalization of 4.1.2 to arbitrary countable domains. In Moschovakis [ 111 we find a counterexample to show that 4.1.2 cannot be lifted from sections to envelopes:

4.1.3. mere exists a Spector theory 0 over w such that en (0)f en (F),for all total type-2 functionals over w . The problem remains to characterize those Spector theories which are equivalent to prime recursion is some total type-2 functional over the domain. This is, of course, only one step toward a full classification of Spector theories. In this connection a Gandy-Spector theorem or, better, a normal form theorem for the class en (0)may be of interest. It is not at all clear how to formulate such theorems in a sufficiently general form. Our proposal for a “weak” version: Definition 12. Let P ( X , u) be a second order relation over 3 .P ( X , u) is called @computable with indexp if whenever {e}@is the characteristic function of some set X , E sc (O), then {PIo(e, 4

0,

if P(X,, u)

1,

ow.

+

400

J.E. FENSTAD

We say that the theory 0 has the weak Gandy-Spector property if whenever R € en (@), there is a @-computableP such that

R ( o ) iff. ( 3 X € s c ( @ ) ) P ( X , o ) 4.1.4. Evely Spector theory has the weak Gandy-Spector property. This has been proved by J . Moldestad. It remains an interesting problem to find other “natural” kinds of normal form theorems which can be used to characterize certain classes of Spector theories. Remark. The original or “strong” Candy-Spector theorem for hyperarithmetical theory provides a P which is first order with respect to the language adequate to describe the domain %. We shall comment on one more topic. The connection between inductive definability and hyperprojectivity was investigated by Grilliot [ 51 . His results was adapted to the present frame by Moldestad IS]. Let Ind(A) denote the class of relations which are inductive in some operator of class A, i.e. reducible to some I’”, where r € A. 4.1.5. Let 0 be a Specto-r theory on \u and R a sequence of @-computable relations on BI. (i) Ind(X2(R)) 5 en(@). (ii) 0 - PR [ R , = , v , E ] , i f a n d o n l y ifInd(Z2(R))=en(@). Here the implication from right to left in (ii) goes beyond Crilliot [5], and gives a certain characterization of the “minimal” Spector theory on a domain yl. A general result would give a necessary and sufficient condition on 0 for the validity of the equation Ind(X,(R)) = en(@). 4.2. Theories over infinite domains. In Moschovakis [ 101 the case of infinite domains is particularly satisfying. Any Friedberg theory in his sense is a recursion theory generated in a “natural” way from an admissible prewellordering of the domain.

ON AXIOMATIZING RECURSION THEORY

401

In our case the situation is more complicated. Our definition of normality does not automatically give us a prewellordering of the domain. In order for us to proceed we must add extra assumptions on the domain. The goal will be to abstract the “natural” or “minimal” recursion theory associated with an admissible set (possibly with urelements, - for this concept see Barwise [2]).

Remark. The more complicated situation in our set up is perhaps not a too serious disadvantage. There seems to be recursion theories over infinite domains (i.e. infinite in the sense of the theory) on which there is no natural associated prewellordering (see [7]). And there seems to be admissible sets where the prewellordering associated with the rank function is not admissible in the sense of [ 101. (Let A be an admissible set such that A is uncountable but A n On is countable.) Such examples should not be denied their proper existence in an axiomatic analysis of computation theories. We admit at once that we do not yet claim to have a “good” definition of computation theories over infinite domains. We shall make some preliminary suggestions, and hope that further work will lead to a “correct“ analysis. Let (@,<)be an s-normal theory on the domain \zI. The basic intention is to express that there is a suitable correspondence between the complexity of the domain 2l and the complexity of the computations in 0. Some partial ordering of the domain seems to be necessary in order to code whole computations in a “natural”, i.e. order preserving way into the domain. We make the following proposal: 4.2.1. mere is a @-computablepartial ordering 5 of A such that the initial segments of 5 are well-founded and (unifomiy) @-finite.

A weak way of expressing the correspondence between 0 and the domain is to assume:

Remark. A more refined analysis would probably postulate the existence of some kind of “coding”-function: 4.2.3. mere is a @computablemapping K : 0 -+A such that if we set

402

J.E. FENSTAD

{ w EA I w 5 K (x)}, then (i) K*(x)is @-finiteforeach x E 0 ; (ii) x ES, implies K*(x)C K * ( ~ ) ; (iii) A = U,,@ K*(x).

K * (x) =

Note that 4.2.2 follows from (iii) in 4.2.3 and that 4.2.3(i) is already implied by 4.2.1. In addition to the coding process we seem to need some sort of “decoding” assumption. As a first approximation we propose: 4.2.4. There is a @-computablenzuppingp(n) such that {p(n)},(u, i f f ( a , (7, z ) 0 A [la,(I,z 11 = Iw I.

U,Z,

w) = 0

In other words, the relation ( a , u,z) E 0lw’is @-computable. The prewellordering associated w i t h 5 is easily seen to be uniformly 0 computable. If we strengthen this to 4.2.5. {w E A I I w I I Iwol} is uniformly @-finitein wo € A , we arrive at the class of “Friedberg theories” in the sense of Moschovakis 10, 5 101. And as he shows these are exactly the class of computation theories associated with admissible prewellorderings. I t remains to isolate the properties which characterizes the recursion theories associated with admissible partial orderings, i.e. with arbitrary admissible sets. A topic of central importance is the relationship between theories over finite and infinite domains. The basic example here is the relationship between hyperarithmetic and meta-recursion (or L,,-recursion) theory. This was generalized in Barwise, Candy, Moschovakis [3] (- see also Barwise [ 2 ] ) . In our context we are looking for a theorem which states that finite theories can be imbedded into infinite ones. The idea behind is simply this. A computation theory on an infinite domain can be a “good” recursion theory, in particular, if we have a suitable correspondence (codinddecoding) between the domain and computations over the domain, and if the semi-computable relations are exactly the X l-definable relations over the domain. The imbedding theorem should say that we can “enlarge” a finite theory to a “good” infinite theory. And, as a possible application, we would expect that fine structure results for, say, the semi-computable relations of the given finite theory could

ON AXIOMATIZING RECURSION THEORY

403

be obtained by “pull-back’’ from the enlargement. The motivating example is again various results for IT: sets obtained via meta-recursion theory. We state the following preliminary version of the imbedding theorem: 4.2.6. Let (@,<) be a Spector theory on a domain %.I t is then possible to construct (i) a domain ( y!* ,I ) where * ’u * extends ‘u and<* is a well-founded partial ordering on (11 *, (ii) a relation R on (a* ), , and I* (iii) an “infinite” theoty (O*, <*) on ( %* >,such that (a) is @*-computableand initial segments of<* are uniformly O*-finite, (b) R is @*-computableand @*-semicomputabilityequals Z,(R)definability over ?l*, and (c) a subset R 5A , where A is the domain of ‘u, is O-semicornputable i f and only i f R is @*-semicornputable.

<*

,<*

Remark. The result and method of proof is clearly inspired by Barwise, Candy, Moschovakis [3] and it uses in the construction of a* the recent developed theory of admissible sets With urelements, see Banvise [2] and his forthcoming lecture notes. There is also some recent work of P. Aczel and Y. Moschovakis which overlaps with the present result. Both Aczel and Moschovakis work in the context of a Spector class (which is equivalent to being the envelope of a Spector theory) and their goal is to relate these classes to the “next admissible ordinal” (Aczel) or “next admissible set” (Moschovakis). Both Aczel and Moschovakis carry their analysis a step further than the above result, showing the existence of a unique, minimal “next” structure. From this also follows the existence of a unique minimal “infinite” extension @* of a given Spector theory 0.- We have deliberately used the word “infinite theory” to describe O*, since it is a bit unclear at the moment how strong properties we can enforce on @*; it will satisfy the properties 4.2.1-4.2.5. We also expect that c can be strengthened to assert the equivalence between 0 and @* 13, but some details remain to be sorted out. 4.3. One further goal is to push the analysis of “computation” so far as to establish the domain of validity for the priority arguments. Only then can we claim to have a reasonably complete axiomatic analysis of generalized recur-

404

J.E. FENSTAD

sion theory. This is very much an open field.

Remark. The concept of a Friedberg theory does not seem to be entirely adequate, see Simpson’s paper “Post’s problem for admissible sets” in this volume. On the positive side see the appendix to that paper and also Simpson ~ 3 1 .

References [ I ] P. Aczel, Representability in some systems of second order arithmetic, Israel Jour. Math. 8 (1970) 309-328. [2] K.J. Barwise, Admissible sets over models of set theory, this volume. [ 31 K.J. Barwise, R. Gandy and Y.N. Moschovakis, The next admissible set, J. Symbolic Logic 36 (1971) 108-120. (41 H. Friedman, Axiomatic recursive function theory, in: R.O. Gandy and C.E.M. Yates (eds.) Logic Colloquium ’69 (North-Holland, Amsterdam, 1971) 113- 137. [ S ] T. Grilliot, Inductive definitions and computability, Trans. Amer. Math. SOC.158 (1971) 309-317. [6] P. Hinman, Hierarchies of effective descriptive set theory, Trans. Amer. Math. SOC. 131 (1968) 526-543. [7] P. Hinman and Y.N. Moschovakis, Computability over the continuum, in: R.O. Gandy and C.E.M. Yates (eds.) Logic Colloquium ’69 (North-Holland, Amsterdam, 1971) 77-105. IS] J . Moldestad, cand. real. thesis, Oslo 1972 (in Norwegian). 191 Y.N. Moschovakis, Abstract f i s t order computability I, Trans. Amer. Math. SOC.138 (1969) 427-464; and 11,138 (1969) 465-504. [ 101 Y.N. Moschovakis, Axioms for computation theories - first draft, in: R.O. Gandy and C.E.M. Yates (eds.) Logic Colloquium ’69 (North-Holland, Amsterdam, 197 1) 199-255. [ 11 ] Y.N. Moschovakis, Structural characterizations of classes of relations, this volume. [ 121 G.E. Sacks, The 1-section of a type n object, this volume. 1131 S. Simpson, Admissible selection operators, Notices Amer. Math. Soc. 19 (1972) A-599. [ 141 H.R. Strong, Algebraically generalized recursive function theory, 16M Jour. Res. Devel. 12 (1968) 465-475. [ 15) E.G. Wagner, Uniform reflexive structures: on the nature of Godelizations and relative computability, Trans. Amer. Math. SOC.144 (1969) 1-41. [ 161 M. Vegem, cand. real. thesis, Oslo 1972 (in Norwegian).

J.E.Fenstad, P. G.Hinman (eds.), Generalized Recursion Theory 0 North-Holland Publ. Comp., I974

DISSECTING ABSTRACT RECURSION Thomas J. GRILLIOT The Pennsylvania State University 1. Introduction

2 . Syntactic description of recursion 3 . Semantic description of recursion: implicit definability

4. Semantic description of recursion: quantifier form 5 . Completeness of C, E, S, A , I schemes

6 . Extending the notion of finiteness

1. introduction We have two prototypes for this discussion: recursiveness on the natural numbers and hyperarithmeticalness on the natural numbers. The latter differs from the former in that a quantification-over-w scheme is admitted. Crudely speaking, to say that F is recursive in Gl, ..., G,, means that F is computable or can be combinatorially generated from the structure ( w ;0, s,=, G,, ..., GJ, where 0 and s are the usual zero and successor function of w . (Note: 0, s,= are sufficient to specify the structure of w as we know from the investigations of Peano.) Similarly, to say that F is hyperarithmetical in Gl, ..., Gn means that F can be generated combinatorially from the structure ( o ; O , s,=,G1, ... ...,G,) with the aid of an oracle that can test quantification over w . I t is natural to replace the structure ( w ;O,s, =, G,, ...,G n ) by an arbitrary structure (A;Gl, ...,G,) where A is a set and G,, ..., G, are functions or predicates on A . This is precisely what our investigation is all about: to find out what recrusiveness on an arbitrary structure means. Unfortunately, the matter is not so easy in that the structure of the natural numbers is fairly unique with respect to other structures. Consequently the abstract study of recursiveness and hyperarithmeticalness may tend to be prejudiced by preconceptions based on usual 4 recursiveness and hyperarithmeticalness on the natural numbers. However, with a little investigation one can isolate these potential prejudices and discover that they seem to be five in number, which we denote by C(constant), 405

Th.J. GRILLLOT

406

E (equality), S (search), A (for all), I (for infinitely many). C-scheme: constant functions should be recursive. This scheme is true for w because it is finitely generated by 0, s. I n general, however, an arbitrary set need not be so combinatorially generated, and hence one cannot argue convincingly that constant functions are ips0 fact0 recursive. E-scheme: equality relation should be recursive. This scheme holds for w because 0, s generate w , s is one-one and 0 can be distinguished from successor numbers. The general situation is quite different. One cannot, for example, convincingly argue that equality on real numbers is ips0 facto recursive. S-scheme: unordered search operator should be recursive. The situation of w is particularly nice in that one has not only an unordered search operator but even an ordered one (the so-called p-operator). This is because w is finitely generated by 0, s. An alternative formulation of the S-scheme is that semirecursive predicates are closed under existential quantification. Again one cannot argue on purely combinatorial grounds that an unordered search operator is ipso facto recursive. A-scheme: the universal-quantifier operator should be “recursive”. This certainly holds for hyperarithnieticity on w . In fact, one of the more elegant formulations of “hyperarithmetical” is based on this idea (see Kleene [5]). The A-scheme is the most natural candidate for formulating an abstract notion of hyperarithmeticalness, but compare with the following scheme. /-scheme: the operator introducing the quantifer “for infinitely many” should be “recursive”. This certainly holds for hyperarithmeticity on w . In fact, the A-scheme and the I-scheme are equivalent on w (in the presence of bounded quantification):

IxP(x)

-

VXP(X)

V y 3 x ( P ( x )& y

< x)

f y V x ( P ( x )& x < y )

In some structures the two schemes are independent. Since the five schemes listed above are independent, we have 32(= 2 9 different formulations of recursion (of varying degrees of interest). To label thcm properly, we will use the terminology (C,E,S,A,I)-recursive, ( C , E , A ) recursive, etc. Thus t o say that F ( o n A ) is (C,E,S)-recursive in ( A ; G )means that F c a n be generated combinatorially from the structure ( A ; G )with the aid of schemes C,E,S. An exact formulation of what this means is given in the next section.

407

DISSECTING ABSTRACT RECURSION

An inquisitive reader will certainly question whether the five schemes listed above are exhaustive. In Section 5 we will show that they are (at least for countable structures) in the following sense: any scheme that holds for recursiveness (respectively, hyperarithmeticalness) on the natural numbers and can be formulated for arbitrary structures is derivable in (C,E,S)-recursion (respectively, (C,E,S,A ,I)-recursion). Historical note. Perhaps the earliest formulation of abstract recursion is due to FrahC [ 11. However, he did not distinguish between the combinatorial and noncombinatorial (schemes C,E,S,A , I ) aspects of recursion. Moschovakis [lo] seems to have been first to isolate the schemes C,E,S and a combined A / I scheme. The completeness of the C,E,S schemes was asserted by Lacombe [8] in a surreptitious way. The completeness of the C , E , S . A , / schemes was proved by Grilliot [2].

2. Syntactic description of recursion Our choice of formulation of recursion is proof-oriented. Besides being fairly versatile, it has the advantage of bridging the computational and the model-theoretic aspects of recursion. Definition. We define 9 t6 , where 0 is a sentence and 9 a set of sentences, inductively as follows. 9 t- 6 i f I9 E \Ir or c p , l cp E 9 for some cp [initial] ; or I9 [&-elimination] ; or 9 U { $ } I- 6 [v-elimination] ; or Vxcp(x) E 9 and 9 U {cp(t)} I9 for some term t [V-elimination] : or 3xcp(x) E 9 and 9 U {cp(y)}f 0 where y is a constant not in 6 [weak 3-elimination]; or ( f ) 3xcp(x) E 9 and 9 U {p(b)} p I9 for all b E B [strong 3-elimination] : or (g) Zxcp(x)E9 a n d q U ( 3 x x,(cp(x,)& ... & p ( x , ) & x l # x 2 & 6 for some n E w [I-elimination]; or x1 # x3 & ... & xnP1# x,)} (h) C x c p ( x ) E 9 a n d \ I r U { 3 x l...x,Vy(p(y) v y = x l v... v y = x , ) } k0 for all IZ E w [C-elimination] .

(a) (b) (c) (d) (e)

cp & $ E 9 and 9 U {p, $ } cp v $ E 9,9 U {p} f 6 and

*,

Clearly Ix represents the quantifier, for infinitely many x, and Cx represents the quantifier, for cofinitely many x ; thus Ix and 1C x l are expected to represent the same thing. Because of our specialized purpose, there is no need

ThJ.GRILLIOT

408

to have a 1-elimination scheme (since 1 ’ s can be driven through the other connectives) nor introduction schemes (since B will always be atomic or negated atomic) nor equality schemes (since they can be incorporated into 9 as desired). To say that Q is recursive in (B;P1,...,P,) will mean that there is a formal description of Q in terms o f P , , ...,P, such that (information about P,,...,P,) U (description)

(information about Q).

The description must be both complete (all true information about Q is derivable) and consistent (no false information about Q is derivable). The description may have intermediary predicate or function symbols in it. For example, the usual description of * (on a)in terms of 0, s = utilizes + as an intermediary. Let us formulate this definition of recursion precisely, Definition. Let (B;P,, ..., P,) be a structure, where B is a set and PI,..., P, are relations (possibly partial) on B. Let Q be another relation (possibly partial) on B . For a language that includes one or more names (constant symbols) for each element of B and one or more names (predicate symbols) for each of P,, ...,P,, Q, define AQ to be the set of formal sentences {Q(bl, ..., b,) : Q ( b l , ..., b,) is true) U{lQ(bl,

..., b,) : Q(bl, ..., b,) is false}

and similarly for Apl, ..., Apn, where we use the same letter for an object and its formal name(s) so long as confusion is avoided. Q is (C,E,S,A,I)-recursive in the structure ( B ; P l ,...,P,) if there is a sentence cp such that

A= u Apl

u _..U Apn U (9) I-

8 for all 8 E AQ

and some expansion of ( B ; P l ,..., P,) is a model of cp. In this definition, some of the relations P,, ...,P,, Q may be replaced by (partial) functions where, if f i s a function, Af is defined to be { f ( b l , ...) b k ) = c : f ( b 1 , ..., b k ) “ C I With regard to the formation of Af, if multiple names are used to denote one element of B , all names must appear as arguments of the function symbol but

DISSECTING ABSTRACT RECURSION

409’

only one name need appear as value. Also multivalued functions are acceptable, only in this case, the symbol = occuring in Af may have to be replaced by another symbol to avoid inconsistency with A=. An n-ary multivalued function amounts to the same as an (n+l)-ary relation in the presence of the E-scheme and S-scheme; but in the absence of the S-scheme the value-place of the function plays a role different from the argument-places as we shall see later. As part of the definition of Q being (C,E,S,A,I)-recursive in ( B ; P l ,...,P, ), we said that some expansion of ( B ; P , , ..., P,) must be a model of cp. This condition can be relaxed by saying some extension of ( B ; P , , ...,P,) must be a model of cp. In other words, it does not matter if the universe of the model is larger than B . This can be seen by the following little trick. Given cp with a model with universe larger than B , form a new description

&

... & vx l... Xk,(Pn(X1, ...)Xk,)+PA(fXl, ...)fXk,))

where P i , ...,PA, Q‘,f are new symbols and cp* is made from cp by replacing each P I , ..., P,, Q by P i , ..., PA, Q‘ and each variable or constant x by fx. ($ + 0 is regarded as an abbreviation for 1$ v 0.) Suppose Q is (C,E,S,A,/)-recursive in ( B ; P )with description cp. Thus

We may think of cp as a program for computing Q from P as follows. Assume that all 1’s in cp are driven through the other connectives so that they occur only immediately preceding atomic formulas. To determine whether or not Q ( a ) is true, one begins with a computation node {p} /-. The connectives in cp are then systematically stripped in accord with clauses (b) through (h) so that one gets a tree of intermediate computation nodes of the form Jr where Jr is a finite set. If a t some node, P ( b ) E Jr or l P ( b ) E Jr or b = c E \I, or b # c E Jr for some b , c E B , then one asks the oracle of P or = whether or not P(b) or b = c is true. If there is a conflict, clause (a) establishes that A- U Ap U \I, Q(a) and A= U Ap U Jr l Q ( a ) ; thus computation node

410

Th.J. GRILL107

\I, is certified. Similarly if 8,18E 9 for some 8, node 9 b is certified. Also if Q ( a ) E 9 (respectively, l Q ( a ) E q),clause (a) establishes 9 Q(a) (respectively, \I, 1l Q ( a ) ) ; thus node 9 is certified for Q(a) (respectively, l Q ( a ) ) . As soon as all terminal nodes are certified for Q(a) (respectively, l Q ( a ) ) , the computation procedure ceases with the answer being Q(a) is true (respectively, false). This computational procedure for (C,E,S,A,I)-recursion has five processes within it that are not purely combinatorial. We label these schemes by C , E , S , A , I .By removing one or more of these schemes as outlined below from the computational procedure, one gets 3 1 other forms of recursion ((C, E, S,A)-recursion, (C, E,S,Z)-recursion, ..., ( )-recursion). Role and irredundance of the I-scheme. C-elimination is an infinitistic rule of proof and hence is not combinatorial. Its role is that of introducing the I x quantifier. Thus the I-scheme is retained or removed by retaining or removing the C-elimination rule from the inductive definition of k.This turns out to be equivalent to allowing or disallowing the quantifierszx, Cx in the description q. To verify the irredundance of the Z-scheme, we can apply a result of I1.C of [4] that there is a nonstandard model of first-order arithmetic in which every (C,E,S,A)-recursive set of standard numbers is finite. In particular, the set of all standard numbers (denoted by w ) is not (C,E,S,A)-recursive in this model. By contrast, w is (C,E,S,A,Z)-recursive in this model since x E o ++ 1 1 y ( y < x ) . In other words, we have the following description of w :

& some equality axioms.

Role and irredundance of the A-scheme. The other infinitistic rule is strong 3-elimination. Its role is to introduce universal quantification over universe B. Thus the A-scheme is retained or removed by retaining or removing the strong 3-elimination rule from the definition of b. Consider the structure ( w X a;<) where (a, b ) < ( c , d ) iff a < c. The set ( 0 ) X w is (C,E,S,A,I)-recursive in ( w X w ; < ) since x E ( 0 ) X w ++ 1 3 y ( y < x ) ; that is, the following is a description of { 0 ) X w :

DISSECTING ABSTRACT RECURSION

411

However, the set cannot be (C,E,S,Z)-recursive in this structure. For suppose p is a description of {0} X w . Pick n E w so that no name for any ( m ,n ) occurs in y.Then y is still a description of {0} X w even when a new element

(-1,n) is added to the structure and < is extended so that (-1,n) < ( m , n ) for all m E w . On the other hand, cp must be a description of the transplant of {0} X w under the canonical isomorphism from (w X w ; < ) to ( w X w U {(-l,~)};<). Thus cp is a description of two different sets, a contradiction. Role and irredundance of the C-scheme. This scheme allows all constant functions as automatically “recursive”. The C-scheme is retained or removed by allowing or not allowing names for elements of B (other than those listed in (B;Pl, ..., P, )) in descriptions. At first glance, allowing names for arbitrary elements of B in a description p i s a purely combinatorial feature. However, in the course of a /--tree such a name would have to be matched with its replicas in Apl U .._U Apn. Such matching cannot be regarded as automatically combinatorial. For example, let w1 be the set of countable ordinals. The predicate P(x) ++ (x is finite ordinal) is (C,E)-recursive in the structure ( w l ; < ) since it has the description: Vx[(o < x

-lPx)&

(w = x

+

1PX)& (x < w -Px)] .

The use of a name (0)for the first limit ordinal is crucial to this description. With a compactness-like argument one sees that P cannot be generated combinatorially from < and =; that is, P is not (E)-recursive in (wl ;<).With slightly more care, one can find a predicate P that is (C)-recursive in (a1 ;<) but not (E,S,A,Z)-recursive in (w,;<). It should be noted, however, that in those structures (B;P,, ...,P,) in which B is generated by P,,...,P, (e.g., ( w ;0,s, =)) the C-scheme is redundant. Role and irredundance of the E-scheme. This scheme says that the equality relation is automatically “recursive”. At first glance, the scheme can be retained or removed by retaining or removing A= in the definition. However, the removal of A= may not be enough in some instances since subtle comparisons between objects of B may be made in the course of determining k.For example, { $ } t- 0 is established when $ is 8 . But to verify that $ is 0 , one must make a character-for-character comparison which often includes a comparison between two names for elements o f B which in turn insinuates a comparison of elements of B themselves. T o avoid such surreptitious uses of equality, one must allow infinitely many names for each element of B. In this

412

Th.J. GRILLIO’I

way, the combinatorial comparison of two names in no way establishes a noncombinatorial comparison of two elements of B. (Conversely, if one accepts the E-scheme, then one might as well use only one name for each element of B.) Consider the structure ( a ;0, s). The predicate Z(a) t-,a = 0 is @)-recursive in ( w ;0, s) (or, equivalently, is ( )-recursive in ( w ;0, s, =)). However, Z is not ( )-recursive in (a; 0, s). For suppose there is a description cp of Z . Let a be a name for 0 different from any name occuring in cp. Then one cannot combinatorially determine that A, U {cp} f Z(a). It is of interest to note that Z and = are of equal strength in the presence of the predecessor function: p ( a ) = a - 1 when a # 0 and undefined otherwise. That is, the structure ( w ;O,s,=) and ( o ; s , p , Z ) are equivalent, and, in fact, the functions ( )-recursive in either structure are exactly the usual recursive functions, and the functions ( A ) recursive (alternatively, (I)-recursive) in either structure are exactly the usual hyperarithmetical functions. Role and irredundance of the S-scheme. This scheme allows one to search through the set B for some object that satisfies a “recursive” predicate. I t is the abstract analogue of the p-operator scheme. This search capability appears in the V-elimination rule in the definition of f . To determine whether \k U {Vxcp(x)} f8 , one must check whether J( U {Vxcp(x), cp(t)} B for some term r. Thus one must search through all terms t , which is tantamount to searching through B. To remove the S-scheme, one must restrict V-elimination to instances where the terms involved can be generated combinatorially from information already given and the oracles of the functions in the structure. To exemplify this restriction, consider the structure ( R ; + .) , where R is the set of real numbers. Suppose that in the course of some computation one encounters

A+ U A. U {Vx(x * x# 2 v P ( d 3 ) ) ) k P ( d 3 ) .

(1)

In the presence of the S-scheme, this can be established by replacingx by d2:

However, the new number 4 2 introduced cannot be found combinatorially from 2 , 4 3 , +, - ;so one could not establish ( 1 ) in the absence of the S-scheme. The restricted V-elimination rule must be stated thus:

DISSECTING ABSTRACT RECURSION

413

for some term t in which no name for an element of B occurs except those generated from the constant and function symbols occuring in 9 U {Vxcp(x), 0). (If 0 is of the form f ( b l , ..., b,) = c, the constant symbol c must be excluded from those “occuring” in 9 U {Vxq(x),O} unless it occurs elsewhere. This exception is made because the value(s) of a function being computed cannot be assumed to be given a priori. In this regard, n-ary multivalued functions are computationally different from (n+1)-ary relations.) It should be noted that in those structures ( B ; P 1 ,...,P,) in which B is generated by PI,..., P, (e.g., (a; 0, s,=)), the S-scheme is redundant. To further exemplify the irredundance of the S-scheme, consider the structure (plane ; compass, straightedge). The functions “midpointing” and “anglebisecting” are ( )-recursive in this structure; but “angle-trisecting’’ is not (C,E)-recursive in this structure, though it is (E,S)-recursive in this structure. More precisely, let A be the set of points in the plane, and let F , G , H be the following 0-, 1. or 2-valued functions on A :

F(a, b , c, d) = point of intersection of line ab and line cd G(a, b, c,d) = point(s) of intersection of line ab and circle cd

H(a, b , c , d ) = point(s) of intersection of circle ab and circle cd (Circle ab denotes the circle with center a passing through b.) Typical functions ( )-recursive in ( A;F , G ,H)are:

M(a, b ) = midpoint of segment ab B(a,b,c) = point d on line ab with the property that angle acd = angle dcb. It can be shown that i f J is (E)-recursive in ( A ; F , G , H ) thenJ(al, , ...,a,) takes on only finitely many values each of which is constructable from a l , ..., a, using only compass and straightedge. It follows that the following function is not (E)-recursive or even (C,E)-recursive in ( A ; F ,G , H ) :

T(a,b,c) = point d on line ab with property angle acb = 3 angle acd. However, T is (E,S)-recursive in ( A ; F ,G , H ) with description

414

Th.J.GRILLIOT

Va b c d e ( B ( a , e , c) # d

V

B (d, b, c) # e

V

T(a, b , c) = d )

& (description of function B ) & (some equality axioms).

In the absence of the S-scheme but in the presence of the A-scheme, one may allow a V-elimination rule of another sort: \Ir

u {Vxcp(x)} k 0 if, for all terms t ,

and, for at least one term

r,

\k U {Vxcp(x),cp(t)} k 8 .

Definition. Q is semi-(C, E,S,A,I)-recursive in ( B ; P , , ..., P,) if, for some cp,

Normally we will talk of only total predicates as being semi-recursive whereas we allow purtiul predicates that are recursive. The removal of the schemes C,E,S.A,I in semi-recursion is just as in recursion. It should be fairly clear that a total predicate is (...)-recursive iff it and its negation are semi-(...)recursive.

3. Semantic description of recursion: implicit definability Each of the 32 variations of recuision that we discussed had something of the following forin: Q is recursive in ( B ; P l ,..., P,) if there is a description cp such that

and an expansion of ( B ; P l ,..., P,) is a model of cp. These variations differed in that the meaning of k was varied. By 1.C and l.E of [4], we see that is complete at least in the case B is countable. Thus we can replace the syntactic k by the semantic I=. However,

DISSECTING ABSTRACT RECURSION

415

simply means that every model of cp that interprets the symbols =, P,,...,P, as the relations (functions) = , P I , ..., P, must interpret the symbo! Q as the relation (function) Q. In other words, by replacing by we get the following semantic formulation of recursion:

Q is recursive in (B;P,, ...,P,) if there is a description cp such that some model of cp interprets PI, ..., P, correctly and every such model interprets Q correctly. In effect, cp isan implicit definition of Q in terms ofP,, ..., P,. Each of the 32 variations of recursion has such an implicit definability form. They differ only in the kind of description allowed and in the kind of models alloweq. These variations can be outlined very easily as follows. Full (C,E,S,A,I)-recursion.cp may have V, 3, I , C quantifiers. Universes of models must be B .

Removal of the I-scheme. I and C quantifiers are not allowed in cp. Removal of the A-scheme. Universes of models need not be B Removal of the S-scheme. Models that expand substructures of (B;P,,...,P,) are allowed. One must then say that every model with universe C that interprets PI1 C, ..., P, 1 C correctly interprets Q 1 C correctly. Removal of the C-scheme.cp may not have names of elements of B in it other than those in the list P,, ...,P,. Removal of rhe E-scheme. Somehow one must allow models that interpret each element o f B as many objects. Historical note: Fra’issC’s [ 13 definition of abstract recursion is essentially the implicit definability one. Kreisel [6] lays great emphasis on implicit definability. Indeed model-theoretic formulations of recursion have an air of completeness lacking in their combinatorial counterparts and thus add great weight to the validity of Church’s Thesis. The equivalence of implicit definability with Moschovakis’ scheme formulation of recursion is proved by Moschovakis [ 1 I ] , and also in the hyperarithmetical case by Grilliot [ 3 ] .

Th.J. GRILL107

416’

4.Semantic description of recursion: quantifier form We wish to generalize the classical results that semi-recursive predicates 1 have Zy quantifier form and semi-hyperarithmtical predicates have quantifier form. The latter result is virtually equivalent to the result that the hyper1 arithmetical functions are precisely the unique solutions of Z1 relationals. However, we must be more careful in the abstract case of describing quantifier forms than just by Ilr and Er,because we have the I and C quantifiers and also the contents of the matrix must be specified. For instance, if only O,s,= are allowed in the matrix (and hence not +, - ,bounded quantifiers), then not 0 1 every semi-recursive predicate is C1 but rather strict n l .Let bold-faced quantifiers (3,U) denote a list of second-order quantifiers (quantifiers over 1 relations of objects) of that kind. Thus every C 1 predicate has the form 3V3 over ( w ;O,s,=) and every strict - Z; predicate has the form ,3V over ( w ;O,s,=). The general results we would like are:

nl

~

( 1 ) Every predicate V’3 over ( B ; P l ..., , P,,)is semi-(C,E,S)-recursive in

( B ; P , , ...,P,l): (2) every predicateV’31 over ( B ; P l , ..., P,) is semi-(C,E,S,I)-recursive in ( B ; P , ,..., P n ) ;

(3) every predicate V3V (also V3V3, V’3V3V, etc.) over ( B ; P l ,..., P,) is semi- (C,E,S , A )-recu rsive in ( B ; P , , ... ,P,, ); (4) every predicate V 3 V (also V (any list of first-order quantifiers)) oveI ( B ; P 1 ..., . P,,) is semi-(C,E,S,A,I)-recursive in ( B ; P , , ..., P, ). We must assume that B is countable. The converses of these assertions will be considered Fbortly. T o see ( 2 ) , suppose Q(a) +-+ V R ...VR,cp(a) where cp in prenex form is 31 2nd R ,. ..., R , are the relation symbols other than PI,.... P,,, Q in cp. I t follows that Q is (C,E,S,I)-recursive in ( B ; P , , ..., P, ) with description Vx(q(x) + Qcx)), which is a VC sentence. For let ‘u be a model of this description. Then % I B is also a model of it because it has quantifier form VC. I f Q ( u ) is true, then cp(u) is true in every model with universe H and so Q(u) is true in ’u I B and hence in “&. The converses of (3) and (4) hold as is seen as follows. Suppose Q is semi( C , E , S , A ,I)-recursive in ( B ; P l . ..., P,,) with description q. First note that cp can be reduced to another description with quantifier form VC by adding new

DISSECTING ABSTRACT RECURSION

41 7

intermediary relations. Indeed, any I-quantifier can be eliminated by noting that Vx 3y (x
VxVyVzVu(R4(x,y,z,u)

+

3u$) .

Drawing quantifiers forward judiciously we get V C 3 (or V3C) quantifier form. In this manner, any description can be reduced to one in quantifier form VC3. Finally note that if cp is such a description of Q in terms of PI,...,P,, then Q(a) ++ VR I ...VR,VQ(q+ Q(a)), where R ..., R k are the ...,P,. Thus Q is x 3 1 V over predicate symbols in cp other than Q,P,,

(B;P,, ...,Pn). We do not know if the converses of (1) and (2) are generally true, but they seem nearly to be true in the following sense. The converse of ( I ) is equivalent to the statement that every description cp for (C,E,S)-recursion can be made into one that is in V-form. We know from the discussion above that each description can be made into one that is in V3-form. However, 3-quantifiers are fairly inert in the absence of strong 3-elimination. Historical note. Montague's [9] formulation of abstract recursion lays heavy emphasis on quantifier form. Moschovakis [ 121 first proved the abstract version of the Kleene-Suslin Theorem (HYP = A:).

5.Completeness of C, E , S , A , 1 schemes We will show that (C,E , S)-recursion is the strongest abstract recursion that generalizes usual recursiveness on the natural numbers, and that (C, E , S , A ,I)-recursion is the strongest abstract recursion that generalizes

418

ThJ. GRILLIOT

usual hyperarithmeticalness on the natural numbers. To see this, suppose Q is recursive, in some sense, in ( B ;PI,...,P,,)where B is countable. Then Q is ..., P,, 0,s, =) for any 0 E B and any successor recursive in that sense in (B;Pl, function s on B (s is a successor function on B if s is injective and B = {O,sO,ssO,sssO, ...}). I t follows that, for any bijection from B t o w , the transplant of Q is recursive in the usual sense in the transplants ofPl, ...,P,. By the following assertion, Q is (C,E,S)-recursive in (B;P,, ..., P,). Let B be countable. Then ( 1 ) Q is (C,b’,S)-recursive in (B;Pl, ..., P,,)iff, for every bijection from B t o w , the transplant of Q is recursive (in the usual sense) in the transplants of PI,...,P, (bijection may be changed to injection);

(2) Q is (C,E,S,/)-recursive in (B;P,, ..., P,l) iff, for every injection from B to w , the transplant of Q is hyperarithmetical in the transplants of P,,...,P,: (3) Q is(C,E,S,A,Z)-recursive in (B;P,, ..., P,) iff, for every bijection from B t o w . the transplant of Q is hyperarithmetical in the transplants of PI,...,P,,. The proofs of (2) and (3) are nearly identical, so let us consider (2). The , PJ, then necessity is straightforward: if Q is (C,E,S,Z)-recursive in ( B ; P 1 ..., Q is (C,E,S,I)-recursive in (C;P,, ...,P,) for all countable C 2 B,and so Q is (C, E, S,I)-recursive in ( C ;P,,...,P,,,0,s, = ) for all countable C 2 B , 0 E C and successor functions s on C ; this is equivalent t o saying that, for each injection from B t o w , the transplant of Q is hyperarithmetical in the transplants of P,,..., P,. Conversely suppose that, for each injection from B t o w , the transplant of Q is hyperarithmetical in the transplants ofP,, ...,P,. Consider the theory A whose symbols include names for elements of B and Ors, < and whose axioms are A = U A p l U ... U A p plus the usual axioms of arithmetic concerning 0, s,< plus the axiom V x l I y ( y < x). A is clearly (C,E,S,I)recursive in (B;P,, ...,P,). Also Q is (C,E,S,A,I)-recursive in every model of A . It follows from 1I.B of [4] that Q is (C,E,S,Z)-recursive in (B;P,, ..., P,,). Historical note. Lacombe’s [8] notion of V-recursiveness is that Q (on B ) is V-recursive in P (on B ) if, for every bijection from B to‘w, the transplant of Q is recursive in the transplant o f P . Lacombe asserted that V-recursiveness is equivalent to FrGsse’s invariant definability notion of recursiveness. Moschovakis [ 111 proved that V-recursiveness is equivalent to his schematic notion of recursiveness. Grilliot [ 21 proved the analogue for hyperarithmeticalness.

DISSECTING ABSTRACT RECURSION

419

6. Extending the notion of finiteness For simplicity, let us ignore the C,E,S,Z schemes. The difference between recursion with and recursion without the A-scheme may be regarded as a difference in the idea of finiteness. The A-scheme says that (for some superbeing) the universe can be comprehended with ease and so in an extended sense of the word the universe is “finite”. There may be structures with sets other than the universe that are t o be regarded as “finite” in an extended sense, e.g., admissible sets. It is of interest to define recursion for such a structure. Let (B;P,, ..., P,) be a countable structure and let C be a countable collection of subsets of B that are t o be regarded as “finite”. The definition of Q being recursive in (B; P I , ...,P,) is just as usual except that the A-scheme is dropped and the inductive definition of allows for clauses that comprehend the elements of C ; namely, for C E C , \I, k 8 if \I, U {ip(c)} k 8 for all c E C and 3x E Cip(x) E \I,. However, a problem of greater interest is the following. Given a structure (B; P,,...,P,) and C a collection of “finite“ subsets of B , what other subsets of B must necessarily be regarded as “finite”? In other words, how does one effect closure under the idea of “finiteness”? For example, if w is regarded as “finite” for the structure ( w ; = ) ,then any finite or cofinite subset of w must be regarded as “finite”, but there is no reason to regard the set of even numbers as “finite”. On the other hand, if the structure is expanded t o (w; O,s, =), then the set of even numbers - in fact, any hyperarithmetical set is readily derivable from the universe and hence must be regarded as “finite” though there is n o reason t o regard a nonhyperarithmetical set as “finite”. To repeat the question: given (B;P,, ...,P,) and a collection C of “finite” sets, characterize all the sets that are thereby “finite”. We cite three possible I answers t o this question. (1) Any subset of B that is both El-definable and I l 1l -definable-on-B in every model of ~

A Pl U . . . U A p n U {~,,,Vx(~~CvV,,~x=c)}.

The definitions of definable and definable-on-B are given on pp. 122-122 of [7]. The definitions are easily adapted to allow second-order quantifiers. (2) Any subset of B that belongs to every model of the set displayed above plus axioms that more or less state that two-element sets are “finite”, “finite” sets are closed under subset. and “finite” unions of “finite” sets are “finite”.

420

Th.J. GRILL107

These axioms are chosen because they are virtually complete for the notion of “countable” (see l.F of 141) and countability is completely degenerated finiteness. (3) Any subset o f B that can be built up from using effectivized versions of the axioms: two-element sets are “finite”, “finite” sets are closed under subset, and “finite” unions of “finite” sets are “finite”. For arbitrary structures we d o not know how these three answers compare.

e

References [ 1 ] R. F r a k t , Unc notion de r h r s i v i t k relative, Infinitistic methods (Pergamon Press, Oxford, 1961) 323-328. [ 2 ] 7.5. Grilliot, Omitting types: application to recursion theory, J . Symbolic Logic, vol. 37 (1972) 81-89. [ 31 T.J. Grilliot, Implicit defjnability and hyperprojectivity, Scripta Mathematica, to appear. [ 4 ] T.J. Grilliot, Model theory for dissecting recursion theory, This volume. [ 5 ] S.C. Kleene, Recursive functionals and quantifiers of finite types, I, Trans. Amer. Math. Soc., vol. 91 (1959) 1 -52. [ 6 J G. Kreisel, Model theoretic invariants: application t o recursive and hyperarithmetic operations, in: J . Addison et al. (eds.) The Theory of Models (North-Holland, Amsterdam, 1965) 190-205. [ 7 ] G. Kreisel and J.L. Krivine, Elements of mathematical logic (North-Holland, A mst erdani , 1967). 181 D. Lacombe, Deus g&n@ralizations de la notion de r&cursivitC relative, Comptes Rendus de I’Academie des Sciences de Paris, vol. 258 ( I 964) 341 0-34 13. [ 9 ] R . Montague, Recursion theory as a branch of model theory, in: B. van Rootselaar and J.F. Staal (eds.) Logic, methodology and philosophy of science 111 (NorthHolland, Amsterdam, 1968) 63-68. [ 101 Y.N. Moschovakis, Abstract first order computability, Trans. Amer. Math. Soc., vol. 138 (1969) 427-504. \ 1 I ] Y .N. Moschovakis, Abstract computability and invariant definability, J. Symbolic Logic, vol. 34 (1969) 605-633. [ 121 Y .N. Moschovakis, The Suslin-Kleene Theorem for countable structures, Duke Math. J., vol. 37 (1970) 341-352.

J.E.Fenstad, P.G.Hinman (eds.), Generalized Recursion Theory 0North-Holland Publ. Comp., I974

MODEL THEORY FOR DISSECTING RECURSION THEORY Thomas J. GRILLIOT The Pennsylvania State University I. Completeness theorems A. Usual predicate calculus with &, V, 3,l B . Predicate calculus with equality C . w-logic D. lnfinitary &’s and V’s E . The infinity quantifier F. The uncountability quantifier 11. Satisfying an infinite sentence A. Compactness ( 3 A) B . Omitting types (VV) C . Omitting compactifiable types (VV 3 A ) 111. Compactness after omitting types A. Application to uncountability quantifier B . Barwise compactness

v,

This paper summarizes some results of model theory that have an impact in examining recursion theory. No attempt has been made to document the results.

I. Completeness Theorems A. Usual predicate calculus with &, V, V,3 , l . First of all we make a blanket assumption that any set of axioms/rules is strong enough to drive 1 ’ s through other connectives so that, if needed, we may assume that 1’s only appear before atoms. We show how a consistent set @ of sentences has a model. The key to this demonstration as well as most of those following is the formation of a set amof sentences with the following properties: (a) @ (b) if cp & $ € @-, then cp, $ € @=; (c) if cpv $ €@=, then cpE@_ or )I E@-;(d) ifVxcp(x)E @= and f is a term formed from the function symbols of am,then cp(t) E @=; (e) if 3xcp(x) € am, 421

422

Th.J. GRILLIOT

then q ( c ) E for some constant symbol c. Assuming that we have such a We let be the set of all atomic or negated atomic sentences in am. amand C am,any can make the following observations: (1) since model of CP_ is also a model of both Q, and a:, and the consistency of am implies the consistency of both @ and (2) since Cp: consists of only atoms or negated atoms, its consistency implies that it has a model;(3) by induction on the number of connectives in a sentence of am, one readily sees that a model \u of CP: with universe {t?,: f is a term formed from the function symbols of am} is a model for all of am.From these three observations it follows that the consistency of CP implies that it has a model provided we can form CPm in such a way that the consistency of implies the consistency of am.We form as a union of sets aO, a1,Q2, ... as follows. For simplicity assunie @ is countable. Let a0 be CP. Pick a sentence from CPo with a connecIf the sentence is y & $, then Q0 U {cp, $ } must be consistive other than 1. tent, so let CPl be this set. If the sentence is y v $, then a0U { y } or a0U { $ } be :he one that is consistent. If the sentence is is consistent, so let Vxcp(x), then a0U { y ( t ) }must be consistent for any term t , so let be Q0 U { q ( t ) }where t is a variableless term. If the sentence is 3xp(x), then a0U { q ( c ) } must be consistent where c is a new constant symbol, so let a1 be this sct. In a similar manner, form a2,a3,Q4, ... in such a way that every connective (and every variableless term in connection with V) is acted upon at some stage. Then CP_ = U CPn is the desired set.

s

B. Predicate calculus with e,quality. The following axioms when added to the usual predicate calculus axioms/rules are complete for =: Vx(x =x) ; x = y + O(x) tf O(y).This is proved just as in the preceding section except that the additional conditions are placed on am: (f) t = t € 9-for all t formed from function symbols of am;(g) if O(t) E @and t = u E CP_ or u = t E CPrn, then O(u) E a_.The axioms mentioned above with these two new conditions. are just enough to permit the formation of C. o-logic. Let A be a countable set whose elements have names in some countable language such that a # b is an axiom when a, b € A and a # b. If one is interested in restricting the meaning of V and 3 t o quantification over A , then the following infinitary rules when added t o usual axioms/rules are complete: { @ ( a ): a E A }/VxO(.x). The proof is as in the preceding section except that

MODEL THEORY FOR DISSECTING RECURSION THEORY

423

condition (e) concerning Q,= must be revised to read: (e) if 3xq(x) E aa, then q(a) E Q,= for some a E A . (One may have to add the axiom Vx3y (x = y ) to Q, to make the details work out readily.) Several adaptations can be made. For example, add A as a unary predicate symbol to the language. T o make A(x) have the meaningx € A in any model, the following infinitary rules are complete: {O(a) : a € A } / V x ( l A ( x ) v O(x)). One can achieve the analogous result using countably many sets simultaneously.

D. Infinitary &’s and V’S. One may wish to incorporate countably many infinitary conjunctions A and disjunctions V in a countable language. For this modification, the following axioms/rules are complete: rules allowing 1 ’ s to be driven through A’s and V’s according to deMorgan’s Laws; On -+ VOi for any n ; { O o , 01, ...}IAOi. The proof is as in the case of the completeness of usual predicate calculus except that conditions (b) and (c) concerning Q,- are revised to read: (b) if Aqi E am, then qi E amfor all i ; (c) if Vqi E Q,_, then pi E Q,- for some i. E. The infinity quantifier. One may wish to add two new quantifierszx and Cx to a countable language with intended meanings of “for infinitely many x” and “for cofinitely many x”. (Note: Zx and 1C x 1 have same intended meaning.) For this modification, the following axioms/rules when added to the usual axioms/rules are complete: rules allowing 1 ’ s to be driven through Z’s and C’s according to deMorgan’s Laws; { 321xO(x), 322xO(x), ...}IZxO(x); V2nxO(x) + CxO(x) for any n , where 32nxO(x) is an abbreviation for 3x, ... 3xn(O(x1)& ...& O(x,) & x1 # x 2 & ...) and V2nxO(x) is an abbreviation for 132nx10(x). The proof is as in the case of completeness of usual predicate calculus except that the following two additional conditions must be added concerning am:(h) ifIxq(x) E am,then 32nxq(x) E Q,- for every n ;(i) if Cxq(x) € Q,-, then V2nxq(x) E Q,- for some n . F. The uncountabaity quantifier. One may wish to add two new quantifiers UCx and CCx to a countable language with the intended manings of “for uncountably many x” and “for cocountably many x”. Surprisingly, the additional rules needed for these quantifiers are finitary, and the completeness proof is quite different from those preceding. To simplify matters, use a two-sorted language with x , y , z

Th.J. GRILLIOT

4 24

denoting type-0 variables and X , Y , Z type-1 variables and E a binary relation relation between type-0 and type-1. Let UCxO(x) be an abbreviation for 13XVy( B(y) c--,y E X ) and let CCxO(x) be an abbreviation for 1UCx7B(x). The following axioms are sufficient to make UC and CC have their intended meanings: 3 X ( y E X & z E X ) ; 3 Y V z ( z E Y - z E X & O ( z ) ) ; Vy E X C C z O ( y , z ) -+ CCzVy E X O ( y , z ) . Informally, these axioms state that all two-element sets are countable, countable sets are closed under subsets and countable unions of countable sets are countable. The middle axioms (subset axioms) can be dispensed with if UCxB(x)is used as an abbreviation for 1 3 X V y ( O ( y ) + y EX) instead of for 7 3 X V y ( O ( y )- y E X ) . Suppose @ is a set of sentences that includes the above axioms and the axiom of extensionality. We want to show that its consistency implies that it has a model in which UC and CC have their intended meanings. By the standard completeness theorem, @ has a countable model %, say with type-0 universe A , and type-1 universe 2,. A, may be regarded as a subset of the power set o f A o . Let { q j ( x ) }be the collection of all formulas in the language of \uo in one free variable x such that 1 ' 1, C C x q j ( x ) ,and let $(x) be any formula such that '$1, UCx $(x). By the compactness theorem, '$4, clearly has an "extension" 211 in which $I13 x [ $ ( x ) & A q i ( x ) ] .The trouble is that the compactness theorem only insures that the type-0 universe A of 211 includes A 0 and not that the type-l universe A , of '$1, includesj,. In order for 2, CAI ,some types omitted in $' 1, must remain omitted in %,. We are thus faced with the problem of superimposing compactness upon omitting-types. This is possible thanks mainly to the countable union axioms, but the details will be left to a more appropriate section (section 1II.A). Thus \uo has a bonefide elementary extension \ul with the property that l?ll 3x[$(x) & A q i ( x ) ] .In like manner one forms 'u2, '$I,, ... and indeed 'u, for any countable successor ordinal U. If u is a limit ordinal, one lets 3, be u7
+

I I. Satisfying an infinite sentence A. Compactness (3A). Let CP be a set of sentences, for simplicity, countable. The problem of compactness is to find a sufficient condition for when @ has a model of an infinite'conjunction or, what is virtually equivalent, an existentially quantified infinite conjunction. The answer is that @ U { 3 x A i E w q j ( x ) has } a model if, for each 12, @ U { 3 ~ A ~ . , ~ q ~has ( xa)model. } (We have simplified the situa~

MODEL THEORY FOR DISSECTING RECURSION THEORY

4 25

tion by considering only countable conjuncts and one existential quantifier.) To see this, let C = @ U {cpi(c) : i E w ) where c is a new constant symbol. Clearly every finite subset of C has a model. Form Zm with the usual closure conditions(see section l.A) plus the condition that every finite subset of C _ has a model. Since C z consists only of atoms and negated atoms, it has a model because every finite subset of it has a model. Therefore, C_ and hence C and hence @ U { 3xAiG,cpi(x)} has a model.

B. Omitting types (VV). Let @ be a countable set of sentences. The problem of omitting types is to find a sufficient condition for when Q, has a model that is also a model of a universally quantified infinite disjunction. The usual answer is that @ U {VxViEwcpi(x)} has a model if cb has a model and, for all $(x) such that cb U { 3x$(x)} has a model, Q, U { 3x($(x) & cpi(x))} has a model for some i. (We have simplified the situation by considering only one universal quantifier.) To see this, form @- from @ so that amhas a model and satisfies the usual closure conditions plus the condition: if t is formed from the function symbols of aW,then qi(t)E am for some i. The hypothesis above is adequate to insure that this new condition on Qm can be achieved. As is noted in section 1.A, if Qm has a model Y f , then it has a submodel whose universe is {fa: t is formed from the function symbols of am}.This submodel is a model of @ U {VxViEwcpi(x)}. Variations can be made for languages with o-rules, infinitary conjuncts and disjuncts and quantifierszx, Cx. Also, adding an infinite conjunct before the universally quantified infinite disjunction is no problem. Thus U {AjG,VxViE,cpq(x)} has a model if @ has a model and, for allj, for all $(x) such that @ U {3x $(x)) has a model, @ U { 3x($(x) & cpii(x))} has a model for some i. Another variation can be made for second-order quantifiers. Let the quantifiers 3 p , Vp vary through all relations of a specified number of arguments. Assume that p does not occur in @. @ U {Vx(ViEw 3pcpi(x) v ViE,Vp$i(x))} has a model if @ has a model and, for all O(x) in which p does not occur - though relation symbols other than those in @ may - with the property that @ U {3xO(x)) has a model, cb U { 3x(O(x) & cpi(x))} has a model for some i or @ U { 3x(O(x) & 1$i(x))} has no model for some i. An important application is the following. Let @ include in its language names for the elements of a countable set A . Then any 1 subset of A that is ll,-definable-on4 in every model of Q, is semi-representable in some finite extension of @.

4 26

'I3.J. GRILLIOT

C. Omitting compactifiable types (VV3A). Let Q, be a countable set of sentences. Q, U { V ~ V ~ ~ ~ 3 y A ~ ~ ~ has cp~~(x,y)} a model if CP has a model and, for all G0(x), $l(x), $*(x), ... such that Q, U { ~ x & < ~ $ ~ ( xhas ) } a model for eachj, Q, U {3~3vA,,~($,(x) & cpi,(x,y))} has a modelfor some i and all j . The proof is quite similar t o t h e one outlined in the preceding section except that the formation of Q,- is made so that the following condition holds: if t is formed from the function symbols of Q,-, then cpij(t,c)E Q,- for some i and allj, where c is a constant symbol. Variations of this result exist also. One important application is the following. Let Q, include in its language names for the elements of a countable set A . Then any subset ofA that is Xi-definable in every model of Q, is finite. In fact, 1 there exists one model of Q, in which every Xl-definable subset o f A is finite.

I I I. Compactness after omitting types A. Application to uncountability quantifier. In section 1.F we were confronted with the following situation. Given a second-order model % in which 91 CCxcpi(x) for each i and % UCx$(x), find an elementary extension in which 3x($(x) & AiE,cpi(x)) is true. Since 3x($(x) & Ai<,qi(x)) is true in % for all n , this would follow trivially from the compactness theorem if it were not that we insist that the type-1 universe of % include the type-1 universe of a. In other words, if W is a type-1 object of 'I(and a o, a l , ... are the type-0 objects of a for which 'u ai E W , then the universally quantified infinite disjunction Vx(x 4 W v ViEwx = ai) must hold in % . Since we are making infinitary requirements on % of the sort VV, the usual compactness theorem is of no value by itself. Compactness must be achieved by using the axioms satisfied by 91, especially the countable union axioms. Let Q, be the set of 0 for which 2! k 6 together with $(c), cpi(c) for i E o ,where c is a new constant symbol. We assume that cpo(x),cpI(x), ... is a complete list of those one-place formulas such that CCxcpi(x) is true in %. We need to show that Q, U (Aw,%Vx(x 4 W v VaEWx = a ) } has a model. If i t does not, then by section 1I.B there exists B(x,c) such that Q, U {3xB(x,c)} has a model, but that, for some W E %, Q, Vx(O(x,c)+x E W ) and Q, Vx(B(x,c) + x # a ) for alla E W. Therefore, we have that C C y [ $ ( y ) ~ V x ( B ( x , y ) ~ xWE ) ] and V z € WCCy[$(y)-+Vx(O(x,y)-+x#z)] are true in 91. Since 8 satisfies the countable union axioms, the bounded quantifier Vz E W can be drawn through the CCy quantifier so that

+

MODEL THEORY FOR DISSECTING RECURSION THEORY

421

CCy [ $ ( y )+ Vx(O(x,y) -+ x 4 W)]is true in ‘91. Combining this with the one above we get that CCy[$(y)+ VxlO(x,y)] is true in %. This means that one of the cpi(c) is $(c)+ VxlB(x,c), contradicting the fact that @ U {3xB(x,c)} has a model. B. Barwise compactness. Let Q, be a countable collection of sentences including, possibly, some with infinite conjunctions and disjunctions. Assume that Q, is closed under components, the finite propositional connectives, and the distribution of 1’s. In other words, cp E @ iff l c p E @, cp & $ E @ iff cp, $ E @, cp v $ E @ iff cp, $ E Q,, Vxcp(x) E @ implies cp(t)E Q, for all variableless terms t formed from a set of function symbols that includes an infinite number of constants, 3xcpfx) E @ implies cp(t)E @ for those same terms, Aiqj E @ implies cpi E @ for all i, Vicpi E Q, implies cpj E @ for all i, lAicpi E Q, iff V i 1 q E @, lVicpi E Q, iff A i l p i E @, lVxcp(x) E @ iff 3xlcp(x) E Q,, 13xcp(x) E @ iff Vxlcp(x) f 41. Barwise compactness gives a sufficient condition for when a subset of @ has a model. Let the notation cp E 9/ denote that $ is a conjunction and that cp is one of its components, and for a subset of Q, let $ C C denote that every cp E $ is an element of 2. Let cp, $, B vary through elements of CP. A subset C of Q, has a model i f

z

(1)

every cp 5 C has a model;

and, for any $, $’, $ a conjunction,

(2)

if, for every cp E $, there is a O then there is B 2 C such that 8

c zsuch that B t=

$’ v cp,

k $’ v $.

Note that if there are no infinite connectives in Q,, then condition (2) is automatic and so Barwise compactness reduces to classical compactness. Condition (2) has the form of a union axiom or a replacement axiom. Indeed condition (2) is automatically satisfied when Q, can be represented by an adset. (In an admissible set, k is missible set in such a way that z becomes a a C, relation.) To prove Barwise compactness, suppose that C satisfies the two conditions above. We form Zm in the usual manner as outlined in section 1.D except that each stage C, must satisfy condition (1). This is obvious for 2, because Zo is To see that it is true for C l , recall the ways that Z1 may be constructed from C,. The difficult one is when Viqj E Co and C, must be zoU {pi} for some i. Such an i can be chosen; for otherwise, for each i, some

z.

428

c

Th.J. GRILLIOT

$i Zo U {pi} has no model which means that $ j - {qi} I= l q j ;by condition (2), for some $ C Zo, $ A i l p i ; thus $ A V i q i C Zo has no model, a contradiction. One proceeds in this manner to show that every C, satisfies condition (1). (One must form the En’s so that each is only a finite extension of C.) I t follows that 22: has a model and hence that C has a model.

J. E. Fenstad, P.G.Hinman (eds.), Generalized Recursion Theory

0North-Holland Publ.

Comp., 1974

AXIOMATIC THEORY OF ENUMERATION Andrzej GRZEGORCZYK University of Warsaw

Sets may be considered to be simpler than functions. Hence I propose to study first an axiomatic theory with a fundamental epsilon-like notion E . ( x E y means: x belongs to the set with number parametery.) The stronger theory of enumeration of functions may be developed later. Besides the relation E we need some other individual constants or functions (primitive recursive in the standard model) as primitive notions. First the pairing function and its inverses:

then the shifting function S and its axiom: Al.

xES(y,z)- ( x , z ) E y .

The next axiom A2 is a collection of six comprehension schemas (ClLC6). I shall use them in two parallel forms as existential formulas and as definitions of new individual constants (whch is the same in the model):

-

p= 8)

c1

VuAx(xEu

c2

V u A x f x E u-p#$)

C'2 x E c - p # $

c3

V u A x ( x E u- p E $ )

C'3 x E c - p E $

c4

VuAx(xEu

C5

Vul\x(xEu -(xEa

C6

VuAx(xEu-(xEa

C'1

xEc-p=\1/

V y ( x , y ) E a )C'4 x E c A

xEb))C'S x E c

--

Vy(x,y)Ea

(xEa A xEb)

vxEb)C'6 xEc-(xEa

429

vxEb).

A. GRZEGORCZYK

430

In C 1-C6 cp, $ , a and b are terms containing no occurrences of the variable u , but possibly containing some variables as parameters. In C'l-C'6, cp and $ may contain only x as variable, a and b must be constant terms, and c is a new individual constant. In order to pass to the enumeration of functions we must postulate the existence of a universal function 21 and the Lachlan function &:

To close the theory we postulate that: A4

AxVw,u(Az(zEx

V y ( z , y ) E w ) Az(zEu

-

1zEw)).

(Every element is the projection of a dual element.) The shifting function S allows us to consider some elements as combinators with respect to the equivalence. I f P ( x l ,...,x,) is a polynomial built o f S and x l , ..., x,, then the combinator associated to P is an element ap such that the following formula is a theorem: (1)

x E S ( ... S(ap,xl),...

)

X,)

-

XEP(Xl, ...)X n ) .

Combinatory Property. For evety P ( x l , ...,x,) there is a combinator associated to P. Proof. By C3 we can define ap to satisfy the formula: (... (x,x,),

...,x I ) E a p- x E P ( x l ,

..., x,,)

Then applying n times A 1 . we get (1). I shall mention some other properties,

Fixed point theorem. There is a function

71 which produces fixed

points:

AXIOMATIC THEORY OF ENUMERATION

(2)

431

zES(x,n(x))-zEn(x) .

Proof. Accordingly to A2 there is an element Q such that:

h t t i n g y = S ( Q , x ) we get:

-

Hence the function: n(x) = S ( S ( Q , x ) , S ( Q , x ) ) satisfies the formula (2).

Definiti0n.x is dual

VyAz(zEy-1zEx).

There are dual elements. There are also elements which are not dual, e.g. the element c satisfying the equivalence:

The supposition that c is dual leads to a contradiction by Russell's argument: if for somey,Az(zEy - l z E z ) , then:yEy - 1 y E y .

Definition.x isfinite - A y ( A z ( z E y

+ z E x ) + y is dual).

The intuition is that every infinite set contains a non dual set. Every finite element is of course dual. Every element defined by alternation of identities with constants is finite:

xEc-(x=al

v ... v x = a , ) .

Considering the Boolean operations U, n, and / as defined by means of E as epsilon, we get that the dual elements constitute a Boolean algebra. On the other hand, having non dual elements we can prove that the complement of c defined above (or defined by: x E c - x # x ) is infinite because it contains a non dual element.

432

A. GRZEGORCZYK

There is a sequence of infinitely many different infinite elements:

xEaO-x=x (3)

xEa,+l -x#aO

A

... A x f a ,

Proof. aoEao. Suppose that aiEai for every i < n . Accordingly to the definition laiEa,+l. Hence ai # a,+l, and by the definition Ea,+l. Definition. x is closed - A y , u(Az(zEy -zEu) Rice’s The0rem.x is closed

-

h aOEx A

+

EX

eyEx)).

l ( b o E x )+ x is not dual.

Proof. Suppose x t o be dual. Hence for some x‘: (4)

zEx‘

l(zEx)

By A2 there is an h such that: ( 5)

(n,y)Eh-((rzEx ~ y = b , )v (nEx‘ ~ y = a ~ ) ) .

The element h considered as a set of pairs is a function which maps { n : n E x } to bo and { H : 1 n E x ) to ao. By A2 for h there is an rn such that:

-

Using the futed point theorem we get no = ~ ( msuch ) that: (7)

zES(nz,nO)

zEnO ,

The element h as a function is total (by (4) and (5)). Hence for no there is a y o such that:

AXIOMATIC THEORY OF ENUMERATION

433

By (6)-(9) and A l we get that:

When x is closed, (10)implies that:

On the other hand (4) and (5) imply that:

(12)

(n,y)Eh

-+

(nEx -1 yEx) ,

and (8) and (12) imply that:

(1 1) and (13) give a contradiction.

Non-extensionality theorem. For every element x and for every n there exist more than n elements which are extensional with x. Proof. Suppose that there are only n elementsxl, ..., x, which are extensional withx. This means that: (14)

Az(zEu e z E x ) - ( u = x l

v ... V U = X , ) .

According to C’1 and C’6 there is an elementy such that: uEy -(u=x1

v ... V U = X , ) .

By (14) y is closed. It is not empty because x E y , and by ( 3 ) there is some u such that 7uEy. According to C’2and C‘5,y is dual. But this contradicts Riceis the orem. Notice that for the above argument we need Rice’s theorem in a uniform formulation. Instead of h take: S(S(S(H,x), ao),,bO)for suitable H , and instead of m take S(M, h) for suitableM.

A. GRZEGORCZYK

434

The axiom A3 enables us to consider the enumeration of functions. Accordingly to A3 the element U may be considered as a partial function and we can write:

U ( z , x )= y instead o f ((x,y),z)E?d. We shall use the abbreviations

and 4.

Kleene’s Sr-theorem. There is a shifting function S’ such that:

U(S ’(a,b), x ) = U(a,( b ,x ) ) . Proof. By A2 there is an element Uo such that:

According to A 1:

By A3a, the elementy is unique. Hence, applying A3b, we get that:

PuttingS’(a,b) theorem.

=

g(S(S(ao,b),a))and applying (15)-(17) we get our

AXIOMATIC THEORY OF ENUMERATION

435

Proof. By A2 there are elements al, a 2 , a3 such that:

Putting (Y = &(a3), by A3 and Kleene’s Sr-theorem we can deduce:

Similarly by A2 there are elements e l - e 4 such that:

Putting J/ = & ( e 4 )we easily verify conditions: c , d, and e . There is of course a standard arithmetical model for AO-A4 in which the universe consists of natural numbers and “xEy” means: the number x belongs to recursively enumerable set having the numbery. The other model consists of the recursive ordinals and metarecursive enumeration. Is it a natural theory, or is it perhaps too weak to give more involved interesting theorems? One can trye to develop the hierarchies in it, but perhaps it may be more appropriate to add the weak second order logic. Another question which seems to be interesting is: taking AO, A 1, how

436

A. GRZEGORCZYK

many instances of A2 can one take and still get a theory compatible with extensionality? If it were possible to define two combinators of the X-calculus, we would have a model for the X-calculus, if the definitions could be compatible with extensionality.

J.E.Fenstad, P. G.Hinman (eds.), Generalized Recursion Theory 0 North-Holland Publ. Comp., 1974

POST’S PROBLEM FOR ADMISSIBLE SETS S.G. SIMPSON The University o f California, Berkeley

In 1944 Post proved that there exists a recursively enumerable subset of w having intermediate many-one degree. Post then asked whether there exists a recursively enumerable subset of w having intermediate degree of unsolvability. In 1956 Friedberg and Muchnik solved Post’s problem affirmatively by proving that there exist two recursively enumerable subsets of w having incomparable degrees of unsolvability. Recently, Sacks and Simpson [ 51 generalized the Friedberg-Muchnik theorem to the context of recursion theory on admissible sets of the form La. Admissible sets of this special form retain the following basic property of w : the universe is well-ordered by a recursive relation. Kreisel [ 2 : p. 1731 asked whether property (W) is in any way essential or “significant” for generalizations of the Friedberg-Muchnik theorem. In the present paper we partially answer Kreisel’s question. Namely we prove: there exists an admissible set M for which both (W)and the FriedbergMuchnik theorem fail. However, our proof has one serious defect: it uses AD, the so-called axiom of determinacy, which is actually not an axiom but rather an unsupported (though pragmatically interesting) hypothesis. We conjecture that this defect can be eliminated. In the meantime, for background material on AD, the reader may consult [3]. Research partially supported by NSF Contract GP-24352. See also Kreisel’s 1973 Zentralblatt review of [ 4 ] in which Kreisel suggests that this question must be answered before it is reasonable to start thinking about axiomatics for post-Friedberg recursion theory. 437

S.G. SIMPSON

438

Our main theorem, Theorem 1 below, is stronger than what was stated above, in two ways. First, while our admissible set M will not have property (W), it will have the following property: the universe is prewellordered by an M-recursive relation whose initial segments are uniformly M-finite. Second, not only will theM-analog of the Friedberg-Muchnik theorem fail, but so will the M-analog o f Post’s weaker theorem mentioned in the first sentence of this paper. Definition. L e t M be an admissible set. B S M is complete- E(M) if (i) B is E(M); and (ii) for each E(M) set A E M there is a E(M) relation C C M XM such that (a) VX3Y C(X,Y> (b) VxVy (C(x,y) + (x € A -y E B ) ) .

Remark. I f M is E-uniformizable then (ii) is equivalent to every E(M).set being many-one reducible to B. In any case, (ii) implies that every X(M) set is A ( ( M ,B ) ) . Theorem 1. Assume AD. Let M = R+, the next admissible set after the continuum. Then every X (M> set is either A (M> or complete.

In particular, the Friedberg-Muchnik theorem fails for R’. Note that R + is a “Friedberg theory” in the sense of Moschovakis [4]. Before proving Theorem 1, we establish some notation. Let R = w w , the real continuum. Let M = R+,the smallest admissible set such that R EM. Put K = On n M . Clearly M = L,(R) where the constructible hierarchy over R is defined by LO(R)

=

transitive closure of R ;

L,+I(R)

=

{XC L,(R) I X is first-order definable over (L,(R), € ) alowing parameters from L,(R)};

POST’S PROBLEM FOR ADMISSIBLE SETS

439

u{ LJR) I a! < A} for limit A;

L,(R)

=

L(R)

= U {LJR)

I a! an ordinal}.

Some Berkeley set theorists have conjectured that AD ‘‘holds” in L(R) in the sense that plausible large cardinal axioms may be found which imply this. Note that our theorem and proof take place entirely within L(R). Let H be the Moschovakis system of notations for the ordinals less than K . Let 11 be the corresponding norm. ThusH is a subset of R and 11 maps H onto K . For eacha
Lemma 1. For evety I, J C R either12 J o r J S R

-

I.

Lemma 2. Let S be a subset of K such that Va!< K (S fla! EM). m e n S is A (M).

Proof. Assume hypothesis. We shall show that S is Z(M). Put K = {x €HI Ix I E S}. It suffices to show that K isE (M). We shall do this by showing that K H . By Lemma 1 it suffices to show that H $ R - K . So suppose H I R - K via f.Then for all x ER we have

I personally do not subscribe to this conjecture. However, I am impressed by the fact that a number of people have tried and failed to deduce a contradiction from ZF + AD.

44 0

S.G. SIMPSON

whence H is n ( M ) a contradiction. Proof of Theorem 1. Let A C M be C(M). Case I : A " M a EM for all a < K . In this Case we shall show that A is A (M). Let A be defined over M by

x EA

-

3 y D ( x ,y )

where D is A (M). For each x E M let h ( x ) be the least 77 such that D(x,y ) holds for some y E Mv. Thus dom ( h ) = A and h is C(M). By the Case hypothesis and the admissibility of M , h [M,] is bounded below K for each a < K . Let g(a) be the least upper bound of h [M,]. Thusg : K -+K and for each x E M we have

By Lemma 2 g is A (M) hence A is H ( M ) q.e.d. Case IZ: negation of Case I. Let a < K be such that A O M , 4 M . Let i E M map R ontoM,. Put I = {r E H 1 i(r) E A } . T h u s Z S R and I is L(M) but not A(M). In particular Z $ R - H hence by Lemma 1 H
Hence there exists a countable admissible set Mo for which the same conclusion holds. The proof that M o exists does not require the assumption of PD outright but only the assumption that PD has an admissible model.

POST'S PROBLEM FOR ADMISSIBLE SETS

441

M has A(M) prewellorderings < and < such that the initial segments of < are uniformly M-finite, and the initial segments of < are M-small. where Y C M is said to be M-finite if Y E M, and M-small if Y ( AlE M whenever A is E (M). We tentatively propose that an admissible set be called thin if it has property (T). Admissible sets of the form L, are thin via the (pre)wellorderings x < y and f ( x )
Bibliography [ I ] K.J. Barwise, R.O. Gandy and Y.N. Moschovakis, The next admissible set, J. Symbolic Logic 36 (1971) 108-120. [ 21 G. Kreisel, Some reasons for generalizing recursion theory, in: R.O. Gandy and C.E.M. Yates (eds.) Logic Colloquium '69 (North-Holland, Amsterdam, 1971) 139-198. [ 3 ] J.E. Fenstad, The axiom of determinateness, in: J.E. Fenstad (ed.) Proceedings of the Second Scandinavian Logic Symposium (North-Holland, Amsterdam, 197 1) 4 1-6 1. [ 4 ] Y.N. Moschovakis, Axioms for computation theories - first draft, in: R.O. Gandy and C.E.M. Yates (eds.) Logic Colloquium '69 (North-Holland, Amsterdam, 1971) 199- 255. [S] G.E. Sacks and S.G. Simpson, The ,-finite injury method, Annals of Math. Logic 4 (1972) 343-367. (61 S.G. Simpson, Degree theory on admissible ordinals, this volume. 171 W. Wadge, Degrees of complexity of subsets of the Baire space, Notices Amer. Math. SOC.19 (1972) p. A-714.

PART V BIBLIOGRAPHY OF GENERALIZED RECURSION THEORY

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

SOME PAPERS ON GENERALIZED RECURSION THEORY ARRANGED ACCORDING TO SUBJECT MATTER A number after an author’s name singles out an item in the Uncritical Bibliography. Thus Grilliot (32) refers to: [32] T. Grilliot, Selection functions for recursive functionals, Notre Dame Jour. Formal Log. X (1969) 225-234. TV after an author’s name refers to his paper in this volume. Recursion in objects of finite type: Aczel and Hinman (TV). Gandy (26, 27). Grilliot (3 1,32,33). Harrington (TV). Kleene (48). MacQueen (75). Moschovakis (80, TV). Platek (92). Sacks (103, TV). Shoenfield (109). Recursion on ordinals: Jensen and Karp (42). Kin0 and Takeuti (43). Kreisel and Sacks (61). Kripke (62). Lerman (68). Lerman and Sacks (69). Owings(88). Platek(92). Sacks(lOl,lO2). Sacks and Simpson(l05). Shore (1 10,111). Simpson (TV). Takeuti (123,124). TuguC (127). Admissible sets: Barwise (8, TV). Barwise, Gandy and Moschovakis ( 1 1). Platek (92). Inductive definability and hyperprojective sets: Aanderaa (TV). Aczel and Richter (TV). Cenzer (TV). Gandy (TV). Grilliot (34). Harrington (TV). Hinman and Moschovakis (40). Moschovakis (81,82). Richter (99). Spector (1 17). Model theoretic, axiomatic and other views of generalized recursion theory: Fenstad (TV). FraissC (21). Friedman (22,23). Gordon (30). Grilliot (TV). Kreisel(58,59). Kunen (63). Lacombe (65,66). Lambert (67). Montague (76,77). Moschovakis (83). Strong (1 18,119). Wagner (128,129).

445

44 6

AN UNCRITICAL BIBLIOGRAPHY OF PAPERS ON GENERALIZED RECURSION THEORY [ 1] P. Aczel, Representability in some systems of second order arithmetic, Israel Jour.

Math. 8 (1970) 309-328. [2] P. Aczel (Abstract) Implicit and inductive definability, Jour. Symb. Log. 35 (1970) 5 99. [3] P. Aczel and W. Richter, Inductive definitions and analogues of large cardinals, in: Conference in Math. Log. - London '70 (Springer, Berlin, 1972) 1-9. [4] J.W. Addison, Some consequences of the axiom of constructibility, Fund. Math. 46 (1959) 337-357. [5 ] J.W. Addison, Some problems in hierarchy theory, Proc. Symp. Pure Math. vol. V (Amer. Math. S O C . , Providence, R.I., 1962) 123-130. [6] J.W. Addison and S.C. Kleene, A note on function quantification, Proc. Amer. Math. SOC.8 (1957) 1002-1006. [ 71 V.I. Amstislavskii, Extensions of recursive hierarchies and R-operations (Russian) Dokl. Ackad. Nauk SSSR 180(1968) 1023-1026. [ 8 ] J. Barwise, Infinitary logic and admissible sets, Jour. Symb. Log. 34 (1969) 22625 2. [ 9 ] J. Barwise, Applications of strict predicates to infinitary Logic, Jour. Syfnb. Log. 34 (1969) 409-423. [ 101 J. Barwise and E. Fisher, The Shoenfield Absoluteness Lemma, Israel Jour. Math. 8 (1970) 329-339. [ 111 J . Barwise, R.O. Gandy and Y.N. Moschovakis, The next admissible set, Jour. Symb. Log. 36 (1971) 108-120. [ 121 S. Bloom, The hyperprojective hierarchy, Zeit. Math. Log. Grund. Math. 16 (1970) 149-164. [ 131 G. Boolos and H. Putnam, Degrees of unsolvability of constructible sets of integers, Jour. Symb. Log. 33 (1968) 497-513. [ 141 R. Boyd, G. Hensel and H. Putnam, A recursion-theoretic characterisation of the ramified analytic hierarchy, Trans. Amer. Math. SOC.141 (1969) 37-62. [15] C.C. Chang and Y.N. Moschovakis, The Suslin-Kleene theorem for V , with cofinality ( K ) = w , Pacif. Jour. Math. 35 (1970) 565-569. [ 161 D.A. Clarke, Hierarchies of predicates of finite types, Memoir Amer. Math. SOC. NO. 51 (1964) 1-95. [ 171 G.C. Driscoll, Jr., Metarecursively enumerable sets and their metadegees, Jour. Symb. Log. 33 (1968) 389-411. [ 18) H.B. Enderton, The unique existential quantifier, Arch. Math. Log. Grund. 1 3 (1970) 52-54. [ 191 H.B. Enderton and H. Putnam, A note on the hyperarithmetic hierarchy, Jour. Symb. Log. 35 (1970) 429-430.

n:

BIBLIOGRAPHY

44 7

[20] S. Feferman and G. Kreisel, Persistent and invariant formulas relative t o theories of higher order, Bull. Amer. Math. SOC.22 (1966) 480-485. [21] R. Frdss6,Une notion de r6cursivit6 relative, in: Infinitistic Methods (Proceedings of the Warsaw Symposium 1959) (Pergamon, Oxford, 1961) 323-328. I221 H. Friedman, Axiomatic recursive function theory, in: R.O. Gandy and C.E.M. Yates (eds.) Logic Colloquium '69 (North-Holland, Amsterdam, 197 1) 113- 137. [ 231 H. Friedman, Algorithmic procedures, generalized Turing Algorithms and elementary recursion theories, in: R.O. Gandy and C.E.M. Yates (eds.) Logic Colloquium '69 (North-Holland, Amsterdam, 1971) 361-390. [24] R.O. Gandy, On a problem of Kleene's, Bull. Amer. Math. SOC.66 (1960) 501-502. [ Z ] R.O. Gandy, Proof of Mostowski's Conjecture, Bull. Acad. Polon. Sci. 8 (1960) 571-575. [26] R.O. Gandy, General recursive functionals of finite type and hierarchies of functionals, Ann. Fac. Sci. Univ. Clermont-FerrandNo. 35 (1967) 5-24. [27] R.O. Gandy, Computable functionals of finite type I, in: J. Crossley (ed.) Sets, Models and Recursion Theory (North-Holland, Amsterdam, 1967) 202-242. [28] R.O. Gandy, G. Kreisel and W.W. Tait, Set existence I, Bull. Acad. Polon. Sci. 8 (1960) 577-583; and I1 9 (1961) 881-882. [ 291 R.O. Gandy and G.E. Sacks, A minimal hyperdegree, Fund. Math. 61 (1967) 215 -223. [ 301 C. Gordon, Comparisons between some generalisations of recursion theory, Compositio Math. 22 (1970) 333-346. [31] T. Grilliot, Hierarchies based on objects of finite type, Jour. Symb. Log. 34 (1969) 177-182. [32] T. Grilliot, Selection functions for recursive functionals, Notre Dame Jour. Formal Log. X (1969) 225-234. [33] T. Grilliot, On effectively discontinuous type-2 objects, Jour. Symb. Log. 36 (1971) 245-248. [34] T. Grilliot, Inductive definitions and computability, Trans. Amer. Math. SOC.158 (1971) 309-317. [ 351 T. Grilliot, Omitting types; applications to recursion theory, Jour. Symb. Log. 37 (1972) 81-89. [36) A. Grzegorczyk, A Mostowski and C. Ryll-Nardzewski, Definability of sets in models of axiomatic theories, Bull. Acad. Polon. Sci. 9 (1961) 163-167. [37] L. Harrington, Contributions to recursion theory in higher types, Ph.D. Thesis, Massachusetts Institute of Technology (197 3). [ 381 J. Harrison, Recursive pseudo-well-orderings, Trans. Amer. Math. SOC.131 (1968) 526-543. [39] P. Hinman, Hierarchies of effective descriptive set theory, Trans. Amer. Math. SOC. 142 (1969) 111-140. [40] P. Hinman and Y.N. Moschovakis, Computability over the continuum, in: R.O. Gandy and C.E.M. Yates (eds.) Logic Colloquium '69 (North-Holland, Amsterdam, 1971) 77-105. 1411 M. Hirano, Some definitions for recursive functions of ordinal numbers, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 10 (1969) 135-141. [42] R.B. Jensen and C. Karp, Primitive recursive set functions, Proceedings of Symposia in Pure Mathematics XI11 Part I (Amer. Math. SOC.,Providence, R.I., 197 1) 143-176.

448

BIBLIOGRAPHY

[43] A . Kin0 and G. Takeuti, On hierarchies of predicates of ordinal numbers, Jour. Math. SOC.Japan 14 (1962) 199-232. [44] A. Kin0 and G. Takeuti, On predicates with constructive infinitely long expressions, Jour. Math. SOC.Japan 15 (1963) 176-190. [45 1 S.C. Kleene, Hierarchies of number-theoretic predicates, Bull. Amer. SOC.61 (1955) 193-213. [46] S.C. Kleene, Arithmetic predicates and function quantifiers, Trans. Amer. Math. SOC.79 (1955) 312-340. I471 S.C. Kleene, On the forms of the predicates in the theory of constructive ordinals 11, Amer. Jour. Math. 77 (1955) 405-428. [48] S.C. Kleene, Recursive functionals and quantifiers of finite type I, Trans. Amer. Math. SOC.91 (1959) 1-52; and I1 108 (1963) 106-142. 1491 S.C. Kleene, Quantification of number-theoretic functions, Compos. Math. 14 (1959) 23-41. [50] S.C. Kleene, Countable functionals, in: A. Heyting (ed.), Constructivity in Mathematics (Proceedings of the 1957 Amsterdam Colloquium) (North-Holland, Amsterdam, 1958) 81-100. [ 511 S.C. Kleene, Herbrand-Godel style recursive functionals of finite types, Proc. Symp. Pure Math. vol. V (Amer. Math. SOC.,Providence, R.I., 1962) 49-75. [52] S.C. Kleene, Lambda definable functionals of finite type, Fund. Math. 50 (1961) 281-303. [53] S.C. Kleene, Turing machine computable functionals of finite type I, in: Logic, Methodology and Philosophy of Science (Proceedings of the 1960 Congress) (Stanford Univ. Press, Stanford, 1962) 38-45; and 11, Proc. Lond. Math. SOC.12 (1962) 245-258. [54] D.L. Kreider and H. Rogers, Jr., Constructive versions of ordinal number classes, Trans. Amer. Math. SOC.100 (1961) 325-369. [55] G. Kreisel, Set theoretic methods suggested by the notion of infinite totality, in: Infinitistic Methods (Pergamon, Oxford, 1961) 97-102. [ 5 6 ] G. Kreisel, Laprkdicative, Bull. SOC.Math. France 88 (1960) 371-391. 1571 G . Kreisel, The axiom of choice and the class of hyperarithmetic functions. Indag. Math. 24 (1962) 307-319. [58] G. Kreisel, Model theoretic invariants: applications to recursive and hyperarithmetic operators, in: J. Addison et al. (eds.) Theory of Models (Proceedings of the 1963 Berkeley Symposium) (North-Holland, Amsterdam, 1965) 190-205. [59] G. Kreisel, Some reasons for generalizing recursion theory, in: R.O. Gandy and C.E.M. Yates (eds.) Logic Colloquium '69 (North-Holland, Amsterdam, 1971) 139-198. [60) G. Kreisel, J . Shoenfield and H. Wang, Number theoretic concepts and recursive wellorderings, Arch. Math. Log. Grund. 5 (1960) 42-64. [61] G. Kreisel and G.E. Sacks, Metarecursive sets, Jour. Symb. Log. 30 (1965) 318-338. [62] S. Kripke (Abstracts) Transfinite recursion on admissible ordinals I and 11, Jour. Symb. Log. 29 (1964) 161-162. [63] K. Kunen, Implicit definability and infinitary languages, Jour. Symb. Log. 33 (1968) 446-451. [64] D. Lacombe, Deux gCnCralisationsde la notion de rkcursivitk, C.R. Acad. Sci. Paris 258 (1964) 3141-3143.

BIBLIOGRAPHY

44 9

[65] D. Lacombe, Deux g6n6ralisations de la notion de r6cursivit6 relative, C.R. Acad. Sci. Paris 258 (1964) 3410-3413. [66] D. Lacombe, Recursion theoretic structures for relational systems, in: R.O. Candy and C.E.M. Yates (eds.) Logic Colloquium '69 (North-Holland, Amsterdam, 1971) 3-17. [67] W. Lambert, Jr., A notion of effectiveness in arbitrary structures, Jour. Symb. Log. 33 (1968) 577-602. [68] M. Lerman On the suborderings of the a-recursively enumerable a-degrees, Ann. Math. Log. 4 (1972) 369-392. [69] M. Lerman and G.E. Sacks, Some minimal pairs of or-recursively enumerable degrees, Ann. Math. Log. 4 (1972) 415-442. [70] A. Levy, A hierarchy of formulas in set theory, Memoir Amer. Math. SOC.no. 57 (1965) 1-76. [71] S.C. Liu, Recursive linear orderings and hyperarithmetic functions, Notre Dame Jour. Formal Log. 3 (1962) 129-132. [72] P. Lorenzen and J . Myhill, Constructive definition of certain analytic sets of numbers, Jour. Symb. Log. 24 (1959) 37-49. [73] M. Machtey, Admissible ordinals and the lattice of a-recursively enumerable sets, Ann. Math. Log. 2 (1970-71) 379-417. [74] M. Machtey, Admissible ordinals and intrinsic consistency, Jour. Symb. Log. 35 (1970) 389-400. 1751 D. MacQueen, Post's problem for recursion in higher types, Ph.D. Thesis, Massachusetts Institute of Technology, 1972. [76] R. Montague, Towards a general theory of computability, Synthese 12 (1960) 429-438. [ 7 7 ] R. Montague, Recursion theory as a branch of model theory, in: B. van Rootselaar et al. (eds.) Logic Methodology and Philosophy of Science I11 (Proceedings of the 1967 Congress) (North-Holland, Amsterdam, 1968) 63-86. [78] Y.N. Moschovakis, Many-one degrees of theH,(x) predicates, Pacif. Jour. Math. 18 (1966) 329-342. [79] Y.N. Moschovakis, Predicative classes, in: Axiomatic Set Theory (Proceedings of Symposia in Pure Math. XIII, Part I, 1967) (Amer. Math. SOC.,Providence, R.I., 1971) 247-264. [ 801 Y .N. Moschovakis, Hyperanalytic predicates, Trans. Amer. Math. SOC.129 (1967) 249-282. [ S l ] Y.N. Moschovakis, Abstract first order computability I, Trans. Amer. Math. Sac. 138 (1969) 427-464; and I1 138 (1969) 465-504. [82] Y.N. Moschovakis, Abstract computability and invariant definability, Jour. Symb. Log. 34 (1969) 605-633. [83] Y.N. Moschovakis, Axioms for computation theories - first draft, in: R.O. Candy and C.E.M. Yates (eds.) Logic Colloquium '69 (North-Holland, Amsterdam, 197 1) 199- 255. [ 841 Y .N. Moschovakis, The Suslin-Kleene theorem for countable structures, Duke Math. Jour. 37 (1970) 341-352. [ 851 A. Mostowski, Development and applications of the projective classification of sets of integers, in: Proc. Inter. Cong. Math. (1965) Amsterdam vol. I11 (E.P. Noordhoff, Groningen) 280-288. [86] K , Ohashi, On a question of G.E. Sacks, Jour. Symb. Log. 35 (1970) 46-50.

45 0

BIBLIOGRAPHY

[87] J.C. Owings, Jr., Recursion, metarecursion and inclusion, Jour. Symb. Log. 32 (1967) 173-179. [ 881 J.C. Owings, Jr., fl; sets, w-sets and metacompleteness, Jour. Symb. Log. 34 (1969) 194-204. sets, can be [89] J.C. Owings, Jr., The metarecursively enumerable sets, but not the enumerated without repetitions, Jour. Symb. Log. 35 (1970) 223-229. (901 J.C. Owings, Jr., A splitting theorem for simple sets, Jour. Symb. Log. 36 (1971) 433-438. [ 91 1 R. Parikh, On the nonuniqueness in transfinite progressions, Jour. Indian Math. SOC.(N.S.) 31 (1967) 23-32. (921 R. Platek, Foundations of Recursion Theory, Ph.D. Thesis, Stanford University, 1966. [93] R. Platek, A countable hierarchy for the superjump, in: R.O. Gandy and C.E.M. Yates (eds.) Logic Colloquium '69 (North-Holland, Amsterdam, 197 1) 257-27 1. [ 941 H. Putnam, Uniqueness ordinals in higher constructive number classes, in: Essays on the Foundations of Mathematics (North-Holland, Amsterdam, 1961) 190-206. [95] H. Putnam, On hierarchies and systems of notations, Proc. Amer. Math. Soc. 15 (1964) 44-50. [96] W. Richter, Extensions of the constructive ordinals, Jour. Symb. Log. 30 (1965) 193- 21 1. [97] W. Richter, Constructive transfinite number classes, Bull. Amer. Math. SOC.73 (1967) 261-265. [ 981 W. Richter, Constructively accessible ordinal numbers, Jour. Symb. Log. 33 (1968) 43-55. [ 991 W. Richter, Recursively Mahlo ordinals and inductive definitions, in: R.O. Gandy and C.E.M. Yates (eds.) Logic Colloquium '69 (North-Holland, Amsterdam, 1971) 273-288. [ 1001 J . Robinson, An introduction to hyperarithmetic functions, Jour. Symb. Log. 32 (1967) 325-342. [ 1011 G.E. Sacks, Post's problem, admissible ordinals and regularity, Trans. Amer. Math. SOC.124 (1966) 1-23. [ 1021 G.E. Sacks, Metarecursion theory, in: J. Crossley (ed.) Sets, Models and Recursion (North-Holland, Amsterdam, 1967) 243-263. [ 1031 G.E. Sacks, Recursion in objects of finite type, in: Proceedings of the 1970 International Congress of Mathematicians (Gauthiers-Villars, Paris, 1971) 25 1-254. sets, Advances in Math. 7 (197 1) 57-82. [ 1041 G.E. Sacks, On the reducibility of [ 1051 G.E. Sacks and S.G. Simpson, The a-finite injury method, Ann. Math. Log. 4 (1972) 343-367. [lo61 B. Scarpellini, A characterization of A; sets, Trans. Amer. Math. Soc. 117 (1965) 441-450. [ 1071 J.R. Shoenfield, The form of the negation of a predicate,.Proc. Symp. Pure Math. vol. V (Amer. Math. SOC.,Providence, R.I., 1962) 131-134. [ 1081 J.R. Shoenfield, The problem of predicativity, in: Essays o n the Foundations of Mathematics (North-Holland, Amsterdam, 1961) 132- 139. ( 1091 J.R. Shoenfield, A hierarchy based on a type-2 object, Trans. Amer. Math. SOC. 134 (1968) 103-108. [ 1101 R.A. Shore, Minimal a-degrees, Ann. Math. Log. 4 (1972) 393-414. [ 11 I ] R.A. Shore, Priority arguments in a-recursion theory, Ph.D. Thesis, Massachusetts Institute of Technology, 1972.

nt

n:

BIBLIOGRAPHY

45 1

[ 1121 S.G. Simpson, Admissible ordinals and recursion theory, Ph.D. Thesis, Massachusetts Institute of Technology, 1971). [I131 C. Spector, Recursive wellorderings, Jour. Symb. Log. 20 (1955) 151-163. [ 1141 C. Spector, On degrees of recursive unsolvability, Ann. of Math. 64 (1956) 581592. [ 1151 C. Spector, Measure theoretic construction of incomparable hyperdegrees, Jour. Symb. Log. 23 (1958) 280-288. [116] C. Spector, Hyperarithmetic quantifiers, Fund. Math. 48 (1959) 313-320. [ 1171 C. Spector, Inductively defined sets of natural numbers, in: Infinitistic Methods (Proceedings of Warsaw symposium 1959) (Pergamon Press, Oxford, 197 1) 97 - 102. [ 1181 H.R. Strong, Algebraically generalized recursive function theory, IBM Jour. Res. Devel. 12 (1968) 465-475. [ 1191 H.R. Strong, Construction of models for algebraically generalized recursive function theory, Jour. Symb. Log. 35 (1970) 401-409. [ 1201 Y. Suzuki, A complete classification of the A; functions, Bull. Amer. Math. Soc. 70 (1964) 246-253. [ 121) M. Takahashi, Recursive functions of ordinal numbers and Levy’s hierarchy, Comment. Math. Univ. St. Paul 17 (1968) 21-29. [ 1221 H. Tanaka, On analytic wellorderings, Jour. Symb. Log. 35 (1970) 198-204. [ 1231 G. Takeuti, On the recursive functions of ordinal numbers, Jour. Math. SOC. Japan 12 (1960) 119-128. [ 1241 G . Takeuti, Recursive functions and arithmetic functions of ordinal numbers, in: Y . Bar-Hillel (ed.) Logic, Methodology and Philosophy of Science I1 (Proceedings of the 1964 Congress) (North-Holland, Amsterdam, 1965) 179-196. [ 1251 S.K. Thomason, On initial segments of hyperdegrees, Jour. Symb. Log. 35 (1970) 189-197. [ 1261 T. Tug&, Predicates recursive in a type-2 object and Kleene hierarchies, Comment. Math. Univ. St. Paul 8 (1 959) 97- 117. [ 1271 T. Tug&, On the partial recursive functions of ordinal numbers, Jour. Math. SOC. Japan 16 (1964) 1-31. [128] E.G. Wagner, Uniform reflexive structures: on the nature of Godelizations and relative computability, Trans. Amer. Math. SOC.144 (1969) 1-41. [ 1291 E.G. Wagner, Uniformly reflexive structures: an axiomatic approach to computability, Informat. Sci. 1 (1968) 343-362. [ 1301 H. Wang, Ordinal numbers and predicative set theory, Zeit. Math. Log. Grund. Math. 5 (1959) 216-239.

INDEX *

Aanderaa, S., 207, 221,245,263, 305,381,445 Aczel, P., 3, 19,41,4345,49,50, 52, 110, 121,212,220,221223,238,263, 266,270-271, 278,287,291,296-297,301, 306,333,340, 381,387,397, 40 1,404,445446 Addison, J.W., 218, 220, 250, 26 1,263,446 Amstislavskii, V., 446 Bachmann, H., 296-297 Barwise, K.J., 97, 107, 110, 112, 114, 120, 122, 196, 198,200, 204, 263,265-266,270-272, 278, 281,287,295,297, 334335,354,381,401404,427, 439,441,445-446 Beth, E., 275,297 Blass, A., 260 Bloom, S., 446 Boolos, G., 446 Boyd, R., 446 Bridge, J., 296-297 Cenzer, D., 221-223,225,229, 233,240,247,256,263, 304,381,445 Chang, C.C., 446

*

Chong, C.T., 190 Clarke, D .A ., 446 Cohen, P., 172 Devlin, K., 123,313-314,316, 38 1 Driscoll, G. C., 446 Enderton, H.B., 446 Feferman, S., 89,93,271,294, 296-297,447 Fenstad, J.E., 220,385,437,441, 445 Fisher, E., 446 FraissC, R., 407,4 15,418,420, 445,447 Friedberg, R., 172,437 Friedman, H., 112,121,385,404, 445,447 Candy, R.O., 4,33,39,40-41,43, 49,50-52,58,78,84,92-93, 107, 110, 121, 200,204,221, 263,265,270-272,297,303, 334-335,339,354,373,381, 402404,439,441,445-447 Gentzen, G., 296 Godel, K., 86,93, 144, 149, 192,258,263

Boldface numbers refer to title pages of authors’ chapters in this volume.

453

454

Gordon, C., 107,270,297,445, 447 Costanian, R., 286, 298 Grant, P., 281 Grilliot, T.J., 49, 50-52, 58, 68, 81-82,93,219, 238, 270-271, 278, 289, 290, 293-295, 298, 340,400,404,405,407,410, 414-415,418,420,421,445, 447 Grzegorczyk, A., 271, 298,429, 447 Hanf, W., 306,381 Harrington, L., 19, 25, 43, 220. 445.447 Harrison, J., 447 Hensel. G.. 446 Ilinman, P.G.,3,41, 43-45. 49, 50, 52,92, 238,293, 295, 297, 387, 397,401,404,445, 447 Hirano. M., 447 Isles, D., 296-297 Jensen, K., 123-125, 140-141, 144, 163, 166, 191-193, 260, 263, 273, 298.313-314,317. 320,322,324,445,447 Jockusc1i.C.. 191 Karp,C., 273,298, 313-314, 317.320, 322, 324,376, 381,445,447 Kechris, AS.,53. 78,419 Keisler, J.H., 113, 115, 121 Kino, A . , 445,448 Kleene, S.C., 3 , 4 , 7 , 2 9 , 33-34, 36, 41, 43, 5 2, 54-55,57,67,

INDEX

71-72,78-79,81-83, 8 6 , 8 8 , 93, 167, 221,263, 266,268269,286,288-289,298, 302,344,364-367,373,381, 386,400,434435,445-446, 448 Kreider, D. L., 448 Kreisel, G., 86, 93, 166, 193, 198,201,204,232,263, 271,289,298,415,420, 437,445,447-448 Kripke, S., 98, 152, 257, 263, 279,302,312,381,445,448 Krivine, J.L., 198, 201, 204, 419,420 Kunen,K., 110, 121, 196,271, 298.445,448 Lachlan, A . , 190, 193,430 Lacoinbe, D., 407,418, 420, 445,448-449 Lambert , W ., 445,449 Lerman, M., 166, 190, 192-193, 445,449 Levy. A . , 8 4 , 9 3 , 9 9 , 105, 115, 121,258,263,269,306-308, 330-33 1 , 3 8 1.449 Linden, T., 318,381 Liu, S.C., 449 Lopez-Escobar, E., 284. 298 Lorenzen, P . , 449 Lyndon, R., 294 Machtey, M., 166, 193,449 MacIntyre, J.M., 190, 193 MacQueen, D.B., 52, 82, 292, 298, 445,449 Martin, D., 208, 220 Martin-Lof, P., 296-298 Moldestad, J., 387,398,400, 404

INDEX

Montague, R., 270,298,397,400, 445,449 Morley, M., 115 Moschovakis,Y.N.,7,41,49, 52, 53, 57-58,60,64,68-69,73-74, 79,84,93, 107, 110, 121,200, 204,208,218,220,263,265266,269,270-271,275278, 288,292,295,297-299,302, 304,334-335,354,381,385392,394-396,398404,407, 415,417-418,420,421-423, 425,44544,449 Mostowski, A., 271,298-299,447, 449 Muchnik, A.A., 437 Myhill, J . , 449 Normann, D., 387,398-399 Novikoff, P.S., 250 Ohashi, K., 449 Owings, J . , 445,450 Parikh, R., 450 Paris, J., 191 Peano, G., 405 Pfeiffer, H., 295, 299 Platek, R., 4-6, 9, 19, 21, 26, 33-35, 41,43-45,50, 52,79,84-85, 93,98,103,152,257,263,279, 286,288,292-293,299,302, 445,450 Post, E., 268,302,437 Putnam, H., 222,264,302-303, 356,381,446,450 Rice, H., 432-433 Richter, W., 22,29,41,4344,48, 52,212,220,221-222,242-

243,263-264,270-271,297, 299,301,302-303,306,333, 340,380-381,445446,450 Robinson, I.,450 Rogers Jr., H., 68, 170, 172, 177, 185, 191, 193,208-210,220, 264,330,381,448 Ryll-Nardzewski,C., 271, 298,447

Sacks, G., 52,53,79,81,92-93, 165-167, 169-170, 182, 189190,193,232,262,292,299, 304,381,399,404,437,441, 445,447-450 Scarpellini, B., 450 Schmidt, D., 291,299 Schutte, K., 296,299 Scott, D., 306,381 Shoenfield, J., 4,9,22,24-28,41, 44-45,52,82,88,93,97, 121, 189-190, 193,233,249-250, 258,264,358,381,445,448, 450 Shore, R., 166, 182, 189, 191, 193,445,450 Simpson,S., 165, 166, 170, 182, 190, 192-193,404,437,441, 445,450-45 1 Smullyan, R.M., 268,299 Solovay, R., 260, 263 Spector, C., 67,79,86,93, 178, 190,220,221,233,264,268, 299,303,381,445,451 Strong,H.R., 385,404,445,451 Suzuki, Y ., 45 1 Takahashi, M., 45 1 Takeuti, G., 165,445,448,451 Tait, W.W., 204,447 Tanaka, H., 305,381,451

455

456

Thomason, S.K., 451 Tugue, T., 445,45 I Vegem, M., 398,404 Ville, F., 195, 271

INDEX

Wadge, W., 439,440-441 Wagner, E.G., 385,404,434, 445,451 Wang, H., 448,45 1