Zcilschr. 1. math. L&k und I3rundhgcn d. Math. Bd. 24, S. 419-425 (1978)

ON THE COMPLEXITY OF ANALYTIC SETS

by KAREL HRBACEK in New York, N.Y. (U.S.A.)

Recently there has been some interest in classification of non-Bore1 analytic (2;) and coanalytic (nt) sets by their complexity. First, A. MAITRA and C. RYLL- NARDZEWSKI [ 7 ] have proved that there exist two analytic non-Bore1 sets which are not isomorphic by a Borel function if the Axiom of Constructibility is assumed. D. MAULDIN [S] then improved this result by showing existence of three analytic non- Borel sets, no two of which are Borel isomorphic, and posed a question whether “three” can be improved to “2‘0’’. (Actually, both [7] and [S] require only existence of some uncountable ZZ; set without a perfect subset.) On the other hand, J. STEEL [lo] has deduced that all analytic non-Bore1 sets are Borel isomorphic from the assumption that all ni games are determined.

Independently, and motivated by quite different questions, R. GOSTANIAX and the author introduced in [3] the notion of L-reducibility for sets of reals and proved, using the Axiom of Constructibility, that there exist two analytic non-Bore1 sets which are not L-reducible to each other. By definition, A is L-reducible to B if A can be obtained from the open sets by applications of the operations countcable union, com- plement, and 23, where

W A,, = ( X E w I { T & E w I X E A,L} E B } .

Since it is easy to see that Borel isomorphic sets are L-reducible to each other, but not necessarily vice versa, this result is an improvement on [ 7 ] . However, methods used in [3] are entirely different from those of [TI.

Another notion of reducibility for sets of reals, st,ronger yet than L-reducibility, has been studied; i t is the “boldface” version of Kleene’s relative recursiveness. We will write A B if A is recursive in B, E and some real X . ( E is a type two object representing existential quantification over integers.) A < B will mean that A B, but not B _I A . A and B have the same S-degree if A A . Clearly, if A is L-reducible to B, then A B. SOLOVAY [ll] has proved that the S-degrees are wellordered by d if one assumes the Axiom of Determinacy.

In this note we extend the method from [3] to show that, assuming the Axiom of Constructibility holds, for any analytic sets S, T with S < T, there exist 2’0 analytic non-Bore1 sets whose S-degrees are intermediate between the S-degrees of S and T and pairwise incomparable with respect to s. More precisely, we prove the following

Then there is a relation R A EW”, S

27*

n m

B and B

Theorem (V = L). Let S E,W” and T cow be Z; in D wW x w“, Zi in D, such that S

and let S < T. T and, for all

RA and RA s R A , where Ra4 = R n ( { A } x w”) and RZI = R - R., =

R - *

= R - ( { A } x w”) .

420 KAREL HRBACEK

Corollaries.

(0) S-degrees of analytic sets are dense. In particular, there is no minimal analytic tioir-Borel S-degree.

(1) Let < be any partial ordering of power at most 2*0 whose initial segments are at most countable. Then < can be represented as a n ordering of a collection of S-degrees of analytic sets by s.

(2) There exist 2 N 0 pairwise incomparable S-degrees of analytic sets. I n particular, there exist 2*0 analytic sets, no two of which are Bore1 isomorphic. (This answers a question of D. MAULDIN in 191.)

(3) There exists a descending chain of X-degrees of analytic sets of length N~ and a n ascending chain of such degrees of length N ~ .

(4) Let < be any A: partial ordering of w , Then < can be represented as an ordering of 8-degrees of 2: spts by $.

( 5 ) There exist N,, pairwise incomparable X-degrees of Z: sets.

Proof of the Corollaries.

(0) If S < T, then S < RB < T. For the remaining corollaries, S = 0 and T is the universal Zi set. Then S < T

by the result of GANDY [2] quoted under (f) below. For every countable F 5 ww, let P E w w x w denote some enumeration of F ; i.e., F := {An P(m, 71.) I m E 0).

(1) We first represent < as an ordering of a collection $ of countable subsets of w“ by inclusion & ; we then assign to each F E $ the S-degree of the set RF = = R n ( F x cow). If Fl F,, RFl is not recursive in RF2, E and any real B : otherwise we could pick A E Fl - F,; R n ( { A ) x w“) would be recursive in RF,, E and the reals A , B ; and so we could conclude that R n ( { A } x cow) is recursive in R - ( { A } x ww), E and the reals A , B and F , , contradicting the Theorem.

A em*’) by in- clusion. In other words, the sets R n ( { A ) x w”) for A E ww have pairwise incompar- able S-degrees.

F,, RF, is clearly recursive in R,, E and F,. If Fl

(2) follows from (1) by letting < be the ordering of $l = { { A )

(3) follows immediately from (1).

(4) For each n E w let f l l E cow be defined by: f J 1 ) = 0 if i = n, f,l(i) = 1 otherwise. F,, iff Assign to each n E O the set Fn = if,,, I m < n) . Clearly m < n iff F,,,

RFm RF9&. Also, the sets RFn are 2: for all n E a). The rest follows from (1).

( 5 ) A trivial consequence of (4). 0 We now proceed with the proof of the main Theorem. First of all, since S-degrees

are closed under complementation. the Theorem is equivalent to a statement obtained from it by replacing the words “Zi” and “analytic” by “D;” and “coanalytic”, resp. We will prove the Theorem in this dual form.

We assume that the reader is acquainted with a few elementary facts about ad- missible sets and primitive recursive set functions. In particular, we will need the following :

ON THE COMPLEXITY OF ANALYTIC SETS 42 I

(a) Let ( M , E ) , E 5 M x M , be a model of the KRIPKE-PLATEK set theory KP. Then the standard part of ( M , E ) ,

sp(M, E ) = {x 1 E r x is isomorphic to E r [n E M I n,Em} for some m E M }

is an admissible set,.

(b) The Godel pairing function p and it,s inverses I, and I, are primitive recursive functions.

(c) There is a primitive recursive one-to-one mapping N of the ordinals onto t,he constructible sets such that L(a) g N"b(a) for all a , where b is a suitable priniit,ive recursive function. @(a) is the a-th level of the construct'ible hierarchy.)

(d) Every primitive recursive function is &-definable and its defining formula is absolute in every admissible set. We refer the reader to the Appendix A6 in [l] for the proof of (a), and to [4], esp. 1.5(3), 3.2 and 2.11, for proofs of (b), (c) and (d).

We will often identify w x w with w (using the Godel pairing function p ) and sub- sets of w 0r.w" with their characteristic functions.

An admissible set M is locally countable if w E &' and for each x E ilir there is f E iM such that f is a one-to-one mapping of x onto (0.

It is well known that every admissible set M has the following property: If < E 111 is a wellfounded relation, then there is f E M such that f maps the field of < into the ordinals of M and x < y implies f ( x ) < f ( y ) for all x: and y in the field of <.

An admissible set M has the /?-property if the following holds: If < E *$f is a non- wellfounded relation, then there is f E M such that, f maps w into the field of < and f(n + 1) < f(n) holds for all n EO.

Our treatment of recursion in type two objects will rely heavily on the expositions by KLEENE [5] and GANDY [2]. In particular, we will use the following consequences of GAKDY'S Selection Theorem :

(e) If T 5 w", R & coo, and both R and oY' - R are semirecursive in E, T and D E ow, t,hen R is recursive in E, T and D.

(f) S E w" is semirecursive (recursive) in E and D E ww if and only if it is I7: ( A : ) in D.

Finally, we will make use of the classical representation theorem for 17: sets (see e.g. SHOENFIELD [12]) :

(g) For every S ww which is II: in D E (ow there is a functional P recursive in D such that, for all A ~ w " , <s,ii = ATL F ( A , n) is a linear ordering of w , and A E S iff +, dl is wellfounded.

Lemma. Let S E w" be IT: in D ~ w " . Then there is a Zl formula rp of set theory such that the following holds: If an admissible set M has the /?-property and D E M , then for all z , a: w E (I) and all C E ow n M

( z } (a , C, E, S) w i f f { z } (a , C , E, S n M ) w

In particular, F r M is Z1,(M) for every functional F recursive in E, S and some real D E M .

i f f (M, E r M ) k rp(z, a, C, W , D).

422 KARELRRBACER:

Proof. The analysis of computations performed in section 5 of [5] shows that { z } (a , C, E, S) E w iff there exists an rj such that

(2.1) [ (Vy E o) (if y is an 11-position, then 11 is locally correct a t y ) A

A 11(Yo) = A (Ye) (3%) (B(etd specifies a tip)] (here 7, e are type one objects, i.e., reals; yo denotes the 0-position; en denotes the n’th position along the branch e, and B(y) denotes the computation description a t the position y ) . Moreover, such an rj is unique, if it exists a t all.

After examining the details of the proof of this fact in [5], it is easy to state a Zl formula y of set theory expressing (2.1), i.e., such that

(2 .2) (2) (a, C, E, S) w iff (311) y(q, 2, a, C, w, Q. For example, to express local correctness a t a position where the rule 58 is being

applied, we write down

q(y) -1 W --f (3x) {x E ow A (Vn E W ) ( X ( n ) = q(yn(n))) A @ ( X ) = W}

where in case aj = E w0 replace a J ( X ) = w by

[w = 0 A (3n E o) ( X ( n ) = 0)] v [w = 1 A (Vn E w ) ( X ( n ) =k O)]

and in case aJ = S we replace a J ( X ) = w by

[W = 0 A Wl] V [W = I A y 2 ] ,

where y1 is a Zl formula expressing “ X E S ” :

(2.3) ( 3 f ) [ ( f maps the field of into ordinals) A

A (VZ, Y E d o m ( f ) (Z +, x Y + f ( 4 < f ( Y ) ) 1 and y2 is a Z1 formula expressing “ X 4 S ” :

(2.4) ( 3 f ) [ ( f maps w into the field of A (Vn E o) ( f ( n + 1) < S , X f(n))]. Similarly, to express the wellfoundedness condition

(Ve) (3n) ( B ( p t i ) specifies a tip) we write down

(2 .5) (If) [ ( f maps the set T, of all q-positions y for which b(y) does not specify a tip into ordinals) A

A (Vz, y E dom(f)) (the position x extends the position y -+ f(x) < f ( y ) ) ] .

We now let v to be the formula (37) y , and prove the following claim by induction on { z } ( - ) :

If { z } (a, C, E, S) is defined and { z } (a, C, E, S) = w, then ( z } (a, C, E, S n M ) = w and ( M , E r M ) k ~ ( 2 , a , C, w, D).

For example, let z specify an application of S8 with a] = S. Then w = S ( X ) where X = hz{z’} ( (a , n), C, E, S) and z‘ is an appropriate index coded in z . By inductive assumption, {z’} ( (a , n), C, E, S) % w iff (z’] ((a, n), C, E, S n M ) s w iff ( M , E r M ) k k (317) y(q, z’, ( a , n), C, w, D). Since (z’} (a , C , E, S) is defined, {z’} ( (a , n), C, E, S) is defined for all n E o, and thus ( M , E r M ) k (Vn E o) ( 3 ! u E o) (311) y. We can now m e Zl-collection and A,-separation in M to conclude that X = ((n, w) I (311) y} E M , and h o X E S n M . Since X = h{z‘} ( (a , n}, C, E, S n M ) by inductive assumption,

ON THE COMPLEXITY OF ANALYTIC SETS 423

we conclude that { z } (a, C, E, S n M) = w. Using Z1-collection (and uniqueness of q's) again, we obtain a sequence (qn I n E w ) E M such that ( M , E r M ) k (Vn E w ) ~ ( q , ~ , z ' , ( a , a ) , C , X ( n ) , D) . From (qn I n EO) one easily constructs 7 E M for which y(q, z , a , C , w, D ) holds. Moreover, q E M and X E M imply that <s,x E M , Tq E M , and we can use admissibility and @-property of M to find, in M , the witnesses f to wellfoundedness or non-wellfoundedness of these relations required by (2.3), (2.4) and (2.5). These considerations show (M, E M) 1 y(q, z, a , C , w) and complete the proof of the claim.

An ent'irely analogous argument shows that, if { z } (a , C , E, S n M) is defined and { z } ( a , C . E, S n M) = 10, then { z } (a, C , E, S) = w and (M, E r M ) 1 p.

Finally, if ( M , E 1 M) k cp, then holds (in the real world), and {z> (a, C, E, S) z w by (2.2).

The lemma follows immediately from these assertions. 0 We will now proceed with the proof of the Theorem. Let then S, T be fixed sets

Let T be the set of all countable admissible ordinals z such that

(i) L, is locally countable and has the p-property; (ii) For every z E w , C EL, n ww and 6 < t there is X EL, n ww such that

T ( X ) $ [ z } (C , X , E, S) and < T , S either is not a wellordering or has order type 2 6. We first' notice that LR1 has the properties (i) and (ii). Clearly LRl is locally count-

able (we are assuming V = L!) and has the @-property. Next, assume that T ( X ) = ( z } (C , X, E, S) for all X E ww such that < T , X either is not a wellordering or has order type 2 6. Since (X E ww I < T , x is a wellordering of order type <S} is A: in D and G E ow, where G is some wellardering of type 6, and therefore recursive in E, D and G by (f), this would show T

It, now follows by an easy downward Skolem-Lovenheim type argument that there exist, arbitrarily large countable admissible ordinals z E T. We let (zY 1 y < N ~ ) be an increasing enumeration of all elements of T and their countable limit points.

Defini t ion. For all A , B E wo, ( A , B ) E R iff B E T and there is y < N~ such that A = N(Z,(y)) and zy 5 the order type of < T , B < T , , + ~ .

We have to prove that R has the required properties.

(I) Let ( M , E ) be a model of KP and let B E T n sp(M, E ) . Then < T , B E sp(M, E ) and, if ,9 denotes the order type of < T , B , also ,4 E sp(M, E ) . If z is any admissible ordinal 5 /3 and W E L , is a non-wellfounded relation, then W has a descending chain in the next admissible set L,+ g sp(M, E ) , and, consequently, in M. Thus the set of all admissible ordinals z 5 p which are locally countable and have the @-property is definable in (M, E ) [uniformly for all ( M , E ) k KP and all B E T n sp(M, E)]. If now T belongs to this set, then both T n L, and ((2, C , X , w) EL, I { z } (C , X , E, S) Z w} are (Zl) definable over L,, uniformly for all such z [we need the Lemma for the second set], so that the verification of the condition (ii) for z can be carried out inside (M, E ) . These considerations show that T n @, and therefore also the sequence (z, I c f y ) where y is the largest ordinal for which zy the order type of < T , B , are definable in ( M , E ) [uniformly for all (M, E ) 1 KP and all B E T n sp(M, E ) ] . The point simply

of reals II,' in D and such that S < T.

S, contradicting our assumption.

424 KAREL HRBACEH

is, that, given B E T , the remaining requirements for ( A , B ) E R can be verified inside any model ( M , E ) of KP such that A , B E M . Now it should be obvious that one can produce an arithmetic formula x with a parameter D such that, for all A , E EW“‘

and all B E T, x ( A , B, E ) iff (0, E ) i= KP, A , B E sp(w, E ) and A = N(Z,(y)) where y is such that t 5 the order type of < T , B < xyfl . We then have ( A , B ) E R iff B E T A (YE) x ( A , B, E ) iff B E T A (3E) x(A, B, E ) . The first equivalence shows that R is IT,’ in D, and also semirecursive in T, E and D [see (f)]. From ( A , B) q! R iff B 4 T v (VE) - x ( A , B, E ) we conclude that coo - R is also semirecursive in T, E and D , using (f) again. By (e), R is recursive in T, E and D .

(11) We now prove that RAi is not recursive in Rg , E, S and C for any C E (I)’’. Assum- ing V = L, we can choose an ordinal y < ~l~ such that A , C E L,, and A = N ( l l ( y ) ) . By definition of R , the order type of < T, is less than ty for all A’, B E <ow such that (A’ , B ) E RAI n L7y+l. Choose a wellordering G E L,,+l n cow isomorphic to tl, (this is possible because of local countability of LTy+I); we then have, for all B E L,

-

(A’ , B ) E i A i f f ( 3 f ) [f is an isomorphic mapping of < T , B into GI A

A A

< T , B is wellfounded A A + A’ A

A (Vf) [f is not an isomorphic mapping of G into < T , ! $ ] A

A’ A (3E) X(A’, B, E ) iff

A (VE) x(A’, B, E l . Moreover, the second and third statements are equivalent for a l l A‘, B E w “ ; and

thus define a set of pairs of reals R* which is A: in GI and D, and such that R.I n Lt,+l = = R* n L7y+l.

Let us assume that R., is recursive in R ,, E‘, S and C E w w ; i.e., R , ( X ) E ( z } (C, X , E, (i,, S ) ) holds for a suitable z E CD and all X E ow. Then RA4(X) z (z> (C, X, E, (R*, S ) ) holds for all X E LTy+l, by our Lemma. But R* is recursive in E, G and D by (f), so we have RA4(X) z (z*} ((C, D, G), X, E, S) for some z* E 01 and all 9 E LTy+l . However, the definition of R implies that

Rd n ( X E LTy+l n cow 1 < T, either is not a wellordering or has order type 2 r,> = = T n { X E LIYil A coo 1 -< T, either is not a wellordering or has order type >= T, ) . So we would have T ( X ) z {z*> ((C, D, G), X, E, S ) for all X E LTy+l n wQ for which < T, is not a wellordering or has order type 2 t, , ,and (C, D, G ) E L,,,, n cow, contra- dicting property (ii) of z,,+~.

(111) The preceding arguments show that R has all the properties required by the Theorem except perhaps S 2 R We modify R by setting

( A , B ) E R‘ iff either B(0) = 0 A ( A , (B(l), B ( 2 ) , . . .)) E R or B(0) + 0 A (B(l), B(2) , . . .> E S .

Then clearly S RL for any A E o” and R‘ shares all the previously proved properties of R.

We conclude with a few remarks and open problems. In view of (f), the Theorem and its Corollaries can be interpreted as t i boldface” analogues of classical theorems about recursively enumerable Turing degrees. E.g., Corollaries (0) and ( 2 ) assert that

ON THE COMPLEXITY OF ANALYTIC SETS 125

the upper semilattice of semirecursive S-degrees of sets of reals is dense and contains incomparable elements (if V = L). One might conjecture that it is actually element- arily equivalent to the upper semilattice of recursively enumerable Turing degrees of sets of integers. but this is not the case, as the next corollary shows:

Corollary. (6) If S and T are 2; sets of reals and S < T, then there exist 2: sets of reals RI and R2 such that S < R1 < T, S < R2 < T, R1 3 R2, R2 2 R1, a id for aiiy set of reals U, if R1 U and R2 U, then T U.

LACHLAN [6] showed that an analogous statement is false for recursively enumer- able sets of integers.

Proof. Define B E R1 [R2, resp.] iff B E T and t’here is y < czl such that ty 5 the order type of < T , B < and y is even [odd, resp.]. Notice that T = R1 u R2. The rest is as in the proof of the main Theorem.

Perhaps the most interesting open problem related to the subject of this paper is the question whether the assumption of the Axiom of Constructibility in tfhe above results can be weakened to that of “Not all I7: games are determined.” The best result in this direction is due to L. HARRINGTON (see [lo]), who proved that the latter assumption implies existence of two analytic non-Bore1 sets which are not Borel iso- morphic. Some structural questions remain open even assuming V = L; for example: Are there analytic non-Bore1 sets S and T such that, if U S and U =( T then U is Borel?

No te (Added July 20, 19%). We have extended and modified methods of this paper to obtain a negative answer to the last question, as well as other structural in- formation about S-degrees. Many of our results (e. g. Corollaries (1) - (5)) follow from an assumption on behavior of admissible ordinals which was shown t o hold in all generic extensions of L (via a set of conditions) by S. SIMPSON. This work will appear in a future publication.

References [l] BARWISE, J., Absolute logics and L m w . Ann. Math. Logic 4 (1972), 309-340. [2] GANDY, R. O., General recursive functionals of finite type and hierarchies of functions. Mimeo-

graphed paper given a t the Symposium on mathematical logic held at the University of Cler- mont-Ferrand in June 1962.

,[3] GOSTANIAN, R., and K. HRBACEK, Propositional extensions of L,,, . To appear. [4] JENSEN, R. B., and C. KARP, Primitive recursive set functions. In: Axiomatic Set Theory.

[5] KLEENE, S. C., Recursive functionals and quantifiers of finite types I. Trans. Amer. Math.

[6] LACIILAN, A. H., A recursively enumerable degree which will not split over all lesser ones.

[7] M~ITRA, A., and C. RYLL-NARDZEWSRI, On the existence of two analytic non-Bore1 sets which

[S] MAULDIN, D. R., On non-isomorphic analytic sets. Proc. Amer. Math. SOC. To appear. [9] MAULDIN, D. R., Non-isomorphic projective sets. To appear.

Proc. Symp. Pure Math. 13, Part I, Amer. Math. SOC. 1971.

SOC. 91 (1959), 1-52.

Ann. Math. Logic 9 (1975), 307 -365.

are not isomorphic. Bull. Acad. Polon. Sci. 18 (1970), 177 - 178.

[lo] HARRINTON, L., and J. STEEL, Analytic sets and Borel isomorphisms (abstract). Notices Amer.

[ll] SOLOVAY, R., Determinacy and type-2 recursion (abstract). J. Symb. Logic 36 (1971), 374. [12] SHOENFIELD, J. R., Mathematical logic. Addison-Wesley Publ. Co., 1967.

Math. SOC. 23 (1976), A447.

(Eingegangen am 3. November 1976)