Chapter 4 Structure Theory for Pointclasses

CHAPTER 4

STRUCTURE THEORY FOR POINTCLASSES We are now ready to plunge into a systematic study of the structure of Il: and 2;. In many ways, this chapter is a continuation of Chapter 2; here too we will establish various interesting properties of Xi and 2 : sets, in fact we will answer several natural questions left open there. What is new and different is that we will use systematically the methods of the effective theory which we developed in the preceding chapter. It turns out that this infusion of ideas from recursion theory creates a more radical change in the flavor of the subject than one might think. It is not just a case of obtaining “finer” results about the lightface pointclasses with a little more computation, as we did in Chapter 3. Even when we prove theorems which are significant only for the boldface pointclasses, we will use recursion theory to great advantage. The most important results of the chapter are uniformization theorems, particularly the Novikov-Kondo-Addison Theorem 4E.4 and the A Uniformization Criterion 4D.4. The latter implies that in many special circumstances we can uniformize a Borel set by a Borel set. As in Chapter 2, we will formulate many of the results of this chapter in a general setting, to ease extension to the higher projective pointclasses. This will lead us naturally to the axiomatic definition of a Spector pointclass, one of the key notions of the subject. Specifically for uniformization results, the notion of a scale will also prove very important. Perhaps this is the most important chapter of this book, because it is the most characteristic of our subject. One could say that Chapter 1 was mostly topology, Chapter 2 was set theory and Chapter 3 was recursion theory; but this chapter would be out of place in anything but a book on descriptive set theory.

4A. The basic representation theorem for Il: sets Most of the results of Chapter 2 depended directly on the fact that 2: sets are Ko-Suslin. Here we will first formulate an effective version of this 190

4A.11

THE BASIC

REPRESENTATION THEOREM FOR

n:

SETS

191

fact and then refine it to a representation theorem for II: sets which is the key to the structure properties of this pointclass. Recall from 3D.8 (*20) that

& ( n )=(a(O), ..., a ( n - 1)). This is a recursive function of a and n.

4A.1. THEOREM. (i) A pointset Pc X x." is a set Q G X x w 1 such that

sy

(1 2 1) is Z?if and only if there

Moreover, i f X is of type 0 or 1 , then Q may be chosen to be recursive. (ii) A pointset P z X is n;,if and only if there is a 27 set Q E X x w such that P ( x )CJ (Va)(3t)Q(x,C ( t > >

and [Q(x,C(t))&f
Moreover, i f X is of type 0 or 1, then Q may be chosen to be recursive. PROOF.(ii) follows immediately from (i). To prove (i), take I = 1 for simplicity of notation and suppose by 3C.5 that

P(x, a ) * (3u)(3v){xE N(X, u ) & a E N ( X , v ) & P*(U, u ) } with P" semirecursive, so there is a recursive R such that P ( x , a ) CJ (3u)(3v)(3n){xE N(X, u ) & a E N ( X , v ) & R(u, v, n)}. By 3B.5, there are recursive functions g, h such that (I!

E N ( N , v ) M ((v)J1 f 0 & (Vi < g ( v ) ) [ a ( i= ) h(v, i)],

so that whenever t 2 g(v), we easily have a E N ( N , v ) H ( ( u ) , )f~0 & (Vi < g(v))[(E(t)), = h(v, ill.

192

STRlJCrURE THEORY FOR POINTCLASSES

[4A.2

Now put

Q ( x , w ) #Seq(w) & (3uIlh(w))(3v5 Ih(w))(3nIlh(w)){xE N ( X , u ) & g ( v )IIh(w)& (Vi < g ( v ) ) [ ( w )= , h(u, i ) ]

R(u, Q, n)) and verify easily that

P(x, a)

@

(3t)Q(x, E ( t ) ) .

If X is of type 0 or 1, then Q is recursive since { ( x , u ) :x E N(X,u ) } is recursive by 3C.3. -I With each irrational a we associate the binary relation on w 5, = {(n,

rn): a((n, rn))

= 1)

and we put a E LO H 5, is a linear ordering e (Vn)(Vrn)[nI,

rn

P,

( n l , n & r n ~ rn)l ,

& (Vn)(Vrn)[(ns, rn & m5, n) 4 n = rnl & (Vn)(Vrn)(Vk)[(nl, rn & rnsak ) =+ n l , k ]

& (Vn)(Vrn)[(ns, n & r n ~ rn) , 9 ( n s , rn v rnsan)], a E WO e I, is a wellordering H

a E LO & < a has no infinite descending chains

*a

8~( V p ) [ ( V n ) [ P ( n+ 115, P(n)l+ (3n)[P(n + 1) = P(n)]].

If a E LO, let la I = order type of sa.

In particular, the mapping a * IaI

takes WO onto the set of countable ordinals and provides a coding for this set in the sense of 3H.

4A.2. THEOREM. The set WO of ordinal codes is IT:. Moreover, there are

THE

4A.31

relations

Sn,

n: SETS

BASIC REPRESENTATION THEOREM FOR

sz in

n: and Z;

193

respectively, such that

p E wo + {a' n p * a 5 z p * [aE wo & la1 5 lpll}.(s) PROOF.That WO is is obvious from the formulas above. To prove the second assertion. take first a5

2P

-

a E LO & ( 3 y ) [ y maps sainto spin a one-to-one

order-preserving manner]

a E LO L?L (3y)(Vn)(Vrn)[n
It is immediate that sris

m

-+

r(n)<,y(rn)l

Z: and for p E WO,

a 5 z p *[(YE

wo & lal5Ip1].

For the relation sn,take a s n p e a E WO & there is no order-preserving map of s ponto an initial segment of sa H

a E WO & (Vy) i ( 3 k ) ( V n ) ( V r n ) [sg n rn *;, [ y ( n )%r(m> < a kll,

where of course we abbreviate s<,

t

s s a t & s f t.

4A.3. THEBASICREPRESENTATION THEOREM FOR n: (Lusin-Sierpinski, Kleene('-'l'). A pointset P c_ X is if and only i f there is a A: function f : X + X such that for all x, f(x) E LO and

(*I

P ( x ) CJ f(x) E wo.

In fact, i f P is n:,then we can choose f : X += X so that (*) holds and the relation R(x,n, m )* f ( x ) ( n )= m

is arithmetical; i f in addition X is of type 0 or 1, then (*) holds with a recursive f . Similarly, P is Il: i f and only if (*) holds with a Borel f, or with a continuous f if X is of type 0 or 1. PROOF.This is an effective and improved version of 2D.2, the representation of complements of x-Suslin sets of irrationals in the form

P ( a ) w T ( a )is wellfounded,

STRUCTURE THEORY

194

r4A.3

FOR POINTCLASSES

where T is a tree on w X x. We might as well give here a direct proof for subsets of an arbitrary space 'X. The last assertion clearly follows from the claims preceding it. Assume then by 4A.1 that

P(x)

-

(Va)(3t)R(x,&i(t))

with R semirecursive, or R recursive if 'X is of type 0 or 1, where R(x,ii((t))& t
For each x, put

T(x)={(uo,...,~ - 1 ) : l R ( x , (~o,..., ~ t - 1 ) ) ) so that T ( x ) is a tree on w and clearly T ( x )is wellfounded.

P(x)

What we must do is replace T ( x )by a linear ordering on w which will be wellfounded precisely when T ( x ) is. Put (U",...,

U,-l>>"

(UO,...,

Y-1)

(UO,..., %I>,

(%,..-,%I)

E T(x)

&{ug>u~v[2)o=u,&u1>ul]v[u,=u,&2)1=u~&2r~~u*]

v[v,= u,&

2)]

= U ]&

-..& u,-1=

where > on the right is the usual "greater than" It is immediate that if (u0,..., us-l), (u, ,..., ut-J (u, ,..., wsPl) is an initial segment of (u, ,..., (u, ,..., 4-J; thus if T ( x ) has an >" has an infinite descending chain. Assume now that >x has an infinite descending vo>*2)

where

2)'

'>X

v2>x... ,

=cub, uf )...,vf,-1),

and consider the following array: uo = (UZ,2): ,..., 1

2)'

1

= (00,Vl,..., vf,-1>

...

...

v' = (uh, 2)';)...,..., V f , - l )

...

...

Us-1&

s< t]}

in w. are both in T ( x ) and (u, ,..., I.+,), then infinite branch, then chain, say

THEBASIC

4A.31

The definition of

REPRESENTATION THEOREM

BXimplies

FOR

SETS

195

immediately that u:r u;r u;r

...,

i.e. the first column is a nonincreasing sequence of integers. Hence after a while they all are the same, say ub = k,

for i 2 i,.

Now the second column is nonincreasing below level i, so that for some

4,k l u; = k,

for iril.

Proceeding in the same way we find an infinite sequence

ko, ki,... such that for each s, (k,, ..., ks-l)E T(x), so T(x) is not wellfounded. Thus we have shown,

P(x),

-

T(x) is wellfounded

CJ >x

has no infinite descending chains.

Finally put usxu

(3t 5 u)(3s Iu)[Seq(u)& lh(u) = t & Seq(u)& lh(u) = s [ u = v((u)o,..., (u)s-l)>x ((u),,..., (~)t-l)ll

and notice that cXis always a linear ordering and

P(x) CJ SXis a wellordering. Moreover, the relation

P(x, u, u ) w u s x u is easily arithmetical for arbitrary 3c and recursive if 3c is of type 0 or 1. The proof is completed by taking 1 if ( n > , I x ( n ) , , 0 otherwise.

The linear ordering sXwhich we used in this proof is variously known in the literature as the Lusin-Sierpinski or the Kleene-Brouwer ~rdering.".'~' Let us prove here just one very useful corollary of this basic result. Put

oyK=suprernurn{lal:a! E WO and a! is recursive}.

196

STRUCrURE THEORY FOR POINTCLASSES

r4A.4

One may think of m y K as an “effective analog” of the least uncountable ordinal K,; w y K is the least ordinal which cannot be realized by a recursive wellordering with field in w.@) 4A.4. THE BOUNDENESS THEOREM FOR n: (Lusin-Sierpinski, Spector‘’-’ l)). Suppose P c X and P satisfies the equivalence

P(x)

-

f(x) E wo

with some A: function f. Then P is A: if and only if supremum(1f (x>l: ~ ( x ) < ) myK. Similarly, suppose

P ( x ) * f(x) E wo

with some Borel function f. Then P is Borel if and only if suprernurn{lf(x>l:P ( x ) }
PROOF.Assume first that for all x, if P ( x ) ,then If(x)lslaI, where a E WO and a is recursive. By 4A.2 then,

P b ) *f(X)%CY, so P is S : and since it is evidently n:,it is A:.

Conversely, suppose supremum{lf(x)l: P ( x ) } r w y K . Let Q E w be any w, so by the basic representation theorem 4A.3 there is a recursive g :w + X and

II: relation on

Q(n) e g(n) E WO. Notice that for every n, g ( n ) is a recursive irrational by (iv) of 3D.7. Hence Q ( n ) - g(n) E WO Lk Ig(n)l< wFK

*( 3 X ) W b )

&L

g(n)%f(x))

which implies that if P is Xi, then so is Q. But Q was arbitrary n: on w and need not be Si,so P is not Xi. -I Proof of the boldface result is a bit simpler.

EXERCISES Put

6 = supremum{(a(:a E WO and a is A i},

4A.71

THE

BASIC REPRESENTATION THEOREM FOR

SETS

197

where a is A: if {(n, m ) : a ( n ) =m} is A:.

4A.5. Prove that 8: = ofK.(Spector [1955]."') Hint. It is enough to establish that 8: Isupremum{(a(:a E WO, a recursive},

so assume towards a contradiction that there is some 0 E WO, P is A: and for every recursive a,if a E WO, then la\< 161. Choose P E o which is l7: but not A: and by 4A.3 choose a recursive f such that

P ( n ) @ f(n) E wo. Now each f(n) is a recursive irrational, so the assumption above implies

P ( n ) * f(n) E wo

If(n)l< IPI

which via 4A.2 shows P to be A:, contrary to hypothesis.

-I

This result is rather surprising, as one might expect to get longer wellorderings in the complicated pointclass A: than one gets in A:.

4A.6. Prove that if A is a S: subset of WO, then there is a countable E such that a E A + (a1< 5.

Hint. If not, then every It: relation P would satisfy P ( x )e &)[a

EA

&f ( ~ ) < Z a ]

with a Bore1 f and would be A:.

-I

The next exercise is an effective version of 1G.5.

4A.7. Prove that for each A: pointset P E S there is a recursive function IT :N--* S and a II: set A EX, such that T is one-to-one on A and n[A]= P. Similarly, if P is A:, then there is a continuous T :y +X and a closed A E X such that IT is one-to-one on A and T [ A ]= P (this is a restatement of 1G.5). Hint.By 3E.6, we may assume X = N . By 4A.3 then, there is a recursive f :X -+ X such that P ( a ) * f ( a )E wo

and by 4A.4, there is a recursive B E WO so that

P ( a ) -f(a)%P.

198

[4A.8

STRUCTURE THEORY FOR POINTCLASSES

Q*(r,a)* Q(r,f(a)). Moreover, easily P(a)* @ y ) Q * ( y , a) w

there exists exactly one y such that Q * ( y , a).

Bring Q* to normal form and let

-

S(6, Y , a)

Q*(y, a)* W n ) ( 3 m ) R ( y ,a,n, m ) (Vn"(y,

a,n, 6(nN

w m
a

9 n 7

m)l.

Now S is a II; subset of N x N x N and the recursive map (6, y, a)* a takes S onto P and is one-to-one on S. The result follows because N X N X N is recursively homeomorphic with N . The assertion about A: sets follows by the same proof, starting with a continuous f such that P ( a ) e f ( a )E wo.

-I

This result is important, particularly because we will prove later that every injective, recursive image of a A: set is A:-see 4D.7. We can also obtain from this result an interesting partial converse to 3E.16.

4A.8. Prove that a point xo is A: if and only if there is a II? singleton {ao}G X such that xo is recursive in ao.

4A.91

THEBASIC

REPRESENTATION THEOREM

FOR

n; SETS

199

Hint. If xo is recursive in some a. with {ao} in Il:, then xo is easily A i. If xo is A:, then the pointset

P(s, i)*[X0ENs

& i = l ] V [ X , ~ N , & i =O]

is also A:, so by 4A.3 choose a Lly set A G X and a recursive T : X + w X w, which is injective on A and such that T[A] = P. Put

and check easily that B E X x X is a Il? singleton and xo is recursive in the unique (Po, yo)E B. Now use the recursive homeomorphism of X X X with X . i It is not true that every A: point is a I l y (or even an arithmetical) singleton-this has been shown by Feferman [19651.

4A.9. Prove that for each countable ordinal 6, the set

4 ={a:a E wo & l a l l [ } is A:, uniformly in the coding for ordinals determined by WO and the canonical coding for A:. Hint. We must show that there is a partial function M :X +X which is recursive on WO and such that

P E WO 4

u(P) is a A:-code of {a:a E WO & l a l ~ [ P I } .

Choose recursive irrationals el, e0 so that alnP

G i ( ~ iP,, a)

~ I S P OG2(~2,P, a) where GI, G, are good universal sets in Il: and Z;respectively by 3H.1 and let

u ( P ) = (S(E17

6 1 7

S(% P I ) .

-I

Of course this exercise is nothing but a restatement of 4A.2 using codings.

200

[4B

STRUCrURE THEORY FOR POINTCLASSES

4B. The prewellordering The basic representation theorem implies easily the so-called prewellorden'ng property for n:,which in turn implies directly many of the nice structural properties of this pointclass. This property can be established for 2; and many other pointclasses more complicated than n:,so it is worth studying its consequences in a general setting. Recall from 2B that a norm on a set P is any function cp :P + Ordinals

taking P into the ordinals. There is a simple correspondence between norms and prewellorderings on P established in 2G.8, where with each cp we associate the prewellordering s*on P, X<*Y

* cp(X>~cp(Y).

Conversely, if < is a prewellordering on P, then < = 5*for some norm cp; moreover cp is uniquely determined if we insist that it be regular, i.e. that cp maps P onto some ordinal A. Let us call two norms cp and $ on P equivalent if l q = d', i.e. cp(X)lcp(Y)

@$(X)S$(Y).

Clearly every norm is equivalent to a unique regular norm. There are many trivial norms on a set, e.g. the constant 0 function, but the concept becomes nontrivial if we impose definability conditions on a norm in the following way. Let be a pointclass, Q :P + A a norm on some pointset P. We call Q a r-norm if there exist relations z:, 5: in and i r respectively such that for every y,

r

(*I

-

P(Y)

r

(VX>{[P(X)

Lk Q ~ X ) ~ Q ( Y ) l *x 5;Y * X S f Y).

It is important for the applications that the definition of a r-norm be precisely that given by (*). Notice that if is adequate and P E then (*) is stronger than simply requiring that the associated prewellordering be in but weaker than insisting that I* be in rn. ir. In cddition to the prewellordering 5*,there are two other relations that are naturally associated with a norm cp. Put

r

r

x 5:Y xY:<

@P(X)

~[~~(Y)Vcp(X)~Q(Y)I,

@P(X)

& [-mY)VQ(X)
r,

THE

4B.21

20 1

PREWELLORDERING PROPERTY

The meaning of these relations becomes clear if we extend the norm P E Z to all of X' by cp (x) = 00 if i P ( x ) ,

Q

on

where 00 is assumed larger than all the ordinals. Then obviously, with this extended cp, X g Y -.P(X)&cp(X)~cp(Y)

x <: Y

-

P(x) Lk d x ) < d y ) .

4B.1.THEOREM. Let r be an adequate pointclass and let Q a norm on some P in r; then rp is a r-norm if and only if both
PROOF.If

Sz,

-

=sz,

<: are in I', we can take x sfy xqy

x 5: y,

e 1 ( y <,*x),

and verify easily that they prove cp to be a r-norm. On the other hand, given such relations S;, s;,notice that

x5:y

e P(x) & [x

S Fy v 1 y I F X I ,

x < , * y o P(x) & 1 y s;x, so that both

I,* and <,*are in r.

r

i

A pointclass is normed or has the prewellordering property if every pointset P in admits a r-norm.

r

n: and II: are n ~ r m e d . ( ' ~ " ~ ) PROOF. Given P in n:,choose a A: function f by 4A.3 such that P(x) * f(x) E wo 4B.2. THEOREM. Both

and for X E P ,put

d x ) = If(x)l. Using the notation of 4A.2, we can take

x %Y

*f(X)Snf(Y),

x 52Y -f(x)%f(y) and verify easily that cp is a Hi-norm. The same proof works for II:, taking a Bore1 f.

202

[4B.3

STRUCTURE THEORY FOR POINTCLASSES

r

4B.3. THEOREM (Novikov, Mos~hovakis"~-'~'). If is adequate, P e r , P E X X X and P admits a r-norm, then 3NP admits an F V " r-norm. Hence, if r is adequate closed under V" and normed, then F r is normed. In particular, 2;, 2 : are normed.

-

PROOF.It is enough to establish the first assertion. Assume that Q(x)

with P in

(3a)P(x,a)

r, let cp be a r-norm on P and define 4 on Q by +(x) = infimum{cp(x,a ) :P(x, a ) } .

Proof that II, is an F V N r-norm is immediate from the equivalences x 5;Y

* (3a)(VP)[(x,4 g Y Y 9

x <;Y

-(3a)(VP"x,

P > 1 7

a><$(y,P)1.

-I

This result is typical of the kind of abstract setting in which the notion of a r-norm proves useful. There will be several opportunities for applying 4B.3 in its full generality. We will study many consequences of the prewellordering property in the next two sections. Here we concentrate on just a few facts which are simple, useful and indicative of the power of this hyothesis about a pointclass. Recall the definition of a uniformizing set P* E P E X x y.

4B.4. THEEASY UNIFORMIZATION THEOREM (Kreisel[1962]). Suppose r is an adequate pointclass, y is a space of type 0, P E X x y is in and P admits a r-norm. Then P can be uniformized by some P* in V"T. In particular, if is adequate, normed and closed under V", then every P E X x y in r with y of type 0 can be uniformized by some P* in

r

r

r.

PROOF.It is obviously enough to prove the result with y=o. Assume then that P G X x o is in I', let cp be a r-n o rm on P and put

P*(x, n>c=,P ( x , n ) & ( v m ) [ ( x n, )s:,*(x, m>l & (Vm)[(x,n)<$(x, r n ) v n ~ m ] , or in other words P*(x, n ) w P ( x , n ) & cp(x, n ) = infimum{cp(x,m ) : P(x, m ) } & n = injimum{m: P(x, rn) & cp(x, m ) = cp(x, n)}.

4B.61

THEPREWELLORDERING

PROPERTY

203

Clearly P* is in V ” r and

P*(x, n ) & P*(x, n’) + P ( x , n ) & P(x, n’) c p b , n ) = d x , n’)

&nln’&n’ln + n = n’,

so P* is the graph of a function. If (3n)P(x, n), let

5 = infimum{cp(x, n ) : P(x, n)}, n = infimum{m : P(x, rn) & cp(x, m ) = t}, and verify easily that P*(x, n). Thus P* uniformizes P. The problem of uniformizing subsets of X x y for arbitrary product spaces y is much harder and cannot be settled using only the prewellordering property. We will deal with it in 4E. Theorem 4B.4 is most often used in the form of the following easy corollary.

4B.5. THE A-SELECTION PRINCIPLE (Kreisel [1962]). Let I‘ be adequate, normed and closed under 7 ,V”, let P c X X y be in r with y of type 0, assume that A G X is in A = rrll r and

(Vx E A)(3y)P(x,Y). Then there exists a A-recursive function f : X + y such that (Vx E A)P(x, f ( x > > .

PROOF. Put

Q ( x , y ) - x g A v [ x ~ A &P(x,y)I

r

and choose Q* E Q by 4B.4 which is in and uniformizes Q. Clearly Q* is the graph of a function f : X + y and (Vx E A)P(x,f(x)). Since y is of type 0, f is r-recursive by 3D.2; now f is A-recursive since

f(x) # Y * @Y’)[f(X)

= Y’

Lk Y # Y’l.

-I

EXERCISES 4B.6. Let I‘ be an adequate pointclass. Prove that a norm cp on some P in is a r-norm if and only if the unique regular norm $ which is equivalent

r

204

r4B.7

S T R U W R E THEORY FOR POINTCLASSES

to cp is a r-norm. Prove also that if cp is a r-norm, then there are relations
r

P(Y) 3, (~X)W(X)8l cpb) < cp(Y)l*

x
-1

r

4B.7. Prove that if is adequate and normed, then the associated -1 boldface class I'is also normed. 4B.8. Prove that for n 2 2 , the pointclasses Zz,Sz are normed. Prove also that every 2; (or S;) pointset of type 0 or 1 admits a 2: (or Sy) norm. Show that the latter result fails for sets of reals. Hint. Given P in 2: so that P ( x >* @m)Q(x, rn)

with Q in 11:-1, put cp(x) =least rn such that Q(x, m).

r

-I

4B.9. Suppose is a 2-pointclass closed under A: substitution. Prove that if every pointset of type 1 in admits a r-norm, then is nonned. -I Hint. Use 3E.6. Recall the definition of reduction from 1C. A pointclass has the reduction property if every pair P, Q of sets in can be reduced by a pair P*, Q* in

r

r

r

r.

r

4B.10. Prove that if r is adequate and normed, then r has the reduction property; in particular, II:, II:, 2; and S; have the reduction property. (Kuratowski, Addis~n."~') Hint. Given P, Q in put

r,

R(x, n ) o [ P ( x ) & n = O ] v [ Q ( x ) & n = l ] , let cp be a r-norm on R and take

A pointclass r has the separation property if when P, Q are in r, P f l Q = 9, then there is some R in A =rn which separates P from Q. We have already proved in 2E.1 that S: has the separation property. 4B.11. Prove that if

r is adequate and has the reduction property, then

4B.121

THE PREWELLORDERING

PROPERTY

205

Figure 4B.1. Separation of P G Q from 3c - Q.

the dual class i r has the separation property; in particular, Zi,Z:, n:, n: have the separation property. (Lusin, Novikov, Addi~on."~')

Hint.Given P, Q in ir,both subsets of X,let P , = X-P, Q1= X- Q, choose P r , QT to reduce PI, Q1 and prove that P f U Q T = X . Take R = a:. -I Many times we use the separation property in the following form: if P is in Q is in -IT and P G Q, then there exists some R E A so that (see Figure 4B.1) PGRGQ.

r,

To see this separate P from X - Q.

r

4B.12. Prove that if is adequate, w-parametrized and has the reduction property, then does not have the separation property. Similarly, if r is adequate, N-parametrized and has the reduction property, then does not have the separation property. In particular, Z:, n:, Z;, Zg, n:, Z: do not have the separation property. (Novikov, Kleene, Addi~on."~') Hint.Let G G w x w be universal for r l w and put

r

r

Choose P * , Q* in

r

which reduces P, Q and assume towards a

STRUCTURE THEORY

206

FOR POINTCLASSES

[4B.13

Diagram 4B.2. Normed Kleene pointclasses.

contradiction that some R in A separates P* from Q*, i.e.

P*cR,

RnQ*=g.

Choose integers e, m such that

R ( n ) o ( e ,~ ) E G , i R ( n ) o ( m , n ) ~ G and let t = ( m , e). Now show that both assumptions t E R, t&R lead to contradictions. -I The second assertion is proved similarly.

r is adequate and w-parametrized, then at most one r, is normed. -I

4B.13. Prove that if of the pointclasses

It follows from the results of this section that the Kleene pointclasses which are normed are exactly those circled in Diagram 4B.2. The circle around 2: is dotted, since only Z y pointsets of type 0 or 1 admit Z:-norms. The diagram for the boldface classes is identical. We have not included here Z;,IZ: and the higher analytical pointclasses, as it is not clear at this point which of 2: or II: is normed, if any. Many of the results in this section have uniform versions which are easy to establish using the methods of 3H. We put down one theorem of this type as an example.

r

4B.14. If is o-parametrized, adequate and has the reduction property, then has the uniform reduction property: i.e. for each !X, there are ( a ,u Z ( a ,p ) such that whenever a, p code recursive functions ~ ( ~p), ', then ul(a, p), u Z ( a ,p ) code sets P", subsets P, Q of 3c respectively in I Q* respectively which reduce the pair P, Q. Hint. All codings are relative to a good parametrization of course, so the hypothesis (for example) means that

r

P(x)

G(a,XI, Qb)* G(P,X )

with G a good universal set.

4 ~ 1

207

SPECTORPOINTCLASSES

Define

ui(a,P, X) U2(a,P, x)

-

so that both U , and U, are in r and let r. There are then recursive irrationals UT
-

G(a,XI, G(P7 x)

G, E,,

reduce the pair U 1 ,U2in e2 so that

G(&i,a,P, XI * G(S(&i,a,P ) , X ) a,P, x) CJ G(S(EZ,a,PI, XI,

where we have used the Good Parametrization Theorem 3H.1. It is easy to check that the recursive functions w(a, P ) = S ( E 1 , a,P ) HZ(a,

P ) = S ( E 2 , a,P )

have the required properties.

-I

4C. spector pointdasses""16'

The consequences of the prewellordering property which we proved in 4B depended on several side conditions on a pointclass e.g. closure under various operations or parametrization. Here we will isolate the most commonly used hypotheses into the basic notion of a Spector pointclass. The simplest Spector pointclasses are II: and S&in fact Il: is the least Spector pointclass. There are, however, many other examples which make it worth axiomatizing the theory. A Spector pointclass is a collection of pointsets which satisfies the following conditions: (1) is a 2-pointclass with the substitution property and closed under V". (2) is w-parametrized. (3) is normed. Recall that (1) implies Zy sr and is closed under &, v ,'3 and 3",and by 3G.1, is also adequate. All the Kleene pointclasses SA,Il!, satisfy (1) and (2), so to prove that one of these is a Spector pointclass we need only verify the prewellordering property. It is also trivial to check that each relativization T ( z ) of a Spector pointclass is a Spector pointclass, see 4C.4. Thus Hi, Xi,n:(z),

r,

r

r

r r

r

r

r

208

STRUCTURE THEORY FOR POINTCLASSES

[4C.1

X:(z) are Spector pointclasses-they

are the only ones we know at this time. In Chapters 5 and 6 we will prove using strong set theoretic hypotheses that some of the higher Kleene pointclasses are also normed and in Chapters 6 and 7 we will introduce many more examples of Spector pointclasses. Here we concentrate on consequences of (1)-(3) above which give us new results about n: and 2:. First, let us prove a strong closure property of Spector pointclasses which implies that every one of them contains every n: relation. Suppose Q(x, w ) is given and we define P(x, w ) by

(*I

P(x,w )H (VaKIt)Q(x,w * G ( t ) ) ,

where w * IJ codes the concatenation of the sequences coded by w and IJ, if Seq(w), Seq(v), see *18 of 3A.6. For each countable ordinal 6, define the set Ps E X x o by the recursion ( * *>

P*(x, W )w Q(x, W ) V (Vs)@q < ~ ) P " ( xw, * b)).

It is easy to verify by induction on 5, that

Pqx,w )* P(x, w); conversely, if (V()iP'(x, w ) then iQ(x, w ) and for some s = so, all 5, iP*(x,w*(so)),so again iQ(x, w*(so)) and for some s = sl, all 6, iP*(x,w *(so, sl)), etc., so finally with a =(so, s1,...) we have (Vt)iQ(x,w * i i ( t ) ) , i.e. iP(x,w). Thus Now define a norm Q

: P + ordinals

by Q(X,

w )= least 5 such that P<(x, w);

-

it is immediate from (* * ) that P satisfies the equivalence

P(x,w )

a x , w)v(Vs)[b,w*>-e(x,w)l.

It is perhaps a bit surprizing that this equivalence completely determines P. 4C.1. LEMMA. Suppose Q(x, w),P ( x , w ) are given and P admits a norm Q

4c.21

SPECTOR POINTCLASSES

209

then

PROOF.First we prove by induction on q ( x , w ) that P(x, W )

(Va)(3t)Q(x,w * E ( t ) ) .

Assuming this for all (x, u) E P with q ( x , u )
Q(x, w ) v ( V s ) [ ( x w , * ( s ) ) < z ( x , w>l. If Q(x, w ) holds, then easily ( V a ) Q ( x , w *E(O)) since w * E ( O ) = w. Otherwise, we have (Vs)[(x, w *(s)) <:(x, w)l,

so that for each s, P(x, w *(s)) and q ( x , w *(s)) < q ( x , w ) . By the induction hypothesis then, (Vs)(Va)@t)Q(x,w * ( s ) * E ( t ) ) from which (Va)(3t)Q(x,w * E ( r ) ) follows immediately. Conversely, assuming i P ( x , w ) , we have i Q ( x , w ) and for some s = so, i ( x , w *(so))
r

r be a Spector pointclass, suppose

P ( x ) w (Va)@t)Q(x,E(tN;

then P is in r. In particular, n: is the smallest Spector pointclass and 2; is the smallest Spector pointclass closed under 3".

210

[4C.2

STRUClWRE THEORY FOR POINTCLASSES

PROOF.The second assertion follows immediately from the first by (ii) of 4A.1. To prove the first assertion using the lemma, it is enough to find some R* G 3c x o in which admits some norm cp so that

r

R*(x,W ) e Q(x, w>v(VS>[(X,w * ( s ) ) < ~ ( xw>l, , since we then have P ( x ) a R*(X, 1). Here is where we will use Kleene's recursion theorem for relations, 3H.3. Let G G N x X X be ~ a good universal set in for r, let 4 : G-* ordinals be a r-norm on G and define

r

R(a,X, Now R is in

W ) a Q(x,

w)v(Vs)[(a,

X,

w * ( s ) ) < $ ( ~X,, ~ 1 1 .

r so by 3H.3 there is a fixed recursive R ( E * X, ,

W) e G(E*,X,

E*

so that

w).

Put

R*(x, W)

e R ( E * ,X, W )

and on R* put the norm d x , w )= 4(&*,x, w).

Computing,

--

R*(X, w )e R(E*, x, w )

Q(x, w>v(VS)[(E*,X, w * ( s ) ) < $ ( E * ,X, Q(x, w )v (VS)[(X,w * (s)) <: (x, w)l

so that R* has the required property.

w)l -1

This theorem is interesting partly because it gives an intrinsic structural characterization of Il:. Of course n: can be easily characterized by its closure properties, e.g. it is the smallest 2-pointclass closed under V" and VN. But nothing very deep can be proved in general about 2-pointclasses closed under V" and VN. We will see that Spector pointclasses have a rich structure theory, much of it giving new results even when we specialize it to n:. There is another practical corollary of 4C.2 which we list together with some simple properties of total functions recursive in a Spector pointclass.

4c.51

211

SPECTOR POINTCLASSES

4C.3. THEOREM. Let I' be a Spector pointcIass, suppose f :!X + y is total and r-recursive; then f is A-recursive, Graph(f)= {(x, y): f ( x )= y } is in A and for every x, f ( x )E A ( X )

=

r(x)n lr(x),

i.e. f ( x ) is a A(x)-recursive point. Moreover, every A: function is r-recursive, so in particular, r is closed under substitution of A functions.

PROOF.The first assertion is easy and uses only the fact that r is a 2-pointclass closed under V". Thus, ((y, s): yf! N,} is in r since it is IIy and r contains all @ sets, hence {(x,s): f ( x )& N,} is in r by closure under substitution of r-recursive functions; thus f is A-recursive. From this follows trivially that f ( x ) E A ( x ) .As for the graph, f(x) = Y * (VS)[Y

E

N,

=, f ( X ) E

-

N,1

* (VS)rf(X)E N,

Y E NS1.

Now if f : S + y is A i, then {(x,s): f ( x )E N,} is A i, hence in hence f is r-recursive.

r by 4C.2, I

EXERCISES

4C.4.Prove that if r is a Spector pointclass, then so is each relativization

&).

-I

With each pointclass where for P E 32,

r we have associated

the boldface pointclass

P e r o f o r some P*E#x!x in and some i.e.

E

r,

r

E N , P = PF,

r= UEr(&).

As usual,

A=rnlr.

r

4C.5.Prove that if is a Spector pointclass, then I' contains ll: and is closed under Bore1 substitutions, 3",V", Am, it is X-parametrized and it is normed.

v",

STRUCTURE THEORY

212

FOR POINTCLASSES

[4C.6

Moreover, every r-measurable function is A-measurable (in fact Arecursive by 3D.22) and has a graph in A. The pointclass r is closed under substitution of r-measurable functions. -1

4C.6. Prove that if r is a Spector pointclass, then i r is closed under the -I operation d. If cp : P + A is a regular norm, we call A the length of cp, IcpI = A.

The length IcpI of an arbitrary norm is (by definition) the length of the unique regular norm equivalent to cp. If cp : P -B Icp1 is a regular norm, then for each [
PS = { x : cp(x)s(}. Clearly

p

r

=

U*
4C.7. Let be a Spector pointclass and let cp : P + IcpI be a regular r-norm on a pointset P in r, where P is of type 0. Prove that for every [
p

= US
with each P E in A. Hint. Choose some y E P such that cp(y) = [ and notice that x E PE

* x 5: y H

1 ( y <:x>.

-I

This result is more useful if we get an estimate on the length (cpl of a r-norm. Given a pointclass r (which need not be a Spector class), put

S = suprernum{l
Clearly S is a countable ordinal, but 6 may well be uncountable-the obvious bound is

6 < (2K-)' = least cardinal > 2N11.

only

4c.101

SPECTORPOINTCLASSES

4C.8. Let

r be an adequate pointclass closed under 3”, V”.

6 = s u p r e m u m { l ~I:

5

213

Prove that

is a wellordering on o,< in A},

6 is a limit ordinal and for every r-norm cp on a pointset P of type 0 in r,

IQl

6.

i

4C.9. Prove that if I‘ is an adequate pointclass, then for every r-norm cp on a pointset P in r, IcplsS. If is a Spector pointclass, then 6 is an ordinal of cofinality >w and every pointset in r is the union of 6 sets in A. Hint. An ordinal A has cofinality >o if for every increasing sequence < < A, limit, 6, < A. This follows here from closure of A under

r

v

i

-*

This is obviously a “soft” version of part of 2F.2, with a very different proof. To get “hard” corollaries of this exercise we must establish a construction principle for the specific A and also get an estimate of the size of 6. Both of these often turn out to be very hard. The traditional notation for 6 and 6 when r is S!, (or n!,) is 6: and 6:. Similarly, for the relativized class x ! , ( z ) (or nX(z)), its ordinal is 6!,(z). (It is trivial to establish that the boldface class corresponding to SX(z) is x:, so the boldface ordinal of Z,!,(z) is again SA.) From the Kunen-Martin Theorem 2G.2 we know that

s: = K1,

s: 5 K z .

This is about all that can be proved about these ordinals in classical set theory, except for 4A.5, that

S

= oYK= least

nonrecursive ordinal.

The next exercise gives an interesting generalization of the Boundedness Theorem 4A.4 to arbitrary n:-norms. 4C.10. Suppose P E X is n: and cp :P + Ordinals is a regular, ni-norrn on P. Prove that P is Borel if and only if (cp(
and each PE is Borel by 4C.7, so P is a countable union of Borel sets and hence Borel.

STRUCTURE THEORY

214

FOR POINTCLASSES

[4C.ll

Conversely, if P is Borel, then the prewellordering x 5Y

@

P(x) c!k P(Y>& cp(X)~cp(Y)

of length A is easily Borel and hence A
-I

This result is often useful in conjunction with the covering lemma which admits a general version.

r r-

4C. 11. Let be a Spector pointclass, let cp be a regular I‘-norm on some P E ‘X in A, let Q be in i r and assume that either X is of type 0 or is closed under V”. Prove that QsP-forsome[
r

Q E P * = { x E P cp(x)s[}. :

Similarly, let be a Spector pointclass closed under VN, let cp be a regular r-norm on some P E ‘X in r - A and let Q be in ir.Prove again that

Q EP

r

-

for some [< Icp 1,

Q E PS

In particular, if is a Spector pointclass closed under VN, G E X x X is universal in and cp : G + Ordinals is a I‘-norm on G, then a pointset P c X is in A if and only if there are irrationals E, and some xoE X such that G(eO,xo) and

r

P = {X E ‘X: G(E,X ) & V(E, X ) 5 ~ ( E O , XO)) (The Covering Lemma; see Figure 4C.1). Hint. By contradiction, see the proof of 4A.4.

-1

The next result is a simple but interesting extension of the A-Selection principle. 4C.12. Let r be a Spector pointclass, suppose P c ‘X X y X y is in of type 0, assume that ‘X,

r with

(VX)(VY)(3Y’)P(X, Y, Y’).

Prove that for each fixed yo, there is a function f : % x w + y which is A -recursive and such that f(x, 0) = Yo, (Vn)P(x, f(x, n), f(x, n + 1)).

(The Principle of r-dependent choices.)

SPECTOR

4c.131

POINTCLASSES

215

Figure 4C.1.The covering lemma.

Hint. By hypothesis and the A-Selection principle 4B.5, there is a A-recursive g : 3c x y + y such that (Vx)(Vy)P(x, y, g(x, y)). Define f by the recursion f(x, 0)= YO? f(x, n + 1)= g(x, f(x, n)).

-I

Another simple but interesting application of the A -selection principle comes up in the next result. This is essentially a representation theorem for A sets which happen to be open-we will need it in the exercises of

4F.

r

4C.13. Let be a Spector pointclass closed under VN, let G G 3c be a pointset in A which is open. Prove that there is some irrational E in A such that

p = U,N(% 4 n ) ) and for each n,

N(%,E(n)) G P. (ff(32,s) is the closure of N(%, s).)

216

STRUCrURE THEORY FOR POINTCLASSES

[4C.14

In particular, under these hypotheses, P is semirecursive in some E E

Hint. Put Q(x, S)

-

P ( x )& x E N , & (VY>[YE fl, =+ P(y)I.

Clearly Q is in I' and (Vx ~ P ) ( 3 s ) Q ( x s), , so by 4B.5, there is a A recursive f :ff + w such that (Vx)Q(x, f(x)). The set

A = {s: (3 E P)[f(x)

= s]}

is in i T and it is disjoint from

B = {s: ( 3 y g P)[y

€

NJ},

since for each x E P, #f(x) E P. By the separation property for some C in A,

lr,there is

AEC, B n C = g . Now it is immediate that

P = U{Ns:s€C}, and for each s E C, N,E P. Take E(S) =

s if s E C, 0 if s&C.

i

The last exercise is an interesting generalization of the fact that Xi relations have countable rank whose proof uses Kleene's recursion theorem for relations, 3H.3 as did the proof of 4C.2.

r

4C.14. Let be a Spector pointclass closed under VN, suppose < is a (strict) wellfounded relation on the perfect product space X which is in -IF, let G E X x X be a good universal set in and let cp :G -+ Ordinals be a r-norm on G. Then there exists a recursive function

r

which is order-preserving from

< into

cp,

i.e.

It follows that if cp is any regular r-norm on the good universal set G,

4D.11

THEPARAMETRIZATION

then

THEOREM FOR

A n 3:

217

Id= 6.

(Moschovakis [19701.) Hint. Put

XI * (VY>[Y< X

Q(a,

so that Q is in

Y ) < Z ( ~ ,x)l

=,(a,

r and by 3H.3 there is a recursive E * E X satisfying Q(E*,

X) H

G ( E *x). ,

Put

f(x) = ( E * , x) and check by a trivial induction that if x is in the field of <, then

f(x>~G&(Vy)[y
=,cp(f(~))
Applying this to each relativized pointclass T ( w ) , we show that 1 ~ exceeds the rank of every strict, wellfounded relation in i r on X, whence IpI=S follows immediately by the fact that every two perfect product spaces are A:-isomorphic and 4C.9. -I

4D.The parametrization theorem for A n9C Most of the results in 4C follow quite directly from the definitions and depend on only few of the axioms for a Spector pointclass. Here we will consider somewhat deeper propositions whose proofs make essential use of the full set of axioms, including o-parametrization. Recall that a partial function f:X+Y is r-recursive if Dornain(f) is in r and f is r-recursive on its domain; if is closed under &, this amounts to saying that the relation

G’ (x,S >

r.

-

f(x1.l 8~f(x) E N,

r

r

is in These partial functions are very useful when is a Spector pointclass. We summarize some of their properties in the next result.

r

4D.1. THEOREM. Let be a Spector pointclass, let f :EC + y be a recursive partial function.

r-

1

218

[4D.1

STRUCTURE THEORY FOR POINTCLASSES

r

PROOF.The set {(y, s): y g N,} is l7: and hence in and the partial function (x, s) H (f(x), s) is easily r-recursive, so by the substitution property there is some Q*(x,) in such that

r

f(x1.l- [Q*k S) fb)g N,1; @

thus

f(x>L&fb)@N,-f(x)L&

and this is in r. The other claims are easier:

f(x>= Y

-

(VS){Y E N s

-

Q*(x,

S)

[f
f(x>L&f(x)# Y *(3s){[f(x)L&f(x)EN,l& Y&N,}.

(ii) Given Q c_ y in r, choose Q* E X in r by the substitution property so that

f ( x ) l d [Q*(x)

and notice that

Q(f(x>>l

(iii) Use 3G.5. We have been using and will continue to use the handy abbreviation y

EA w

y is A-recursive

-?L(y)

is in A,

THEPARAMETRIZATION

4D.21

THEOREM FOR

A nx

219

and similarly for A ( x ) . It is also convenient for any pointclass A to put

A f l X = {x E X:x is A-recursive}. For example 2: n = A: n&=the set of recursive real numbers. Using partial functions we can formulate simply an easy to prove but very powerful parametrization theorem for the points in a Spector pointclass.

4D.2. THEPARAMETRIZATION THEOREM FOR THE POINTS IN A, A ( x ) . Let r be a Spector pointclass. For each product space y , there is a r-recursive partial function d:o-+Y such that for every y y

E

y,

E A e for

some i, d ( i ) l & d(i)= y.

Similarly, for each X, y there is a r-recursive partial function d:wxX+=y such that for all x, y , y

E

A ( x ) w for some i, d(i, x ) && d(i, x ) = y.(I7)

PROOF.Take first the case y = X . We prove the second assertion, the first being simpler. Choose a set G c o x 3 C x o x o which is universal for r ~ ( 3 C x x x x ) and let G* E G be in r and uniformize G by the Easy Uniformization Theorem 4B.4. Here we are thinking of G as a subset of (w x X x o)x o, i.e. we uniformize only on the last variable. Now put

d(i, X ) & H (Vn)@rn>G*(i, x, n, m ) and if d(i, x ) & , let

d( i, x ) = a where for all n, rn

a ( n )= m

H

G*(i,x, n, rn).

We omit the trivial computation which establishes that d is in r. From this it follows that each d(i, x) is in A ( x ) by 4D.1. Conversely, if a E A ( x ) ,

STRUCTURE THEORY

220

FOR POINTCLASSES

[4D.5

choose i so that a ( n ) = m w G(i,x, n, m)

so that a ( n ) = m w G*(i, x, n, rn) and hence d( i, x ) & & d( i, x) = a. If y is of type 0, the result is trivial. Otherwise, there is a A bijection

T : x"y with A: inverse T-' by 3E.7, so let d as above parametrize the A ( x ) points in N and define d* :w x X + by d*(i, x) = v(d(i, x)); clearly d*(i,x)&&d*(i,x)EN,, d ( i , x ) & & ~ ( d ( i , x ) ) ~ N ,

so d* is r-recursive. In particular, each d*(i, x) is in A ( i , x ) = A ( x ) . Conversely, if y is in A ( x ) , then a = v-'(y) is in A ( x ) since T-' is A : and hence r-recursive, hence a = d(i, x) for some i and y = ~ ( a= d*(i, ) x). -I

There are many interesting corollaries of this theorem and we will leave most of them for the exercises. Two deserve special billing. 4D.3. THE THEOREMON IIESTRICTED QUANTIFICATION (Kleene [1959b]"S'). Let be a Spector pointclass, assume that Q E X X y is in and put

r

r

P ( x >w ( 3 E~A ) Q ( x , Y).

Then P is in r. Similarly, if Q E X x Z X y is in P ( x ,Z )

then P is in

r.

-

-

r and

( 3 E~A ( z ) ) Q ( x , 2, Y),

PROOF.Taking the second case,

P(x, Z)

so P is in r by (ii) of 4D.1.

(3i){d(i,Z>L& Q(x,

2, d(i, z))},

-I

THE PARAMETRIZATION

4D.41

THEOREM FOR

A 17I

22 1

The next result gives a very powerful method for uniformizing Borel sets by Borel sets in the special circumstances when this is possible.

r

4D.4. THEA-UNIFORMIZATION CRITERION. Let be a Spector pointclass closed under V N , let P E X X Y be in A and assume that each section P, ={y: P(x, y)} is either f3 or contains some points in A ( x ) n y , i.e.

(*I

(3Y)P(X, Y) * ( 3 Y E A(X))P(X, Y).

is in A and P can be uniformized by some P* in Then the projection A. Conversely, if P c X x y is in A and can be uniformized by some P* in A, then each non-empty section P, has some point in A(x)."~' PROOF.Assume (*) and let Q =gyp, i.e.

Qb)*( ~ Y ) P ( xY,) * (3Y E A(X))P(X, Y). Clearly Q is in A by closure of Now put R (x, i)

i r under w

3N and 4D.3.

P(x, d(i, x))

where d parametrizes A ( x ) n y by 4D.2. By the A selection principle 4B.5, since (Vx E Q)(3i)R(x, i), there must be some g : X + o in A such that (Vx E Q ) R ( x ,g(x)). Put P*(x, Y)* Q(x)

d(g(x), XI = Y.

It is immediate that P* uniformizes P and that it is in A follows by 4D.1, since P*(x, y ) w Q(x) & (3i)[d(i, x)J & d(i, x) = y & g(x) = i ] ,

iP*(x, y) w iQ(x)v(3i)[d(i, x ) J & d(i, x) # y & g(x) = i]. For the converse, suppose P* E P is in A and uniformizes P and assume that (3y)P(x, y). Then there is a unique y* such that P*(x, y*) and Y*E

N, * (3Y"*(X,

Y)

* (VYXP*(X, Y) so y* E A(x).

-

Y E Ns1

Y E N1, -I

We leave the application of this result for the exercises of this and the next two sections.

222

STRUClWRE THEORY FOR POINTCLASSES

[4D.5

EXERCISES 4D.5. Let r be a Spector pointclass. Prove that a partial function f : X -+ o is r-recursive exactly when its graph, {(x, i ) :f(x)&& f(x) = i } is in r. Similarly, a partial function f : X + X is I'-recursive exactly when the associated f* :X x o + o is r-recursive, where

Prove also that the collection of f-recursive partial functions is closed under composition. Hint. For the last assertion, compute: f(g(x))&dt f(g(x))E N,

-

( 3 E~A(x))Cg(x)&& g(x) = Y

f(Y>J& f ( Y ) E Ns1.

Use 4D.1 and 4D.3.

4D.6. Let P G X x y be a pointset in some Spector pointclass that there exists a r-recursive partial function

-

r. Prove

f:X-,Y such that (i) f(x1.l w (3y E A(x))P(x, Y), (ii) (3YE A(X))P(X, Y) + P(X, f(x)). (The Strong A-Selection Principle.) Hint. Put Q(x, i )

-

P(x, d(i, x))

where d parametrizes A(x) n y by 4D.2 and let Q* E Q uniformize Q ir f by 4B.4. Now Q* is the graph of a F-recursive partial functior: g :X + w by 4D.5 and the partial function we need is given by

4D.7. Let r be a Spector pointclass closed under V N ,let P c X be in A and assume that f : X + y is A -recursive and one-to-one on P. Prove that f[P] is in A and that there is a A-recursive function g : y + X which agrees with the inverse function f-' on f[P]. Hint. If P(x) & f(x) = y, then x is the unique point in P whose image is

4D.91

THE PARAMETRIZATTON THEOREM FOR

y ; hence

s E Q(x)

-

An

223

x E N,

e (3x’)[f(x’) = y e (VX‘)lf(X’) f

& P(x’)& x’ E N,]

y v lP(X‘)V

X’E

N,]

andQ(x) is in A(y), i.e. x ~ A ( y ) Hence .

-

Y E f[PI@( 3 X ) [ P ( X )

Y =f b ) I

(3x E A (Y ) ) [ P ( x )8 L Y = f(x>l

and f[P] is in A by closure of i r under T and 4D.3. To get the inverse function, notice that (Vy E f[P])(3x and apply the strong A-selection principle, 4D.6.

E A(y))cf(x) = y]

-I

r=

Taking n:,this is a lightface version of 2E.7 with a very different proof. The classical result follows easily from this, by “relativization.”

4D.8. Let r be a Spector pointclass closed under V”, let P G X be in A and assume that f :X + y is A-measurable and one-to-one on P. Prove that f[P] is in A and there is a A-measurable function g :y .+ X which agrees with the inverse f-’ on f[P]. Hint.If P is in A, then P is in r ( c o )and in ir(&J for some e0, in N, so easily P is in A ( & ) for some E , say with ( E ) = ~ E ~ (, E ) = ~ E ~ Similarly, . if f is A-measurable, then f is A(&’)-recursive for any E ’ such that ((x, s): f(x)E N,} is in A(&’).Thus we can find some E * such that P is in A(&*) and f is A(&*)-recursive and apply 4D.7 to r”= r ( ~ it* follows ); that f[P] is in A(E*)GA and similarly for the inverse. -I This technique of obtaining boldface results from lightface, finer theorems is very easy. We will not always bother to put down the boldface consequences, unless they give well-known classical theorems and we want them to stand out. It is worth putting down for the record the characterization of A which follows from 4D.7 and 4A.7.

4D.9. Prove that a set Q E X is A if and only if Q is the recursive, injective image of some n: set P EN. -I Before using 4D.4 to establish some interesting uniformization results, we point out that not every Borel set can be uniformized by a Borel set. First a lemma which is interesting in its own right.

224

STRUCTURE THEORY FOR POINTCLASSES

[4D.10

4D. 10. Prove that there is a Il: set A c X , such that A # pl but A has no A :-recursive member; similarly, for each x, there is a I l ? ( x ) set A EX, A # pl, such that A has no A :-recursive member. (Kleene [1955~]."~.*~') Infer that not every set A E X is a recursive image of N. Hint.Towards a contradiction, assume that every non-empty Ily set has a member in A: and let P ( n ) be a 2: relation on w which is not Il:. There is a Lly set Q(n, a) such that

P(n>

-

(3a)Q(n,a)

and by our assumption, we then have

P(n)

(3a E A:)Q(n, a)

which implies that P is in Il: by 4D.3. If A = f[X] with a recursive f, then A would have recursive members, -1 namely any f(a)with recursive a.

4D.11. Prove that there is a Ily set P E X X K which cannot be uniformized by any S: ~ e t . " ~ * * ~ ' Hint. Assume the contrary and let G(n,E , a) be a universal Ll: subset of w X X X X . Since w X X is recursively homeomorphic with X , the assumption implies that G can be uniformized by some 2: set G* G G, say G* is Xi(&*) for a fixed E * . Now every @ ( E * ) set A G X is of the form A ={a:G(n,E * , a)}

with a fixed n ; if A # pl, then (3a)G(n,E * , a),so A contains the unique a* such that G*(n,E * , a*).But this a* is in A:(&*), since a * ( t ) = w~ ( 3 a ) [ G * ( n , ~ * , a ) & ( t ) = ~ ] w

contradicting 4D. 10.

(Va)[G*(n,

E*,

a)+ a ( t ) = w ] , -I

Nevertheless, there are many special circumstances in which Borel sets can be uniformized by Borel sets. The next exercise gives a simple topological condition which is often easy to verify and implies the more subtle definability condition of 4D.4.

4D. 12. Let r be a Spector pointclass closed under V", let P E X X y be in A and assume that for each x, the section P, has at least one isolated

THE PARAMETRlZATION

4D.161

THEOREM FOR

An

225

p0int-e.g. it may be that each P, is finite, or countable and closed. Prove that P can be uniformized by some P* in A. Infer the same result for P in A, with P* in A. Hint. If y is isolated in P,, then for some s, P, fl N, = {y}, so that the singleton {y} is in A(x) and y is easily A(x)-recursive. For the second assertion recall that each P in A is in some A(&*)and use the result on -I the Spector pointclass

r(~*).

In 4F we will improve this result substantially by showing that it is enough to assume each P, to be a countable union of compact sets. The next exercise is simple but amusing.

4D.13. Prove that if P E (R" x (Rm is a conuex Borel set, then the projection Q = {x E (R" : (3y E 6im)P(x,y)} is Borel and P can be uniformized by a Borel set. Hint. For rn = 1, each section P, is either a singleton or contains a -1 whole line segment. Use induction on m. We now establish some interesting definability results about A

nX.

4D.14. Prove that if r is a Spector pointclass, then for each X the set A n X is in r. Similarly, the relation {(x, y): x E A(y)} is in r. (Upper classification of A.)(18) Hint. x E A (3i){d(i) i &d(i) = x}. -I

r

4D.15. Let be a Spector pointclass and let d : o + 3c be a r-recursive partial function which parametrizes A nX = {x E 3c: x is A -recursive}. Prove that there is a r-recursive partial function c:X-+o such that

c(x)J CJ x E A and for x E A, d(c(x)) = x. Hint. Use the Easy Uniformization Theorem 4B.4 or 4D.6.

-I

4D.16. Prove that if r is a Spector pointclass closed under either VN or F,then for every perfect product space X the set A fl X is not in 7 r. (Lower classificatibn of A.)(18)

226

-

STRUCTURE THEORY

FOR POINTCLASSES

[4D.16

In particular, A t nX is not Z: and A; r l X is not I:. Hint. If 7 :X X is a A isomorphism, then clearly x EA

@

W 1 ( x ) E A,

so it is enough to prove the result for X . For simplicity in notation put 9=ArlX.

Case 1.

r

is closed under VN. Let j E Je (3Cr)CaE a>& c(a)= j ] w

so J is in

(3i)[d(i)l&c(d(i>)= j ] ,

r. Also i$J-(Va)[a$

s v [ c ( a > l& c ( a ) # i l l ,

so that if a> were in A, then J would be in A. Then the irrational

would be in A and different from all d ( j ) . Case 2. is closed under Y. Let

r

iEIod(i)k and let

cp

be a r-norm on I. Put P ( a ) e ( V i ) [ a ( i )= 0 + i E I ] & ( V i ) ( V j ) [ ( a (= j ) 0 & i ‘Ti)

Clearly P is in

P(a)

9

a ( i )= 01.

r and

{i : a ( i )= 0 ) = I v (3j)[jE I & {i: a ( i )= 0 ) = {i: q ( i )c cp( j ) } ] .

Since I $ A , or else we get a contradiction as before, we have

i $ I * ( 3 a ) [ a $ B & P ( a ) & a(i)#O] which proves I E A and yields a contradiction.

-I

The definition of a Spector pointclass was a bit complicated, because it involved the subtle substitution property. We give here an elegant characterization of Spector pointclasses in terms of a closure property much simpler than substitution.

4El

THEUNIFORMlZAllON

THEOREM FOR

n:,m:

221

4D.17. Let r be a 2-pointclass closed under V", o-parametrized and normed. Prove that r is a Spector pointclass if and only if it satisfies the following property of closure under restricted quantification : if Q E X X Z X Y is in and

r

P ( x ,2)

r.

-

( 3E~A(z))Q(x, 2, Y),

then P is also in (Kechris.) Hint. Spector pointclasses are closed under restricted quantification by 4D.3. Conversely, to establish the substitution property for some r satisfying the hypotheses above, suppose Q G 'J is in f and f :X --., 'J is computed on its domain by some P E X x w in r. Put Q*b) clearly Q" is in

@

( 3E~A(x))[Q(y) 8~(VS)[Y E N + P ( x , ~111;

r and

f(x>.l&f(x) E A(x) * [Q*(x)

-

Q(f(x>)l.

Thus to complete the proof it will be sufficient to check that under the conditions on fb1.l f(x) E A ( X I . Suppose f(x) = y and put

r,

S ( n , s) w P ( x , s) & radius(N,)5 2-". Clearly S is in T(x), which is adequate, closed under 3",V" and normed. Also (Vn)@s)S(n, s), so by the A-selection principle 4B.5 there is a A(x)-recursive function g :w w such that (Vn)S(n, g ( n ) ) . It is now immediate that y E N , -(3n)[Y

N Y , g(nH N Y , gb)) c N1

so that y E A ( x ) .

-1

Unfortunately this elegant characterization is not useful in practice since it is usually much easier to establish that a given r satisfies the substitution property rather than prove directly closure under restricted quantification.

4E. The uniformkition theorem for IZ:, 2; (19-221 We now proceed to establish one of the central results in the subject, that IZ: sets can be uniformized by IZ: sets. The key tool for the proof is the notion of a scale.

228

STRUCrURE THEORY FOR POINTCLASSES

r4E.1

A scale on a pointset P is a sequence @ = {qn},,=- of norms on P such that the following limit condition holds: if xo, xl, x2,... are in P and limiti, xi = x and if for each n, the sequence of ordinals q n (xd,

(xJ,q n (x2)>.**

c ~ n

is ultimately constant, say cpn

(xi 1= An

for all large i, then P ( x ) and for every n, qn(x) 5 An*

Thus a scale is just a semiscale (in the sense of 2B) which satisfies an additional lower semicontinuity property. As with norms, there are many trivial scales on a pointset, at least if we use the axiom of choice: choose a one-to-one norm cp :P w K and set for each n, cp,(x) = cp(x). Again as with norms, we get a nontrivial concept by imposing definability conditions. Let r be a pointclass and = {cpn},,E,,, a scale on some set P. We call @ a r-scale if there are relations Sr(n, x, y), Sp(n, x, y) in r and i r respectively such that for every y,

+

P(~)=>(tln)(tlx){[P(x)&cp,(x)acp,(~)l*SI.(n,x,~) * S r h x, Y)),

(*)

In other words, (p is a r-scale if all the norms qn are r-norms, uniformly in n. It is trivial to verify as in 4B.1 that if is adequate and (p is a scale on some P in r, then (p is a r-scale exactly when the relations

r

r.

are in A pointclass r is scaled or has the scale property if every pointset in r admits a r-scale. It is often sufficient for our purposes to prove that pointsets of type 1 in r admit r-scales, whether or not the stronger scale property holds in r (see 4E.6). 4E.1. THEOREM. Every Il: pointset of type 1 admits a Ili-scale; similarly, every Hi pointset of type 1 admits a Hi-scale.

THE

4E.13

UNIFORMIZATION THEOREM FOR

ni,Pi

229

PROOF.Let us first develop a bit of notation. If a codes a linear ordering ,I

i.e.

QE

LO as we defined this in 4A, then for every integer n put

5,In

= {(s, t ) : s 5,t

& t <,n}

= {(s, t ) : a((& t ) ) = 1& a((t,n)) = 1

a ( ( n , t>># 1).

Clearly I, In is also a linear ordering-it is the initial segment of 5, with top n, if n is in the field of 5, and it is the empty relation otherwise. If is a wellordering with rank function p, then for each n, I , I n is a wellordering and p b ) = l%bl.

In particular, for n, rn in the field of

n 5,rn

I,,

1

I

w I , In1 I5,Em\.

Given a pointset P G X of type 1 in n:,choose a recursive f :Ec +X by 4A.3 such that for every x, f(x) E LO and

P ( x )w f ( x )

E

-

Let

wo.

(5,711 (5,d

be an order-preserving map of N1X N , (ordered lexicographically) into the ordinals, i.e.

(Z,

T>((Z’,

.1’>(.,r5<5’Ivr5=5’&rl~7)’1.

Finally, for x E P put pn(x) =

(I g f ( x ) l , I 5 f ( x , l n l > *

We claim that (p ={cp,,},,, is a n:-scale on P. To prove this, assume that limiti x, = x with xo, xl, each n and all large i,

... in P

and that for

C P ~ ( X= , ) (A, An).

This implies immediately that for each n and all large i,

I ‘rcqdnl

=

a.

The key to the proof is the fact that f is continuous, being recursive. Let us first use this to prove that the mapping

n *An

STRUCTURE THEORY

230

-

is order-preserving from

n
FOR POINTCLASSES

[4E.1

into the ordinals. This holds because

f(x)((n, m))= 1& f(x>((m, n>)# 1 +for all large i, f(q)((n, m))= 1& f(q)((rn,n)) # 1 (by the continuity of f ) I*

for all large i, n cf(%) m

+.for +. A n

all large i, I s f ( J n l < I sf(qi)bml


where the last implication is justified since for all large i, 1 sf(,)Enl = A,. Since n H A, is order-preserving, If(x) is a wellordering, i.e. f(x) E WO and we know P ( x ) . The same fact implies that for every n,

4E.21

THE

UNIFORMIZATION THEOREM FOR

ni,2;

231

As with semiscales in the proof of the Kunen-Martin Theorem, here too we often need scales with very special properties. A scale (p = ((P,,},,~~ on P E X is very good if the following two conditions hold: (1) If xoyxl, ... are in P and for each n and all large i, q , ( x , ) = A , , then there exists some x E P such that limit,, xi = x (and hence for each n, qn(x) 5

(2) If x, y are in P and cp,(x) 5 cp,(y), then for each i 5 n, c p i ( x ) 5 cp,(y). Condition (1) implies that (p is a good semiscale in the sense of 2G.

4E.2. LEMMA.Let r be an adequate pointclass. If a pointset P of type 1 admits a r-scale, then P admits a very good r-scale. P c X is a set of irrationals and let G= {$,,},,€,,, be a r-scale on P. Choose A 2 w and large enough so that all the norms $, are into A. For each n, wellorder the sequences of length 2n of

PROOF.Assume at first that

the form

(to,ko, el, k , ,..., en, k,) ( t o , ko,*.*,5rv

kn)

(5,
5 (qo, 10,--.,q n ,

1,)

"5o
To & ko=

101

10 & 51


v .*. v

[to= qo&

..*

& 5, = q,, & k,

I l,,]

and let (50,

ko,.*-,t n , kn) * (50, ko,..., tn,

kn)

be an order-preserving map of this ordering into the ordinals. Finally put q n ( a ) = (4o(a),a(O),41(a),~(11, ..., +,,(a), dn)).

We will show that (p = {(P,},,~,,, is a very good r-scale on P. Suppose first that ao,a,,... are in P and for each n and all large i, cp,(a,)is constant, q n ( a i > = ($o(ai), a i ( O ) ~ * *$,(ail, *~

ai(n))

=(to", G..., t:,k : ) . Since by the definition

ky

= ai(j)

( j 5 n, all large i ) ,

232

STRUCTURE THEORY FOR POINTCLASSES

[4E.2

it follows that

k ; = ki is independent of n and ai= a = (k,,, k , ,...).

Similarly,

6; = $i(ai)

( j In, all large i ) ,

so that

6; = ti is independent of n and for all large i, $j (ail

= 6j-

Since 6 is a scale, we thus have a E P and for each j , follows immediately that for each n, c~,(a)S(Eo;o.ko,

ti,

kiY-a-9 t-7

+,(a)IG;from this

k).

It is also immediate from the definition that for x, y in P, cP,(x)rcp,(y)=+foreachi l n , c P i ( X ) l c P i ( Y ) , so that (p is a very good scale. To prove that (p is a r-scale, let a -+, p

and put

R(n, a,P ) - a

w

a 5Ztp & p -=:,a


v ..-

v[a-*J

4 0 )< & a ( O ) = p ( O ) & . * * &a-+J & a ( n ) 5 P ( n ) ]

~ ( 3 i ~ n ) { ( V j < i ) [ a - +&,ap( j ) = p ( j ) ]

&b
Clearly R is in

r and

a s:"P

-

R(n, a,P I ,

so (p is a r-scale, since the argument for < ": is similar.

4E.31

THE

UNIFORh4IZATION THEOREM FOR

ni,zi

233

Finally, if Q c S is of type 1 with X##, let

7T:N-X be a recursive isomorphism, let

P = m-'[Q] and verify easily the following two propositions: if then the sequence

4 is a r-scale of

Q,

= Gn(ma)

is a r-scale on P and if (p is a very good r-scale on P, then the sequence

There are many interesting results about scales and we will look at some of them in the exercises and again in Chapter 6. Here we concentrate on the relation between scales and uniformization.

4E.3. THEUNIFORMIZATION LEMMA.If r is adequate, S is of type 0 or 1 and P G Z x N admits a r-scale, then P can be uniformized by some P* in

vXr.

PROOF.By 4E.2, let

(p = {

-

v ~ } be , , a~very ~ good r-scale

R ( n , x, a) and put

on P, let

( V P ) [ ( Xa) , ~ ? ~ ( PI1 x,

P*(x, a ) w (Vn)R(n,x, a ) . It is sufficient to show that P" uniformizes P, since R is obviously in V N r and hence P* is in V N r . To begin with, clearly P*(X, a)4 P(X, a),

since P*(X, a)=+ (x, a)S?"(X, a) =+

P ( x , a).

Assume now that for some fixed x, (3a)P(x,a);we must show that in this case there is exactly one a such that P*(x, a).

234

r4E.3

STRUCTURE THEORY FOR POINTCLASSES

i I

I

-

7

-

A3

A2

I

I I

I

I

I I

I

I

I

I

I

I

I I I

I

A1

I

J

Po

Figure 4E.1. Uniformizing via a very good scale.

Keeping x fixed, put A, = infirnum{cp,(x, a):P(x, a)} and let (see Figure 4E.1) A,

= {a:P(x, a)}

An+I={a:

P(x,a)&~n(x,a)=An}

={a:(VP)[(X,a ) s w P , ( x P)1} , ={a:R(n, x, a)}.

Clearly each A, is non-empty and

so it is enough to prove that Notice that

n A,

is a singleton.

A0 2 A, and by the second condition on a very good scale,

(VP",

a)swn+, (x, PI1 + (VP)[(x,a)%(X9

@)I,

THE

4E.41 SO

UNIFORMIZATTON THEOREM FOR

n:,

235

that in fact Ao?A,2A22... .

Choose now some ai E Ai, one for each i. We then have (P,(x, a i )= A, for each i > n, so by the first condition on a very good scale, there is some a such that ff

=h i t , ,

ai,

P(x, a),and for each n, qn(X,

a)lAn;

by the definition of A, then, q,(x, a)= A,,

so a E n , A n and this intersection is non-empty. Moreover, if also /3 E A,, then the sequence

n,

ffo,

P,

a 1 9

P, a 2 3 P Y . = Yo, Y1, Y z , Y 3 Y .

has the property that for each n and all large i, (P,(x, y i ) is constant, so that lirniti-yi must converge, presumably both to p and to a = h i & , ai, so that p = a.Hence A, is the singleton {a},which is what we needed to show. -1

n,

r

A pointclass has the uniformization property if every P E I x y in can be uniformized by some P* in r.

r

4E.4. THENOVIKOV-KONDO-ADDISON UNIFORMIZATION THEOREM. (Kondo [1938].“”22’) The pointclasses Ili, Hi, S;, all have the uniformization ProPerty. PROOF.Suppose first that P E I X y is in follows from 4B.4.If not, let

n:.If y is of type 0, the result

7T:K-Y

be a A: isomorphism of X with 9, let

a:x*- I be a A: isomorphism of I with some space I * of type 0 or 1 and define

236

S T R U m R E THEORY FOR POINTCLASSES

[4E.5

Now Q is Il: and by 4E.3 we can find a Zl: set Q * r X * x X which uniformizes Q. It is immediate that the Il: set

P*(x, Y)

-

Q*(a-'(x>,

T-'(Y))

uniformizes P. The argument for IIi is identical. If P E X X y is in Z:, then

P ( x ,Y)

-

(3a)Q(x, Y, a)

with Q in II:. Applying the result about II: to Q E 32 x (3x X ) , we get a n: set Q* G X x (yx X ) which uniformizes Q. Then P*(X, Y) -(3a)Q*(x, Y, a)

is easily seen to uniformize P.

-1

We will see in Chapter 5 that this is just about the strongest uniformization theorem which can be proved in Zermelo-Fraenkel set theory; it is consistent with the axioms of Zermelo-Fraenkel (including choice) that there exists a Il: set which cannot be uniformized by any "definable" set-in particular, it cannot be uniformized by any projective set. Among the many important consequences of the uniformization theorem, perhaps the most significant is the basis result for 2: which we now explain. A set of points (in various spaces) (l3 is called a basis for a pointclass I', if every non-empty set in has a member in (33, i.e. for P E 3c in

r

-

(3x)P(x)

r,

(3x E B)P(X}.

We also say that a pointclass A is a basis for points is a basis for r, i.e. for P in

r,

r if the set of A-recursive

(~x)P(x) w (3 E A)P(x).

In 4D10 we proved that A : is not a basis for Zly, and hence it is not a basis for 2;or II:.

4E.5. THEBASISTHEOREM FORZ:. The pointclass A: is a basis for 2; and more generally, for each x, A : ( x ) is a basis for Z:(x). Thus, if P E X X y is

THE

4E.61

UlrlIFORMIZATION THEOREM FOR

ni,2 ;

237

in S:, then

PROOF.The second assertion immediately implies the first. To prove it, given P G X X Y , choose P * c X x ? j in 2; which uniformizes P. If

-

(3y)P(x,y), then there exists exactly one y which satisfies P*(x, y), call it y*; clearly Y * E RT, @Y)[P*(X, Y) c!k y E N,I

* (VY)[P*(X, Y) so y* is A:(x)-recursive.

-

Y E Ns1, i

Again this result is best possible in Zermelo-Fraenkel set theory, i.e. we cannot prove in this theory that every non-empty Il: set must contain a “definable” element.

EXERCISES

r

4E.6. Suppose is an adequate pointclass closed under substitution of A functions. Prove that if every pointset of type 1 in admits a r-scale, then every pointset in admits a very good r-scale. In particular, n: and ll: are scaled. Hint. Suppose P G X is given, P in Using 3E.6, let

r

r

r.

?T:X+X be a recursive surjection of X onto 3c such that for some I Z y set A G X , n [ A ]= I and .rrf(x) = x for every x E I , with a A: function such that ~ c % I = APut . Q(a) E A c!k P(?T(cu)), so Q is in

tj={+JnEU.

-

r and by hypothesis and 4E.2, Q admits a very good r-scale On P set

c ~ n

= JI, Cf(x))

and show that (p is a very good r-scale. The key point is the continuity of n ; this implies that if xo, xl, ... are in P and cxi = f(&), then limiq, = -I limit,; w ( a i )= ~ ( awith ) ~ ( aE P. ) The analog of 4B.3 also holds for scales, i.e. if I‘ is scaled, adequate and closed under V”, then PI‘is also scaled. There is a bit of computation to this and we will postpone it until Chapter 6 when we will need it.

238

STRUClURE THEORY FOR POINTCLASSES

[4E.7

The next result is implicit in the proof of 4E.4, but we put it down for the record.

r

4E.7. Prove that if is adequate and closed under substitution of A: functions and V N and if every pointset of type 1 in r admits a r-scale, -I then both and yr have the uniformization property.

r

Every non-empty Zi set has a A: member by 4E.5 but need not have a A: member by 4D.10. The correct basis for Zi is a small part of A:, by the next result. With each relation P E 32 = ok on a space of type 0 we associate its contracted characteristic function ap, =

I

1 if P ( ( n ) l ..., , (n)k)r 0 if i P ( ( n ) ]..., , (n)k).

We call a set, function or point recursive in P if it is recursive in ap.Notice that we only define these notions here for P of type 0-the correct concept of recursion relative to an arbitrary pointset is quite complicated and we will not go into it now. 4E.8. Prove that there is a fixed 2 ; set P of type 0 such that {x: x is recursive in P } is a basis for 2 ; . (Kleene's Basis 'Theorem, Kleene [ 1955b Hint. Suppose Q ( a )w (3P)(Vt)R(6(t), P(t))

is a typical Z: set of irrationals with R recursive. As usually, we can think of Q as the projection Q = p[TI of the tree T on w x o,

T = {(a,, b ~a,, , h,..., at-,, bt-d: (Vi.ct)R((ao,...,a,),(b~,...,bi)). Any infinite branch of this tree will determine an element of Q, so our aim is to find a definable infinite branch. The basic idea of the proof is that the leftmost infinite branch (see Figure 4E.2) of T is recursive in some 2 ; set P of type 0. Recall the function u * v from *18 of 3A.6 and suppose that we can find integers a,, b,, a , , bl ,... such that for every n,

(*I

(3a)(3P)(Vt)R((ao,..., a , - h * W ) ,(b,,..., b,-,)* P ( t ) ) ;

4E.81

THEUNIFORMIZATION

THEOREM FOR

I

ni,zi

239

I

I Figure 4E.Z. The leftmost infinite branch.

choosing a, p to witness this, and taking t = 0 , we have in particular

( v n ) R ( ( a o ~an-l), - - . ~ (bo>*.-, bn-l)), i.e. with a ( n )= a,, p ( n )= b, now, we have (Vn)R(E(n), P ( n ) ) ,

so

~ E Q .

It is clear that (*) simply asserts that an infinite branch of T starts with the finite sequence (ao,bo,..., a,-,, bn-l). We now choose for each n the leftmost finite sequence which is the beginning of some infinite branch. To be precise, put

P(u, v ) 4=3 (3a)(3P)(Vt)R(u* a t > , clearly P is

tJ

*m;

2;.It is easy to verify that P(u, v ) =i* ( 3 n ) ( 3 n ) P ( u* ( n ) ,v * ( m ) ) .

Thus we can define

(Y

= (ao,ul,...),

P = (bo, bl,...) as above, recursive in P,

240

STRUCTURE THEORY FOR POINTCLASSES

[4E.9

by the simple recursion a(t>= (cLsP(a!(t)*((s)o), P(t)*((s)1)))07

P ( t ) = (PSP(m*((S)O),P ( t > * ( ( S ) 1 ) ) ) 1 .

This shows how to assign to each Zi set Q G X a 2: set P of type 0 such that { x : x is recursive in P} is a basis for the single pointset Q. To get a single P so that {x: x is recursive in P} is a basis for 2: subsets of N, apply this procedure to some Q E o x N which is universal for 2 i b N . Moreover, to see that this yields a basis for Zi,use the fact that for every X there is a recursive surjection T :X * ‘X and that if a is recursive in P -I and T is recursive, then ~ ( ais )recursive in P. It should be quite obvious by now that every basis result implies some uniformization result, at least implicity, as a corollary of its proof. The uniformization theorem that comes out of the preceding exercise is a bit messy, but it is worth putting down because it implies that we can always find measurable uniformizations for 2: sets. 4E.9. Prove that every 2: set Q c X x y can be uniformized by some Q * E Q which can be constructed from 2: and II: sets using the operations &, v, 3”,V”. Infer that if Q E X x y is 2: and D = aYQ is the projection of Q on X, then we can find a function f :X y which is Baire-measurable, absolutely measurable and such that (Vx E D ) Q ( x , f(x)). (rite uon Neumann Selection Theorem, von Neumann [1949].‘23’) Hint. It is enough to prove the result for X X X with X of type 1, since the smallest pointclass containing 2: and II: and closed under &, v, 3”, V” is easily invariant under Bore1 isomorphisms. Suppose then that

-

Qb,a >

--

with R clopen and put P(x, u, u ) P*(x, t, u, u )

( 3 P W t ) R ( x ,E ( t ) , P ( t > )

(3a)(3P)(Vt)R(x,

*w, *m, ‘v

Seq(u) & Seq(u) & lh(u)= lh(u) = t

P(x, u,v ) & (Vu’)(Vu’){[Seq(u’) & Seq(u’)& Ih(u’)= lh(u‘)= t & (u’, u’)<(u, u)] d l P ( X , u, v ) } .

THEUNIFORMIZATION

4E.91

THEOREM FOR

n:, 2 ;

24 1

It is clear from the proof of 4E.8 that the relation Q*(x, a)e ( V t ) @ u ) P * ( x , t, c ( t ) , U )

uniformizes Q . For the second assertion, assume first Q G X X N and define P, P", Q* as above, choose a fixed a . E X and put

f(x>=

a.

if ( V a ) i Q ( x , a )

a

if ( 3 a ) Q ( x , a ) and Q*(x, a).

For any closed F c X we have f ( x ) E F w [a0E F & ( V a )

7Q(x,

a ) ](%)[a ~ E F & Q*(x, a)]

and since II: sets have the property of Baire and are p-measurable for each o-finite Borel measure p, it is enough to prove that the set B = { x : @a)[aE F & Q * ( x , a)l) has the same properties. Computing, x EB

@

( 3 a ) ( V t ) { ( 3 ~ ) P * ( x ,t, E ( t ) , U ) ( 3 P > [ P ( t >= d t )

@

P E FB

( 3 a ) ( V t ) [ x E S&(t)I,

where SE(t)= { x : ( 3 U ) P * ( X , t, E ( t > , u ) & ( 3 P ) ( P ( t )= E ( t ) & p E

a}.

Now each S, is absolutely measurable and has the property of Baire by 2H.8 and 2H.5 and B = Sa,S,, so by the same results,

B is absolutely measurable and has the property of

Baire. In the general case, if P c X x y with y perfect, let Borel isomorphism, let Q G X X X be defined by

T

: N--y Y be a

Q ( x , a)* P ( x , da))

and choose f :X +X as above. Take g ( x ) = .rr-'(f(xN

and verify easily that g has the required properties, since for any Borel -I = { x : f(x) E T [ A ] } .

A E 3, g - l [ A ]

This is the strongest result we can prove in Zermelo-Fraenkel set theory in this direction. We will see in Chapter 5 that it is consistent with

STRUCTURE THEORY

242

FOR POINTCLASSES

[4E.10

this theory that there exists a function f :(R + (R whose graph is II: and which is not Lebesgue-measurable; the graph of f, then, is a ll: set in the plane which cannot be uniformized by the graph of a Lebesgue measurable function. By 4E.5, every non-empty H i set has a A: member. The next exercise gives another basic result for H i which is stronger, at least superficially. For any pointclass a point x is a r-singleton if the set {x} is in

r,

r.

4E.10. Prove that the collection of Hi-singletons is a basis for H i . Hint. Given P G Iin H i , let a n , x) * P(X> and let Q* E Q uniformize Q in Hi-singleton in P.

n:.The unique x such that Q*(O, x) is a

-I

On the other hand, if we impose the weakest natural closure property on a basis for H:, then this basis must include all of A:. 4E.11. Suppose B is a set of points which is a basis for H i and which is closed under Turing reducibility sT, i.e. y

E

B and x is recursive in y

4

Prove that 63 contains every A: point. Hint. If a: is a A: irrational, then the set P P(P)

Let

-

(3Y){(VS)[Y

P(P>

E

N

4

x E B.

= { p : P = a:}

is easily xi,

a E N1%I Y = P).

@r)Q(P, 7 )

with Q in H i and let Q* E Q be in n: and uniformize Q. Now Q* is non-empty, so it must contain a point of @, which must be (a,y ) for some y. Since a is recursive in (a,y), a E B.It follows easily that 63 contains the -I A: points in all spaces.

4E.12. Prove that for each perfect I, the collection of Hi-singletons in 9c is a H i pointset-and hence a proper subset of A : f l I by 4D.16. Hint. Choose a universal H i pointset G E o x I, let G* uniformize G and notice that x is a Hi-singleton w (3e)G*(e,x).

-I

4F.11

243

ADDITIONAL RESULTS ABOUT IT:

4F. Additional results about Il: Most of the results in this chapter have been about a general Spector pointclass perhaps with an additional hypothesis that is closed under VN or that it has the scale property. Here we will look at some very specific properties of Il: which do not follow easily from neat, axiomatic assumptions. These results too will be extended to some of the higher Kleene pointclasses in Chapter 6, using strong set-theoretic hypotheses, but we will need new proofs from them. First, an effective version of the Perfect Set Theorem 2C.2.

r,

r

4F.1. THEEFFECTIVE PERFECT SETTHEOREM (Harrison [1967]; this proof due to Mansfield [1970]). If P G X is a Zi pointset which has at least one member not in A:, then P has a non-empty perfect subset. Similarly, if P is Z:(z) with some member not in A : ( z ) , then P has a non-empty perfect subset. In particular, if P c X is Z:(z) and countable, then P c A :(z) n X.

PROOF.The argument for Z:(z) is identical with that for Zi,so we only prove the absolute version. We may assume that P has no A members, since {x E P: x $ A :} is also 8: by 4D.14. Suppose X = X , to begin with, choose a recursive R such that P ( a >CJ ( ' l P ) ( V t ) R ( W P, W ) , and let

clearly T is a tree on w x w and in the notation of 2C,

P=p[T].

Look up the proof of the Perfect Set Theorem 2C.2. We claim that in the notation used there,

T=S; because if not, then there is some u = (ao,bo,..., a,-,, bn-,) E T with p[TJ

STRUCTURE THEORY

244

a singleton {a}and

a ( n )= m

-

[4F.2

FOR POINTCLASSES

(3a’)(3P){(tlt)R((a,, ..., an-l)*c5’(t), (bo,*.-,bn-l)* P ( t > ) & a ’ ( n )= m } ,

so easily a is A i. Now p[S] = p[T] # 8, so P = p [ q has a perfect non-empty subset as in the proof of 2C.2. -I The result follows for arbitrary X as in 2C.2, using 3E.6. This theorem implies in particular that Borel sets with countable sections can be uniformized by Borel sets, see 4F.6. The next result is a converse to 4D.3 for the case = H i . First a lemma.

r

4F.2. LEMMA.There is a H y relation S ( a , 0, y ) , such that whenever P E WO and a E LO, a E wo & la1 5 IPI ( 3 y ) S ( a ,P, 7 ) @

w

(37E A :(a9P ) ) S ( a , P, Y ) .

PROOF.The notation is that of 4A. As in the proof of 4A.7, put Q(a,P, y )

y maps 5,onto an initial segment of sD in an order-preserving fashion

and y

=0

outside the field of

Sa,

where we allow “initial segment” to include all of s-.As in that exercise, Q is easily H:, say Q(a,P, Y ) * ( t l n ) @ m ) R ( a ,P, Y , n, m )

with R recursive. Put further,

Q*b, P, Y , 6)

@

( V n ) R ( a ,P, Y , n, a ( n ) )

and notice that Q(a,P, Y ) * (36)Q*(a,P, Y , 6)

* (38E A : ( a , P, r))Q*(a,P, Y , 61, since if (3S)Q*(a,P, y, S), we can choose S(n)=least m R ( a , 0, y, n, m ) , where this 6 is clearly in A : ( a , P, y ) .

4F.31

ADDITIONAL RESULTS

ABOUT

Ilt

245

Finally, take S ( a , P, Y ) * Q*(a,P. ( ~ 1 0 (Y)I) ,

and verify easily that the lemma holds with this S. 4F.3. THE SPECTOR-GANDY THEOREM (Spector [1960], also Gandy [1960]). For every n: set P E X there is a ll: set R G X x X such that P ( x )w (3E 1A : ( x ) ) R ( xa). ,

PROOF.Suppose first that P is A:. By 4A.7 there is a n':set A GJV and a recursive T :X + X which is injective on A and T [ A ]= P. Hence,

P ( x )e+ (3a)[aE A & T ( ( Y )= XI

* (3a E A~(x))[(YE A & T ( ( Y )= XI, where the second equivalence holds because if ~ ( a= x) and a E A, then a is the unique irrational satisfying these conditions and it is easily A : ( x ) . Thus for a A: set P we have the stronger representation: P(X)* (3a)R(x,a) e+

(3a unique a ) R ( x ,a)

* (3aE A Xx))R(x,a), where R is some n(:set. Towards proving the result for n: pointsets of type 0 or 1, recall first 4D.14 according to which {(a,x): a E A :(x)} is n:.Hence, for 3c of type 0

246

STRUCTURETHEORY

FOR POINTCLASSES

or 1, there is a recursive function g :X g(a,x) E LO and OL E

A :(x)

H

X

g(a,

[4F.3

X -+ X such that for each a, x,

X) E

WO.

For each x, let o;= supremum{(@(: p is recursive in x, @ E WO};

by the relativized version of 4A.4, easily, for each x (1)

supremum{lg(a, x)l: a E A :(x)} = ot,

or else {a:a EA:(x)} would be A : ( x ) , contradicting 4D.16. Suppose now P E X is n:,with X of type 0 or 1, so there is a recursive f :X 3 X such that for each x, f(x) E LO and

P ( x ) w f ( X ) E WO. From (l),we get immediately

since for each x, f(x) is recursive in x. We claim

(2)

P(X) *

E A b ) ) ( 3 Y E A : ( x ) ) s ( f ( x ) g, ( a , X I ,

Y),

where S is the n':set of the lemma. To prove direction ( 4 )of (2), assume P ( x ) ;then f ( x ) E WO and f(x) is recursive in x, so by (1) there is some a E A :(x) such that I f ( x ) l s Ig(a,x)(. By the lemma then, there is some - y ~ A : ( f ( x ) ,g(a,x)) such that S(f(x), g(a,x), y); but clearly, Y E A:(x) by 3E.17 since f(x) is recursive in x and g(a, x) is recursive in (a,x) and hence A : ( x ) . To prove direction (*) of (2), suppose there is an a E A : ( x ) and some y such that S ( f ( x ) , g ( a , x), y ) . Now g(a,x) E WO and ~ ( x ) ELO, so by the lemma we have ~ ( x ) EWO, i.e. P ( x ) . This completes the proof of (2). From (2) we get the theorem for any pointset of type 0 or 1 by a trivial contraction of quantifiers. Finally, suppose P c_ 3c where X is not of type 0 or 1, so there is a A: isomorphism

a:#-X. If P is

n:,then the inverse image Q ( a )w P(.rr(a))

4F.41

ADDITIONAL RESULTS

ABOUT

Il:

247

The Spector-Gandy theorem does not have many applications but it is undoubtedly one of the jewels of the effective theory. It gives a very elegant characterization of II: in terms of a (restricted) existential quantifier which is particularly significant in the case of relations on w : P E w is If: i f and only if there is a II? set R E w X .N such that

P ( n ) e ( 3 a E A : ) R ( na , ). This corollary says in effect that the collection of A irrationals somehow “determines” the collection of II; relations on w. The third main result of this section is also peculiar to the effective theory, like the Spector-Gandy theorem. It differs from it in that it says something most significant about perfect product spaces. A set P E ‘X is thin if P has no perfect subsets other than 9. Countable sets are thin and by the Perfect Set Theorem, every S: thin set is in fact countable. As we will see in the next chapter, it is consistent with the axioms of Zermelo-Fraenkel Set theory that there exist uncountable, thin II: sets. 4F.4. THELARGEST THINIIi SETTHEOREM (Guaspari [19??], Kechris [1975], Sacks [1976].). For each perfect product space ‘X, there is a thin, II: set C , = C,(’X)E ‘X which contains every thin, II: subset of 32.

PROOF.Fix 3c and let G E w x ‘X be universal for the IT: subsets of ‘X, let cp : G 4 Ordinals

248

STRUCNRE THEORY FOR POINTCLASSES

-

[4F.4

be a ni-norrn on G.Put

(1) R(n, x) We claim:

G(n,x) & [{y : G(n,y ) & cp(n, y ) 5 ~ ( nx)}, is countable].

(2) R(n,x)*G(n,x)&(Vy)(CG(n,

y > & c p ( n , y ) ~ c p ( n , x ) J = *EA:(x)}. y

To prove direction (=.) of (2), notice that if R(n, x), then the set

A ={Y: W n , Y) 8~cp(n, y)'cp(n, = {Y : (a, = {Y

XI}

Y 15: (n,X I }

: 1 ( ( n , X I <:(n,

Y))}

is A:(x), so if A is countable, we must have A=A:(x)n'X by 4F.1. Conversely, assuming the right hand side of (2) we immediately infer that A is countable, since A:(x) f l 'X is countable. Now (2) implies that R is n:,since it yields

R ( n , x ) * G ( n , x ) & ( V ~ ) [ ( n , x ) < ~ (Yn) ,V Y EA:(x)I. We define C , = Cl(3C) by

* (3n)R(n,X I . Clearly C , is n:,so it remains to show that C , is thin and that it contains every thin, If: subset of 'X. Assume first that Pc 'X is thin and n:,so that for some fixed no,

P(x)

G(no,x).

For each x in P, {y: G(no,y ) & cp(no,y)scp(no, x)} is A:(x) as above; in particular it is Borel, so it must be countable, since it is a subset of P and cannot have a perfect subset. Hence P(x>

-

Wn0, XI =* Rho, x)

-

Cl(x>

and P E C,. Suppose now, towards a contradiction, that F f @, F is perfect, F G C,, Put Q(x, n) * F ( x ) & R(n,x).

The relation Q is Il: and (Vx ~ F ) ( 3 n ) Q ( xn), , so by the A-selection principle 4B.5, there is a Borel function g :3c + w such that ( V X E F)R(g(x),x).

ADDITIONAL RESULTS ABOUT ll:

4F.51

249

The map x * (g(x), x)

is also Bore1 and maps F into G. Now G is not in S: by 3E.9 (otherwise every Il: subset of X would be in S:), hence by the Covering Lemma, 4C.11 there exists an ordinal A < 191, such that ~~F==.cp(g(x),~)rA. The ordinal A is countable, since Iq 15S

= K1.Letting

An,E={X:R ( n , x ) & ~ ( n , x ) = 5 ) , this means that

F E Un,Es;h An,<, However, each An,Eis countable, since An.Edy: G(n, Y ) & P ( ~Y, ) ~ V ( ~ , X ) ) with x any point such that R(n, x) and q ( n , x) = 5, so F is countable, -I contradicting the assumption that it is perfect. This theorem has led to an interesting theory of the structure of countable and thin Il: sets which we will not pursue here beyond 4F.7 and 4F.8. See Kechris [19751, [19731.

EXERCISEB

4F.5. Prove that if P c N is a countable 2: set of irrationals, then there exists a A : irrational

E

such that

p E {(E)O,

(&)I, (E)*,...).

Hint. By 4F.1, P s A i n N and by 4D.14 the set A : n N is Il:. It follows from the Separation Theorem, 4B.11 that there exists a A : set Q, PsQGAinN.

Let c : N + w be the II:-recursive partial function of Exercise 4D.15 and notice that

A

= {i: &)[a E Q = {i:

= i])

(3aE A :)[aE Q & c ( a )= il)

is easily A and the parametrizing partial function d :w -+N is defined on

250

STRUCrURE THEORY FOR POINTCLASSES

[4F.6

A. Put &((i,j))=d(i)(j) if ~ E A , & ( k ) = O if k # ( i , j ) forall ~ E A .

-1

4F.6. Prove that if P G X X Y is a Borel set such that each section P, = {y : P ( x , y)} is countable, then the projection 3yP is Borel and P can be uniformized by a Borel set P*. (Lusin [1930a], Novikov [1931]."9-21') Hint. Suppose P is A:(&). Each section P, is easily in A:(&, x), so

p, # B ==+ (3Y E A:(&, XI"

Epx1

by the Effective Perfect Set Theorem 4F.1. Now apply the A -I Uniformization Criterion 4D.4, taking r = H i ( & ) . 4F.7. Prove that a set P is thin if and only if every Borel subset of P is countable. Infer that the notion of being thin is preserved by Borel isomorphisms. Hint. Use the Cantor-Bendixson Theorem. -I The result implies that if

7~ : X w

y is a A i isomorphism, then

C,(Y>= d C I ( X ) l . Hence all the sets C,(Z) for perfect X are determined by the set the largest thin set of irrationals. 4F.8. Let C c X be a thin II: set and on C define xsy~xisA:(y). Prove that I is a prewellordering, so C ramifies into a wellordered sequence of sets of points which are A:-equivalent. Prove that the length of 5 is no more than K1.(Kechris [1975].) Hint. Let cp : C + Ordinals be a Hi-norm on C. Suppose x, y are in C and cp(x) 5 cp(y). Since the set

A

= {z : z E C & cp( z ) I cp(

y)}

is easily A:(y) and has no perfect subsets, A G A:(y) n X by 4F.1; hence x is A :(y). This proves comnarability and transitivity is already known from

ADDITIONAL RESULTS

4F.101

ABOUT

n:

25 1

3E.17. To prove that Iis wellfounded, assume that x, y are in C and y is -1 not A : ( x ) and prove as above that c p ( x ) s c p ( y ) . In the remaining exercises we outline the proofs of several uniformization theorems for Bore1 sets. Let us first recall a simple fact about trees which will be needed below. 4F.9. Let T be a tree on a set X which is finitely splitting, i.e. every sequence U=(X,,...,X,-~) in T has at most finitely many one-point extensions in T, (x, ,..., xnP1,y l ) ,..., (x, ,..., x,-~, y k ) . Prove that T is infinite if and only if it has an infinite branch. (Konig's Lemma.) Hint. If (x,, x1,...) is an infinite branch, then for each n, (x, ,..., x,-~)E T, so T is infinite. If T is infinite, then for some xo the subtree T(%)must be infinite, since T = T(,,,U U T(,,, for some y l ,..., y k . Again, for some x1 the subtree T(,+,) must be infinite, so inductively we get an infinite branch (x,, xl,...). -I

Fix a product space X and for simplicity of notation let N ( s )= N ( X , s) be the s'th basic nbhd of X. A finitely splitting tree of nbhds of X is a tree T on w which is finitely splitting and such that if (so,..., s , - ~ ) E T, then for each i = 0,..., n - 1 radius(N,,)~ 2 - For ~ . simplicity we will call these nbhd funs (see Figure 4F.1). With each nbhd fan T we associate the subset of S

K = K ( T )={x :(3a)(Vn)[(a(O), ...,a ( n - 1 ) )T&x ~ E N ( a ( n- 1))3). It is not hard to verify that each K ( T ) is a compact subset of S and each compact set K is K ( T ) with a suitable nbhd fan T. In the next result we get an effective version of this. Let us say that a nbhd fan T is in u pointclass A if the set of codes of the sequences in T is in A, i.e. if

T" ={(SO

is in A.

r

,..., s,-,):(s~ ,..., s , - ~ ) E T }

4F.10. Let be a Spector pointclass closed under V N and X a fixed perfect product space. Prove that a set K E S is compact and in A if and only if there exists a nbhd fan T in A such that K=K(T).'24'

252

STRUCrURE THEORY FOR POINTCLASSES

Figure 4F.1. Neighborhood fan.

Figure 4F.2.

[4F.10

ADDITIONAL RESULTS

4F.101

ABOUT

ll:

253

Hint. If T is in A, then x E K(T)-(Vn)(gu)[u

E

T'&lh(u) = n&(Vi < n)[x E N ( ( u ) ~ ) ] ] ;

the implication (-) is immediate and the converse implication follows easily using Konig's Lemma. Thus K ( T ) is in A and its compactness can be proved by a simple topological argument. Conversely, suppose K is compact and in A. Recall from 4C.13 that there is an irrational E E A such that

X -K and for each n,

= N(E(O)) U

N ( E ( ~U) )

N(E(n))s X - K .

To construct a nbhd fan T such that K = K ( T ) , intuitively we first find to,..., t k as in Figure 4F.2, such that

and each N ( t i )has radius 11. Then for each i, we find so,..., s, such that

n&(ti) E N(S
and each N ( s i ) has radius s;, etc. The key to proving that this can be done in r i s the A-Selection Principle 4B.5. Put

P(n, s, u )

Seq(u)& (Vi < Ih(u))[radius(N((u),))~2-"1

n N(s)

(3 < I ~ ( u ) ) [ Ex N ( ( u ) ~ ) ] ] 8~( V X ) [ X E N ( E ( n ) )+ (vi < ~ h ( u ) ) [ xN((uIi)]I. &

& (VX)[X E K

Since K f

l N(s)is compact

and disjoint from N ( E ( n ) ) ,easily

r,

Moreover, P is easily in so by 4B.5 there is a A-recursive function f(n, s) such that (Vn)(Vs)P(n,s, f(n, s ) ) . Choose once and for all to,..., tk such that each N ( t i )had radius 5 1,

254

STRUCTURE THEORY FOR POINTCLASSES

[4F. 11

and put

T = {(so ,..., sn-,):so is one of to,..., tk & (Vi < n - 1)(3j< Ih(f(i

+ 1, si)))[si,, = (f(i + 1, s~))~]}.

Clearly T' is in A and it is simple to check that K ( T )= K .

i

This representation of compact sets in A allows us to prove that each of them (if # (a) must have a member in A. 4F.11. Let r be a Spector pointclass closed under V". Prove that if K is a compact, non-empty set in A, then K has a member in A.'"' Hint. Choose T in A such that K = K ( T ) , let

R ( u ) H (Vn)(3u)[Seq(u)& Ih(v) = n & u * u E T'], R(u,u ) @ 1 R ( u )

v [ R ( u )& R ( u )& u is a one-point-extension of u ] and by 4C.12 let f: w+w be in A and such that

f(0) = 1= code of the empty sequence, ( V n ) R ( f ( n )f(n , + 1)). It is now easy to check that is clearly in A.

n

N(f(n)) contains a single point in K which -1

4F.12. Let r be a Spector pointclass closed under V", suppose P G 'X X ?I is in A and for each x, the section P, is (a or contains a compact set in A(x). Prove that the projection of P is in A and P can be uniformized by some P" in A. Verify that the hypothesis holds if each section P, is compact.'24' i Hint. Use 4F. 11 and the A-Uniformization Criterion 4D.4. This result implies immediately that Borel sets with compact sections can be uniformized by Borel sets. We proceed to show that for this it is enough to assume that the sections are a-compact. First a purely topological lemma. 4F. 13. Suppose A E X is closed, F :X + 'X is continuous and

4F.131

ADDITIONAL RESULTS

ABOUT

lli

255

with each K,, closed. Prove that for some n and for some basic nbhd N, in

x,

P, # F [ A fl N J c K". (Kunugui's Lemma.) Hint. Towards a contradiction, suppose no F [ A fl N,] is contained in some K,,. In particular, there is some x = F ( a )6 K O ,so there is a nbhd M of x such that M f l KO= P,. We can then find a basic nbhd NO of a such that F[A nN"]E M, in particular

F[A f l No]flKO= fl as in Figure 4F.3. Assume now that F(A n p ] is not a subset of K1, so there is some x = F ( a ) with a E A nNO, x $ K 1 and repeat the argument. Thus we get a sequence of basic nbhds NO

=, N' =, ...

in X such that each F [ A f l N'] is non-empty and f l A f l N'] f l Ki= $3. If we also make sure that N =, N + ' and radius(") + 0 as i + a,we find a point a E A f l N ' for each i, so that F ( a ) $ Ki, all i, which contradicts F ( a )E U" K". -1 The next lemma isolates part of the construction that we need for the main result here.

Figure 4F.3.

256

STRUCTURE THEORY

4F.141

FOR POINTCLASSES

4F.14. Let f be a Spector pointclass closed under V N , let ‘X be a fixed product space, suppose AcKsB, where A is in ir,K is compact and B is in fan T in A such that

r. Prove that there is a nbhd

A E K ( T )E B.‘24’ Hint. Notice first that the closure

A

of A is also in

i r since

and of course A G K, so A is compact. By the Separation Theorem for (4B.1 1) choose a A -set C such that

-

7r

AGCGB. Following the method of proof of 4C.15, put

P(x, s) H x $ C & x E N, & N,nA = @ (Figure 4F.4); P is clearly in f and (Vx$ C)(3s)P(x,s) since A is a closed subset of C. By the A-Selection Principle then, there is a A-recursive function f such that (Vx$ C)P(x,f(x)). The set

Figure 4F.4.

4F.141

ADDITIONAL RESULTS

ABOUT T :

251

the separation theorem gives us a A set I between these two such that the open set G = {Ns: SEI} clearly satisfies

u

so in particular,

x&C==+.XEG

x ~ B ~ x E G , see Figure 4F.5. We now imitate the construction of 4F.10 above to get a nbhd fan T such that the associated set K ( T ) satisfies -

A sK(T) K(7')f-l G =fJ. This will complete the proof, since we evidently have -

AcK(T)cB.

Figure 4F.5.

STRUCTURE THEORY

258

FOR POI~CLASSES

14F.15

Briefly, we first write

G = N(E(O))U N ( E ( ~ ) ). *U * with some that

E

in A and N(E(rn))E G, all m and then we find to,..., t k such

{N(to)u u N(tk)}nN ( E ( o ) )

= 8,

and each N ( t i )has radius 5 1. Then for each i, we find so,..., s,, such that

A nN(ti)E N(s,) u ..-u N ( s , ) , {N(s,) u u N(s,)} n{N(E(o)) uN(E(I))}= p, and each N ( s i ) has radius<& etc. The proof that this procedure determines a A, compact set K ( T ) is via the A-Selection Principle as in 4F.10 and we omit it. -I A set is cr-compact if it is a countable union of compact sets. (In every Z: set is cr-compact.)

a''

4F.15. Suppose L E X is a nonempty, A: set which is a-compact. Prove that L has a nonempty A: and compact subset; infer that L has a A: member. Similarly with A i(x) substituted for A ; t h r o ~ g h o u t . ' ~ ~ ' Hint. The argument for A :(x) is identical with that for A ;. Suppose L = U, K,, where each K,, is compact. By 4A.7, there is a L!: set A E X and a recursive F: X + X,injective on A, such that nA]= L. By 4F.13 then, for some s and some n, the set

B=F[AnN,] is contained in some K,,. Now B is A:, by 4D.7, B is compact and

B f BGBGL; by the preceding exercise then, there is a compact set K in A such that

~#BGKEL and then L has a A: member by 4F.11 above.

-I

We put down for the re:ord the uniformization theorem that follows from this exercise.

4F.16. Prove that if P E X X y is A:(z) and every section P, ={y: P ( x , y)}

4F.181

ADDITIONAL RESULTS

ABOUT

259

Ill

is o-compact, then the projection 3yP is A : ( z )and P can be uniformized by some P* in A ; ( z ) . Similarly, if P c X x y is Borel and every section P, is a-compact, then 3’P is Borel and P can be uniformized by some Borel P*. (Arsenin [19401, Kunugui [19401; see also Larman [19721.) Hint. Use 4F.15 and the A-Uniformization Criterion 4D.4. If P is -I Borel, use the fact that P must be A:(&) in some E , 3E.4. The uniformization theorems of Lusin-Novikov (4F.6) and ArseninKunugui can be turned into interesting structure theorems about Borel sets with “small” sections which we now proceed to show. 4F.17. Suppose Pc_ X X y is in A ; ( z ) and every section P, = {y : P(x, y)} is countable. Prove that there exists a set P* E o X X X y in A : ( z )such that P(X, Y)

-

(3nP*(n, x, Y)

and such that for each n E w, the set

P: = {(x, y ) : P*(n, x, y ) ) c P uniformizes P. In particular, if P c Z X Y is Borel and each section P, is countable, then P = U,PZ where each P: is Borel and uniformizes P. (Lusin [1930a], Novikov [19311.) Hint. Let P * E P uniformize P in A:(z) by 4F.6, and put Q(x, Y, n )

-

d(n, 2, x1.l

d(n, 2, X )

=Y

n

where d is the partial function which parametrizes A i ( z , x) y by 4D.2. ~ y, n), so by the A-selection Theorem 4B.5 Now (V(x, y ) P)(3n)Q(x, there is a A :(z)-recursive f : X X y .+w so that

4F.18. Prove that if P c X is A:(z) and o-compact, then P ( x )M (3K)CK is A:(z) and compact, K c P and X E K ] ; moreover P is S,”(a)for some

260

r4F.18

STRUCTURE THEORY FOR POINTCLASSES

OL E A : ( z ) , in

fact P satisfies an equivalence

(*I

P ( x ) e (Vn)P*(n, x)

where P* is in n:(a) for some a EA:(z) and such that each section P z = {x: P*(n, x)) is compact. (~ouveau.)('~) Similarly, if P E X x y is in A i ( z ) and each section P, is a-compact, then there is some P* E o x X x y in A : ( z )such that P(X, Y)

@

(VnP*(n, x, Y)

and such that each section P:,x is compact. In particular, if P E !X x y is Borel and each section P, is a-compact, then there exist Borel sets P: such that

P = u,P: and each P: is Borel with compact sections. (Saint Rayrn~nd.)"~' Hint. To simplify notation suppose P is A: so that by 4A.7,

P

-

= flA]

for some n: set A E K and a recursive F :N Put cx E A* ( ~ s ) { oELN, & (Vp)[p E N, f l A

+ X,with

F injective on A.

=, ( 3 K ) [ K is compact in A i, K E P and F ( P )E K ] ] } .

It is easy to verify that A* is a ZIf set, using the representation of compact A: sets via nbhd fans and it is obvious that A* is open. We will prove that A E A*. Assume towards a contradiction that

B=A-A* is nonempty and notice that B is 2 : and closed. Since

F [ B ]E F[Al

= U,K,

with each K,, compact, 4F.13 above implies that for some s, n

so that for some s, F[B n N , ] is a nonempty compact subset of P. Since F [ B n N S ]is in Zt, 4F.14 then guarantees that there is a A:, compact set K such that

F [ B nN,] E K E P.

4F. 181

ADDITIONAL RESULTS

ABOUT

26 1

II:

Fix a E B nN, and suppose p E N, nA. If p E A*, then F ( p ) is a member of some A i, compact set by definition; if p$ A*, then p E A - A* = B,so F(P)EK, so again F ( p ) is a member of some A:, compact set. This establishes that ar E A*, contradicting ar E B. Thus we have shown AgA*. It then follows immediately that for some A i, compact set K E P, x E K .

x EP

Call a E K u code of the nbhd fun T if

-

(SO

and put

Q(x, i)

,..., s,-

1)

E T e (Y ((SO

,..., s,-

1))

=1

x E P & d ( i ) l & d(i) codes a nbhd fan T in A t

such that K ( T )E P and x E K ( T ) ,

where d parametrizes A : n N by 4D.2. Easily Q is l7: and ( V x ~ P ) ( 3 i ) Q ( xi), , so we can find a A: function f : P - + o such that (Vx E P)Q(x, f(x)). Put Put also

RJi)

-

Rl(i) H (3x E P)[f(x) = i].

d(i)J,and d(i) codes a A i, compact subset of P

and notice that R1 is set R such that

-

Xi, R , is II: and R 1 s R2,so we can find some A: R 1 s R E R,.

It is now obvious that iER and

so letting

d(i)l and d(i) codes some A i, compact set Ki

P=

u {Ki: ~ E R }

P*(n, x) w R(n) & x E K,, we have (*) in the theorem with P* in A t and such that each section P: is compact. The same argument relativized to a fixed but arbitrary x gives the second assertion and then the third assertion follows trivially. To get the full strength of the first assertion in the theorem choose p E A: so that

{(P),,: n = 0, 1,...}={d(n): R(n)},

262

STRUCKIRETHEORY

r4F.19

FOR POINTCLASSES

choose Y E A : so that for each n and m

(-y),(m)= largest u so that Seq(u) & lh(u)= m (P>,(u) = 1

and let

P * ( n , x ) o x is in K(T,,) where T, is the nbhd

-

fan coded by (p),

(VmI(3u ~ ( ~ ) , ( r n ) ) [ ( P ) ,=( u1>& h ( u )= m & (Vi< m ) E N((~>i)ll*

Again we have (*) with this P* and this pointset is clearly IIy(a) with a = ( P , ?)Ed:. -I The basic idea in these uniformization results about Borel sets is that we can find a Borel uniformization when the sections of the given set are “topologically small,” i.e. o-compact. We now proceed to show that we can also find Borel uniformizations when the sections are “topologically large,” i.e. not meager. The key to this type of result is a basic computation of the category of 2; and 2: sets. 4F.19. For each set P c Z x Y , put Q(x) -P,

={y: P(x, y)} is not meager.

Prove that if P is Zy,then Q is also Z;,if P is Zj, then Q is Zi and if P is n:,then Q is also II:. (Kechris [1973].) Hint. If P is 2: and hence open, then by the Baire category Theorem 2H.2 P, is not meager e P, # 8 w (3i)P(x,ri) where {r,,, rl, ...} is the recursive presentation of y, so Q is Z :. Now if R is E - 1 ,

{y: ( 3 m ) R ( x ,y, m)}is not meager M

(3m)[{y : R(x, y, m)}is not meager]

w

(3m)(3s)[Ns-{y: R(x, y, m)}is meager]

and by induction hypothesis the relation in the brackets is are done. Suppose now P(x, Y ) (-Ja)F(x,Y, a)

-

n”,_,, so we

263

ADDITIONAL RESULTS ABOUT l l i

4F.191

with F in Ily and fix x for the discussion. By the von Neumann Selection Theorem 4E.9 we can find a Baire-measurable f :y + X which uniformizes F, and then by 2H.10 we can find a comeager Gs set A G y such that the restriction fEA of f to A is continuous as in Figure 4F.6. Choose E E X so that

(*I

E((s,

t)) = 1w f[Ns ~

A ] G N, & N, n A # 8

and check first that (1)

E ((s,

t ) ) = 1 + F,

n (N, x Nl># 8,

since if y E N, nA, then (y, f(y)) E F, fl (N, x N,). Finally put

B, ={y

(2)

E y: (3s)@t)[E((s,

-

t))= 1 & y

EN,]

& (Vs)(Vt)(Vk)[ [ ~ ( ( s , t ) ) = 1& Y E

Nsl

[

E ( ( s ’ , t’)) = 1& y E Nst (%’)(3’)

& N,,G N, & N , , G Nl

1 & radius(N,,)
& radius(N,,)<-

and check easily that (3)

I1

A nP, cB,.

Jv

I I

f

I

Figure 4F.6.

264

STRUC~URE THEORY

r4F.20

FOR POINTCLASSES

If P, is not meager, then (3) implies that B, is not meager. Thus we have shown that

P, is not meager

-

( ~ E ) { E satisfies

(1) and the set B,

defined by (2) is not meager}.

On the other hand, the definition of B, makes sense for arbitrary E E N, in fact the relation B ( E 9 y ) w Y E B E

is easily II:, hence 2:. Moreover, if for some fixed x and implication (1) holds, then

E

the

y E BE+there are sequences so, s1,..., to, t1,... such that Ns02 N,, 2

em.,

Nb z N,, 2 -..,and

for each n, (N," x N,) r l F, # fl,

radius(NJ < l/(n + l), radius(N,) C l/(n + 1)and y EN,,

+ there is a sequence of points (y,,, a,) in F,

-

such that limir,,-= y,

=y

and limit,,--

a, = a exists

(3a)(y,a)E F, (since F, is closed)

+ y E P,. Thus

P, is not meager

tj

(a&){&satisfies (1) and the

set Be

defined by (2) is not meager} and this relation is immediately 2;. The claim for II: sets follows from the remark which we have already used, that for any set P with the property of Baire,

P is not meager w (3s)[Ns- P is meager].

-I

4F.20. Prove that if P c X is Z:(z) and not meager, then P has a member in A:(z). (Thomason [1967], Hinman [1969]; see Kechris [1973].) Infer that if P c_ X ' x y is in A :(z) and each section P, is not meager in 3, then P can be uniformized by a set in A : ( z ) ;similarly, every Borel set P c _ X X y with non-meager sections can be uniformized by a Borel subset. Hint. Assume P is II:(z) and not meager and suppose cp: + x is a

4F.201

ADDITIONAL RESULTS ABOUT I l i

265

regular n:(z)-norm on P. Now

so if x
(*I

for some A < K , {x E P: cp(x) = A} is not meager.

If K = N ~then , P is not Bore1 by 4C.10. Choose a G,, nonmeager set A E P by 2H.4 (applied to X - P ) and then use the Covering Lemma 4C.11 to infer that for some 5 < x , A E U ~ . = ~ { X E P : ( P ( X )so = ~that } again (*) holds. We have thus shown that for any regular, n:(z)-norm on P, (*) holds. Fix now a very good n:(z)-scale (p ={cp,} on P, with all the norms regular and put for each n = least

A,

A such that {x E P: cp,(x)

P, = {x E P:cp,(x)

= A}

is not meager,

= A,,}.

Putting this another way, x E P, w {y E P: cp,(y) = cp,(x)} is not meager

-

& (vw)[cpn(w)
{y E P:cpn(y)5 cp,(x)} is not meager & {Y E

P:p n ( Y ) < q n ( X ) } is meager,

where in the second equivalence we have used again the countable additivity of the ideal of meager sets and the fact that cp,(x)
cpn+l(Y)scpn+l(x)

(y) < qn

( ~ n

so that immediately

* cpn(y)5pn(x) * pn+l(Y) < ~ n + (XI, l

Po 2 PI 2 P2... .

If xo, xl, ... is any sequence of points with xn E Pn, then by the definition of a good scale again, there exists some x such that

. .

hrnrt,,,

x,

=x

and P ( x ) .

Moreover, if yo, yl, ... is another such sequence converging to some y E P,

266

[4F.20

STRUCrURE THEORY FOR POINTCLASSES

the sequence XO,

Yo, x1, Y I,...

would have to converge to a unique point, so that x = y. Hence: there is a unique point x* which is the limit of some sequence xo, xl, x2,... with x, E P, and this x* E P. Since obviously x =X*

H

(VS){XEN, 4 (Vk)(3n 2 k)(3y)[P(n, y ) & y E Nsl)

it will be enough to prove that the relation

P*h7Y 1

@

y E p,

is in ,Z;(z),since this would show that x * is ,Zi(z)-recursive and hence (easily) in A:(z). Finally this follows easily from the preceding exercise and the equivalence XEP,

-{y:

-

i ( x < z n y ) } i s not meager

& {y : y


{y : i (x

& (Vs)[(N,-{y: y
which is easy to verify. This completes the proof of the first assertion and the rest follows from the A-Uniformization Criterion 4D.4. -1 There is a similar result for measure which we will not prove here-if P c _ ' X X y is Borel and all sections P, have p-measure > O for some a-finite Borel measure p on y, then P can be uniformized by a Borel set. (The basic lemma for this is due to Tanaka [1968] and Sacks [1969].) Kechris [1973] has an excellent discussion of these and related results as well as additional references. We will end this section with a negative uniformization result-an obstruction to improving the von Neumann Selection Theorem. First a computation. Recall from 4F that w ; = suprernurn{lal: a E WO & a S

~ X }

and for any two irrationals a, P let (a,p ) be the irrational coding their pair as in lE, (a,P>((O, 4) =dn), (a,P ) ( K

4)= P b ) ,

(a,P ) ( t ) = 0 if t is not of the form (0, n) or (1, n).

4F.221

ADDITIONAL RESULTS

ABOUT

ni

267

4F.21. Prove that the relation P ( a , p ) H ,:a.@)=

OJy

is S: and that for each a,there is a perfect nonempty set C such that

p E c e -:,@)

= aJy.

Hint. By a direct relativization of 4A.5, for each x, o;= 6:(x) = suprernum{lal: a E WO & A : ( x ) } .

Compute :

, i a * @ ) > ~ y * [ 3 y ~ A : ( a P, ) ] { ~ EWO & (VS)[(S

E LO

& 6 <=a)e ( sycannot

be mapped in an order preserving way onto

s8)l).

This implies immediately that {(a, p ) : O J : ~ ~ @ ' >isOZI: ~ ) and hence P is

2:.

Since p E A :(a) P ( a , p ) , if the converse implication held we would have that A : ( a ) n N is Z i ( a ) for each a, contradicting 4D.16; thus for each a,the Z : ( a ) set { p : P ( a , p ) } has members not in A : ( a ) and hence it contains a perfect set by the Effective Perfect Set Theorem 4F.1. -1 Lush [1930a] claimed that every 2: set can be uniformized by a set which is the difference of two S: sets. This is not true. 4F.22. Prove that there exists a S: set which cannot be uniformized by the difference of two S: sets. (J. Steel, D. A. Martin.) Hint. Take P(a,P ) H O ~ * * ~ ' = O & ~P + A i ( a ) which is 2 ; by the preceding exercise and 4D.14 and suppose that P is uniformized by P*(a, P )

Q(a,0) 8~i R ( a ,

P)

where (equivalently with the hypothesis that P* is the difference of two

2: sets) we assume that Q and R are n:(a*) for some fixed a*.Let p" be the unique p such that

Q(a*,P * ) & i R ( a * ,p")

holds. By 4A.3, there is a recursive function f such that Q(a,P ) - f ( a ,

P ) E WO;

268

STRUCTURE THEORY FOR POINTCLASSES

[4G

since Q(a*,p*) holds, f(a*, P*)E WO so that for some (,

-

If(a*, p * ) l I ~ < o ' 1 " * ' ~ * 'm=y * ,

since P ( a * , p*) holds and hence S(P)

wT*. The relation

fb*, 6) E wo c!?c If(a*,P)ls 6 & -lR(a*, P )

is easily in Z:(a*) and obviously S ( p * ) holds; but p * $ A : ( a * ) , so by the Effective Perfect Set Theorem 4F.1, S contains a perfect set of irrationals. This contradicts the inclusion

s G { p : P*(a*, p)}={p*}.

-1

46. Historical remarks

The results of this chapter are the hardest to credit, partly because we have presented them in a modern form which is the end product of the work of many researchers. In addition to this, there has been considerable duplication, and rediscovery of ideas, as the recursion theorists often did their work in ignorance of the classical theory. Since the writing of a detailed and documented history of the subject would be a formidable (though fascinating) task, I have confined myself below to a few remarks which indicate the origin of the main ideas (when this is clear from the literature) and point to the most significant papers. 'Let us begin with a brief summary (in somewhat modernized terminology) of the results of Lusin-Sierpinski [1923], surely one of the most important early contributions to the theory of analytic sets. Lusin sieves (cribles) were introduced here and they were used to obtain a representation theorem for II: sets quite similar to our 4A.3. A sieve is a map r H F, which assigns to each rational number r a subset F, of a space X.The set sifted by the sieve is defined by x

E Sieve, F,

-

{r: x E F,} is not well ordered,

where we use the customary ordering on the rationals. The basic result of Lusin-Sierpinski [1923] is that the sets of the form Sieve, F, with each F, closed are precisely the analytic (2;) sets; thus every coanalytic (n:)set P satisfies an equivalence P(x) { r : x E F,} is well ordered

-

with a sieve of closed sets-a representation very similar to that of 4A.3. 4Lusin and Sierpinski [1923] used this characterization of II: sets to give a new proof of the Suslin Theorem (A: = Borel) and also to prove

4Gl

HISTORICAL REMARKS

269

that S: sets are both the union and the intersection of K1 Bore1 sets, our 2F.2. (Half of this result was first shown in Lusin-Sierpinski [1918] which anticipated somewhat this later joint paper.) They also established the Boundedness Theorem 4A.4 for n: sets with the natural ordinal assignment that comes from their representation. ’Fix an enumeration ro, rl, r2,... of the rationals and define the set WO * E C3 of binary infinite sequences by a

E

WO*={r,,: a ( n ) = 1)is wellfounded;

this is (essentially) the set of codes for ordinals introduced in LusinSierpinski [19231 and used extensively in the classical development of the theory. Lusin and Sierpinski showed that WO* was ll: but not 2: and Kuratowski [1966] 038 Lemmas 2, 5 gives the essential content of 4A.2. The effective version of the Basic Representation Theorem 4A.3 (for X = o ) was proved in Kleene [1955a], one of the most significant contributions to the effective theory. Kleene’s result asserted that each II:

where f is recursive and 0 is a set of (integer) notations for the so-called constructive ordinals. (Incidentally, these had been introduced by Church and Kleene and their supremum is wFK,sometimes read “Church-Kleene ol.”)Kleene’s main motivation was the study of these ordinals rather than II: and A:; but he used his representation theorem in his [1955b] and [1955c] to study II: and A: and in the second of these papers he established the effective version of the Suslin Theorem (for ‘X = o).We will prove this result in Chapter 7, where we will also cover in some detail the very fundamental method of definition by effective transfinite recursion introduced in Kleene [1955a]. A representation theorem for II: (with I= o)which is much more similar to 4A.3 was established in Spector [1955], another basic source of ideas for the effective theory. Spector used integer codes of recursive wellorderings of o (for ordinals below o f K )but , other than that his basic notions were quite close to ours. He also proved 4A.2 (essentially) and 4A.4 (for ‘X = o)as well as 4A.5. ‘Kleene and Spector worked in almost complete ignorance of the classical theory and there is no apparent lead from the classical work to theirs-except (possibly) for one slender thread. ’The ordering on finite sequences of integers which we introduced in the proof of 4A.3 was first defined in Lusin-Sierpinski [1923], where it

270

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was used in almost exactly the same way in which we used it. Kleene [1955a] used the same ordering (essentially for the same purpose) and credits Brouwer [1924] for the definition and some of its basic properties-this is Brouwer’s famous intuitionistic proof that every (constructive, totally defined) real function must be uniformly continuous on closed intervals. Now, Brouwer has no list of references in his paper, but he might have seen Lusin-Sierpinski [19231; the publication dates make this barely possible. In any case, Brouwer’s background in topology makes it quite likely that he knew the early papers in descriptive set theory (including Lusin-Sierpinski [1918]) and he might have been led to the ordering along the same path followed by Lusin and Sierpinski. 10 Recursion theorists are apt to refer to the Kleene-Brouwer ordering, while someone versed in the classical theory would naturally call this the Lusin-Sierpinski ordering. 11 As we remarked in the introduction, the relationship between classical descriptive set theory and Kleene’s theory of the arithmetical and the analytical pointclasses on w was first perceived as a list of analogies between the two theories, to begin with by Mostowski and later (and more accurately) by Addison in his Thesis [1954] and later in his [1959a]. (Addison and Spector were graduate students of Kleene during the same general period 1951-1954; it is interesting that Addison’s deepening interest in and knowledge of the classical theory at that time was not effectively transmitted to Kleene and Spector.) The general, unified theory which we are studying in this book evolved slowly irr the years since 1955 from these analogies. l2 The prewellordering property was first isolated explicitly by Moschovakis in 1964, in an effort to find common proofs for theorems about II: and 2; (on w); see Rogers [1967]. (The original version was somewhat more complicated and this present definition is due to Kechris.) On the other hand, arguments which involve ordinal assignments to points (like the index in Lusin-Sierpinski [19181) pervade the classical literature both in descriptive set theory and in recursion theory, so many of the results in 4B-4D are best viewed as elegant and strengthened versions of their classical, concrete special cases. The credits given in the text refer to these special cases. l 3 In particular, Novikov [1935] assigned ordinals to the points of a Xi set precisely as we did in 4B.3, starting with an ordinal assignment from a sieve on the given n: matrix. Novikov [1935] used 4B.3 to settle the problems of separation and non-separation for n:, Xi and lli-the

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separation theorem for S: is already in Lush [1927]. Kuratowski [1936] inferred the separation property for from the reduction property for X i which he introduced and established. Finally, Addison [1959a] put down the lightface results in 4B.10, 4B.11 and 4B.12, following both the classical work and Kleene [1950], where the failure of separation for 2: was proved. 14 The further step of using the prewellordering property as the key tool in studying the structure theory of collections of relations was taken in generalized recursion theory, particularly in recursion in higher types and inductive definability; Moschovakis [19671, [1969], [1970], [1974a], [1974b] and the present work are successive stages in the development of what is sometimes called prewellordering theory. 15The present notion of a Spector pointclass is the natural generalization to the context of Polish spaces of the Spector classes of Moschovakis [1974a]. Theorems 6B.3 and 9A.2 in that monograph correspond to the substitution property and 4C.2 here. l6 The study of collections of relations with arguments in several spaces (and specifically 2 -pointclasses and Spector pointclasses) as opposed to studying collections of subsets of a fixed space (often a-fields) is one of the chief methodological differences between our approach to descriptive set theory here and the classical work. We are forced to look at relations since the effective pointclasses are not closed under countable unions but they are closed under projection along w, to take an example. At the same time, the use of relations makes the logical computations of complexity (which were also used in the classical theory) much simpler, so that there is an advantage, even if one is only interested in the projective pointclasses. 17 The parametrization theorem for A, 4D.2 is an abstract version of the various “hierarchies” for the hyperarithmetical sets, for example the sets Hain Kleene [1955b] or the sets W, in Spector [1955]. Similar abstract parametrizations were constructed directly from the prewellordering property in Moschovakis [1967], [1969] and [1974a] whose Theorem 5D.4 is the basic model for 4D.2 here. 18Kleene [1959b] has the basic version of 4D.3, for n:,with a proof based on the hierarchy of the H,-sets. The upper classification of A (4D.14) is a trivial consequence of this. As for the lower classification of A (4D.16), it has been rediscovered by several people at various times, with the proof for A: usually depending on the Uniformization Theorem 4E.4. The simple argument for A: that we gave is due to Kechris.

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l9 Lusin [ 1930al introduced the fundamental problem of uniformization and announced four results. I. Every 2; set can be uniformized by the difference of two Xi sets. (This is actually false, see 4F.22.) 11. There is a 2: set which cannot be uniformized by a II: set. 111. Every Borel set can be uniformized by a ni set (joint result with Sierpinski). IV. If P E X x Y is Borel and every section P, is countable, then P is the union of countably many Borel sets P z , each of which uniformizes P. (There was a similar result for analytic P.) 20 Theorem I1 is equivalent to 4D.11, which we obtained as an immediate corollary of Kleene’s 4D.10. Novikov [1931] also gave a proof of this result, as well as a proof of a weak version of IV,that for Borel P with countable sections the projection qYP is Borel and there exists a Borel uniformization, our 4F.6. The complete IV is 4F.17 here. 21 Sierpinski [1930] established I11 and asked whether every n: set can be uniformized by some projective set. It was a bold question, because Lush had published an example which purported to show that one could not “effectively” (in what he called “realistic mathematics”) uniformize Il: sets. This uniformization problem was soon recognized as the outstanding problem of descriptive set theory, until Kondo [1938] solved it using the basic idea introduced by Novikov and published in LusinNovikov [1935]. (Kondo gives additional credit to another Novikov paper where apparently II: sets with finite sections were uniformized.) The lightface version was worked out by Addison in the late fifties. 22 Kondo’s solution of the uniformization problem was in many ways harder than the problem-his proof appeared to be so complicated that few people ever read it. But the difficulty is only a matter of style, as there is basically only one natural proof of this result. The present treatment in 4E.1-4E.4 via scales was worked out in 1971 by Moschovakis who was attempting to generalize the result using strong axioms. We will look at this generalization in Chapter 6. 23For the applications of descriptive set theory to analysis, the most important uniformization result is von Neumann’s Selection Theorem 4E.9. This was proved before the war, despite the late publication of von Neumann [1949]. Here we obtained it as a direct corollary of the Kleene Basis Theorem for Z;,4E.8. 24 There are many results in the literature on Borel uniformizations of Borel (or even 2: and n:) sets with special properties, many of them set in wider contexts than the category of Polish spaces. We have concentrated on just the basic theorems here which illustrate the applicability of

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the effective theory to this kind of problem-and particularly the usefulness of the A-uniformization criterion 4D.4. These “effective proofs of boldface results” have been part of the folklore of the subject for a long time and there is nothing basically new in our treatment here; the final versions of 4F.9-4F.18 owe much to the seminar notes of some lectures given by Louveau after he had seen a preliminary version of this chapter. *’ Louveau has obtained recently a very beautiful extension of 4F. 18 which in particular implies the following: if P G 3;: is A and 22 for some n, then P is Z z ( a ) for some a E A:.