The Normed Induction Theorem
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The Normed Induction Theorem Yiannis N. Moschovakis UCLA and University of Athens The Soskov meeting, Gjuletchica, 20 September, 2014 and the meeting in honor of Costas Dimitracopoulos, Athens, 3 October, 2014 What it is about ² An operator © : P(X ) !P(X ) on the powerset of a set X is monotone if S ⊆ T =) ©(S) ⊆ ©(T )(S ⊆ T ⊆ X ); and every monotone © has a least ¯xed point © characterized by ©(©) = ©; (8S ⊆ X )[©(S) ⊆ S =) © ⊆ S] ² This set © built up by © is de¯ned explicitly by T © = fS ⊆ X j ©(S) ⊆ Sg; (Exp) and inductively by the ordinal recursion S S © = » ©»; where ©» = ©( ´<» ©´): (Ind) ² The Normed Induction Theorem gives simple|often best possible|classi¯cations of ©, especially in Descriptive Set Theory Yiannis N. Moschovakis: The Normed Induction Theorem 1/16 Outline ² The arithmetical and analytical hierarchies on ! and N (3 slides) ² Borel sets and their codings (2 slides) ² Norms and the prewellordering property (2 slides) ² The main result and some of its consequences (5 slides) ² The story of O (2 slides) ² Results by several people will be discussed ² The basic references for proofs are the Second Edition of my Descriptive Set Theory book and the article Kleene's amazing 2nd Recursion Theorem, both posted on www.math.ucla.edu/»ynm Yiannis N. Moschovakis: The Normed Induction Theorem 2/16 Notation and terminology ² ! = f0; 1;:::; g; s; t 2 !, N = (! ! !), the Baire space, ®; ¯ 2 N ² A space is a product X = X1 £ ¢ ¢ ¢ £ Xk where each Xi is ! or N ² A pointset is any P ⊆ X = X1 £ ¢ ¢ ¢ £ Xk and we write synonymously x 2 P () P(x) () P(x1;:::; xk ) ² A pointclass is any collection ¡ of pointsets ² Restriction: For each X ,¡¹X = fP ⊆ X j P 2 ¡g ² Relativization: ¡(®) = fP® j P 2 ¡g where P®(x) () P(®; x) t0+1 t1+1 tn¡1+1 ht0;:::; tn¡1i = 2 3 ¢ ¢ ¢ pn¡1 ®(t) = h®(0); : : : ®(t ¡ 1)i; n(t) = n if x = (x1;:::; xk ) 2 X ; then x(t) = hx1(t);:::; xk (t)i 0 ® = ¸t®(t + 1) (®)i = ¸s®(hi; si) ² The results we will discuss hold for all recursive Polish spaces |and \boldface versions" of them hold for all Polish spaces Yiannis N. Moschovakis: The Normed Induction Theorem 3/16 The arithmetical and analytical pointclasses 0 0 0 §1 §2 §3 ( ( ( ( ( ( 0 0 0 ¢1 ¢2 ¢3 ¢ ¢ ¢ ( ( ( ( ( ( 0 0 0 ¦1 ¦2 ¦3 1 1 1 §1 §2 §3 ( ( ( ( ( ( 1 1 1 ¢ ¢ ¢ ½ ¢1 ¢2 ¢3 ¢ ¢ ¢ ( ( ( ( ( ( 1 1 1 ¦1 ¦2 ¦3 0 k ² §1, semirecursive: P(x) () (9s)R(x(s)) with recursive R ⊆ ! 1 0 ² ¦1 : P(x) () (8®)Q(x; ®) with Q in §1 () (8®)(9t)R(x(t); ®(t)) with recursive R ⊆ !k+1 1 1 ² §2 : P(x) () (9®)Q(x; ®) with Q in ¦1 () (9®)(8¯)(9)R(x(t); ®1(t); ®2(t)) with recursive R 1 1 1 1 1 ² The relativized pointclasses:§n(®); ¦n(®); ¢n(®) = §n(®) \ ¦n(®) S ² The boldface pointclasses: §1 = §1(®), etc. e n ® n Yiannis N. Moschovakis: The Normed Induction Theorem 4/16 Closure properties and !-parametrization 0 ² A function f : X ! ! is recursive if f(x; w) j f (x) = wg is §1 ² f : X !N is recursive if f (x) = ¸tg(x; t) with a recursive g ² f : X ! Y1 £ ¢ ¢ ¢ £ Yl is recursive if f (x) = (f1(x);:::; fl (x)) with recursive components f1;:::; fl ² The sequence-coding and projection functions are recursive i i i ? All §n; ¦n; ¢n are closed under recursive substitutions &; _ and bounded number quanti¯cation (9s · t); (8s · t) 0 0 §n is closed under (9s 2 !); ¦n is closed under (8s 2 !) 1 1 §n; ¦n are closed under (9s 2 !), (8s 2 !) 1 1 §n is closed under (9y 2 Y ); ¦n is closed under (8y 2 Y ) i i ² The pointclasses §n; ¦n are not closed under : ? ¡ is !-parametrized if for each X , there is a G ⊆ ! £ X in ¡ such that ¡¹X = fGe j G(e; x)g, where Ge (x) () G(e; x) 0 0 1 1 ² Theorem A.§n; ¦n; §n; ¦n are all !-parametrized Yiannis N. Moschovakis: The Normed Induction Theorem 5/16 The Borel pointsets ² B¹X = the smallest σ-algebra of subsets of X which includes §0 ¹X e 1 S ² Codes for Borel sets: B = » B», where by ordinal recursion, 0 ® 2 B» () ®(0) = 0 _ [®(0) 6= 0 & (8i)[(® )i 2 [´<»B´]] i.e., B is the least ¯xed point of the monotone operator 0 ©(S) = f® j ®(0) = 0 _ [®(0) 6= 0 & (8i)[(® )i 2 S]]g (S ⊆ N ) ² For each space X and each ® 2 B we de¯ne B(®) = BX (®) by an (easy) ordinal recursion, so that if ®(0) = 0; B(®) = fx 2 X j (9t)[®0(x; t) = 0]g; S 0 if ®(0) 6= 0; B(®) = i (X n B((® )i )) Basic (easy) fact. For each X , B¹X = fBX (®) j ® 2 Bg Yiannis N. Moschovakis: The Normed Induction Theorem 6/16 How complex are B and the operation ® 7! BX (®)? ² The explicit de¯nition of B = © gives no useful complexity bound 1 ² The inductive de¯nition of B = © gives (easily) that B is §2 1 Theorem 1. B is ¦1 Theorem 2. For each X , the relation MX (®; x; w) () ® 2 B ³ ´ & [w = 1 & x 2 BX (®)] _ [w = 0 & x 2= BX (®)] 1 X 1 is ¦1; and so each B (®) is uniformly ¢1(®) ² Theorem 2 is old (not trivial) but Theorem 1 had not been noticed ² Getting (easy) proofs of these was a motivation for this work Yiannis N. Moschovakis: The Normed Induction Theorem 7/16 Norms and the prewellordering property ² A norm on a pointset P ⊆ X is any function σ : P ! Ordinals and it is a ¡-norm if the following two relations on X £ X are in ¡: ³ ´ ¤ x ·σ y () P(x)& :P(y) _ σ(x) · σ(y) ³ ´ ¤ x <σ y () P(x)& :P(y) _ σ(x) < σ(y) ? A pointclass ¡ is normed (or has the prewellordering property) if every pointset in ¡ admits a ¡-norm 0 ² §1 is normed: if P(x) () (9t)R(x(t)), put σ(x) = ¹tR(x(t)) 1 1 ² ¦1 and §2 are normed |and various versions of this fact were used in the early 20th century to derive much of the structure theory for these pointclasses Yiannis N. Moschovakis: The Normed Induction Theorem 8/16 Normed pointclasses 0 0 ² §n is normed for every n (same proof as for §1) 1 ² If the Axiom of Costructibility holds, then every §n+1 is normed ² If the Axiom of Projective Determinacy holds, then 1 1 ¦2n+1 and §2n are normed for every n Fact: If ¡ is closed under recursive substitutions, !-parametrized and normed, then the dual pointclass :¡ is not normed This is proved by Kleene's construction of recursively inseparable disjoint r.e. sets and implies that the three results above identify all the arithmetical and analytical pointclasses which are normed, under either of the conflicting hypotheses of constructibility or projective determinacy ² There are many interesting pointclasses which are closed under recursive substitutions, !-parametrized and normed Yiannis N. Moschovakis: The Normed Induction Theorem 9/16 ? The main result ² Let ¡ be a pointclass. An operator © : P(X ) !P(X ) is ¡ on ¡, if for every relation P ⊆ Y £ X in ¡, the relation Q(x; y) () x 2 ©(fx0 j P(y; x0)g) is also in ¡. Theorem (The Normed Induction Theorem, ynm 1974) Suppose ¡ is closed under recursive substitutions, !-parametrized and normed: If © : P(X ) !P(X ) is monotone and ¡ on ¡, then © is in ¡ ² In most applications, the hypotheses are either known or trivial ² The theorem is proved by a simple 2nd Recursion Theorem argument ? Once you prove it, you almost never need to use again the 2nd Recursion Theorem in Descriptive Set Theory Yiannis N. Moschovakis: The Normed Induction Theorem 10/16 Borel sets and their codes The set B of Borel codes is the least ¯xed point of 0 ©(S) = f® j ®(0) = 0 _ [®(0) 6= 0 & (8i)[(® )i 2 S]]g (S ⊆ N ) 1 Theorem 1. B is ¦1 1 Proof. Since ¦1 is closed under recursive substitutions, 1 !-parametrized and normed, it is enough to prove that © is ¦1 on 1 1 ¦1, i.e., to verify that if P(y; ®) is ¦1 and 0 Q(®; y) () ®(0) = 0 _ [®(0) 6= 0 & (8i)P(y; (® )i )]; 1 then Q(®; y) is also ¦1; but this is obvious from the closure 1 properties of ¦1 ² The NIT gives an equally trivial proof of Theorem 2, that every 1 B(®) is ¢1(®), uniformly for ® 2 B ² The Suslin-Kleene Theorem. Every ¢1 pointset is uniformly Borel e 1 0 This can also be proved using NIT on §1, but not quite trivially Yiannis N. Moschovakis: The Normed Induction Theorem 11/16 The original motivation for the theorem ² ¡ is a Spector pointclass if it is !-parametrized, normed and closed under recursive substitutions, &; _; (9s 2 !) and (8s 2 !) 1 ² ¦1 is the smallest Spector pointclass 1 ² §2 is the smallest Spector pointclass closed under (9® 2 N ) ² The pointclass IND of all (positive, elementary) inductive pointsets is the smallest Spector pointclass closed under both (8® 2 N ) and (9® 2 N ) ² The pointclass Envelope(3E) of all sets Kleene-semirecursive in the type-3 object which embodies existential quanti¯er 3E over N is the smallest Spector pointclass which is closed under 3E ² A pointset is inductive if it is §1-de¯nable over the smallest admissible set which contains N ² Envelope(3E) is closed under (8® 2 N ), but it is not closed S 1 3 under (9® 2 N ), and so n §n ( Envelope( E) ( IND Yiannis N.