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Effective Descriptive Set Theory E®ective descriptive set theory Yiannis N. Moschovakis UCLA and University of Athens Mostowski100, October 13, 2013 Outline (I) A bit of history (3 slides) (II) The basic notions (7 slides) (III) Some characteristic e®ective results (6 slides) (IV) HYP isomorphism and reducibility (Gregoriades) (3 slides) I Descriptive set theory, ynm, 1980, Second Edition 2009 I Classical descriptive set theory as a re¯nement of e®ective descriptive set theory, ynm, 2010 I Kleene's amazing second recursion theorem, ynm, 2010 I Notes on e®ective descriptive set theory, ynm and Vassilios Gregoriades, in preparation (The ¯rst three are posted on www.math.ucla.edu/»ynm) Yiannis N. Moschovakis: E®ective descriptive set theory 1/20 The arithmetical hierarchy ² Kleene [1943]: The arithmetical hierarchy on subsets of N 0 0 recursive ( §1 (rec. enumerable) ( §2 ( ¢ ¢ ¢ (:§ = ¦; §\¦ = ¢) I Tool for giving easy (semantic) proofs of GÄodel'sFirst Incompleteness Theorem, Tarski's Theorem on the arithmetical unde¯nability of arithmetical truth, etc. ² Mostowski [1947]: Reinvents the arithmetical hierarchy, using as a model the classical projective hierarchy on sets of real numbers 1 1 Borel ( §1 (analytic) ( §2 ( ¢ ¢ ¢ (:§ = ¦; § \ ¦ = ¢) I He grounds the analogy on the two basic results 0 1 Kleene: ¢1 = recursive; Suslin: ¢1 = Borel I Mostowski is unaware of Kleene [1943]: the only post 1939 paper he cites is Post [1944] (He refers to Kleene [1943] in a Postscript added \in press" saying that it \just became available in Poland") Yiannis N. Moschovakis: E®ective descriptive set theory I. A bit of history 1/3 2/20 Mostowski's de¯nition of HYP on N ² Mostowski [1951] introduces the hyperarithmetical hierarchy I In modern notation, roughly, he de¯nes for each constructive CK ordinal » < !1 a universal set for a class P» of subsets of N I The analogy now is between HYP and the Borel sets of reals. 0 M. mimics closely Lebesgue's classical de¯nition of §» sets, replacing countable unions by projection along N and using e®ective diagonalization at limit ordinals I There are technical di±culties with the e®ective version. M. does not give detailed proofs and refers to the need for \a rather developed technique which we do not wish to presuppose here" To make the de¯nition precise, one needs e®ective trans¯nite recursion on ordinal notations. This depends on the 2nd Recursion Theorem and was introduced in the literature by Kleene [1938], not cited in this paper. Most likely, Mostowski is referring to a version of this method Yiannis N. Moschovakis: E®ective descriptive set theory I. A bit of history 2/3 3/20 The de¯nition of HYP on N by Kleene and Davis ² Kleene 1955a: HYP = recursive in some Ha, a 2 O Crucial case: 0 jaj = jbj + 1 =) Ha = Hb (the jump of Hb) I Kleene outlines the details needed to make Mostowski's de¯nition rigorous and establishes the relation between the CK two de¯nitions Roughly, for ! · » < !1 , P» = the class of sets recursively reducible to Ha (jaj = »+1) I Kleene credits Martin Davis who introduced essentially the same de¯nition and at the same time in his Thesis I He alludes to Kleene [1955b] in which he proves the basic result 1 1 ¢1 = HYP : compare to Suslin's Theorem ¢1 = Borel 0 1 ² Kleene 1950: The \analogy" §1 » §1 fails, because there exist 0 1 recursively inseparable r.e., sets Does not mention §1 » ¦1 0 1 0 1 r.e. = §1 » ¦1 open = §1 » ¦1 HYP » Borel ² John Addison established these analogies ¯rmly Yiannis N. Moschovakis: E®ective descriptive set theory I. A bit of history 3/3 4/20 Recursive Polish metric spaces I A metric space (X ; d) is Polish if it is separable and complete I A presentation of X is any pair (d; r) where r : N !X and r[N] = fr0; r1;:::g is dense in X I A presentation (d; r) is recursive if the relations d;r d;r P (i; j; k) () d(ri ; rj ) · qk ; Q (i; j; k) () d(ri ; rj ) < qk ; are recursive, where q = (k)0 k (k)1+1 I Recursive Polish metric space: (X ; d; r) with recursive (d; r) I N = f0; 1;:::g as a discrete space; the reals R and Baire space N = NN with their standard metrics; products of recursive Polish metric spaces, etc. I Every Polish metric space is recursive in some " 2 N I Computable space: assume only that Pd;r and Qd;r are r.e. There is a computable metric space which does not admit a recursive presentation Yiannis N. Moschovakis: E®ective descriptive set theory II. The basic e®ective notions 1/7 5/20 0 Open and e®ectively open (§1) pointsets Fix a recursive Polish metric space (X ; d; fr0; r1;:::g) I Codes of nbhds: For each s 2 N, let n o Ns = N(X ; s) = x 2 X : d(x; r(s)0 ) < q(s)1 S S ¡ ¢ I A pointset G ⊆ X is open if G = n N"(n) = n N X ;"(n) with some " 2 N Any such " is a code of G I 0 G is semirecursive or §1 if it has a recursive code 0 Lemma (Normal Form for §1) 0 P ⊆ X ,Q ⊆ X £ Y are §1 if and only if P(x) () (9s)[x 2 N(X ; s)& P¤(s)] Q(x; y) () (9s)(9t)[x 2 N(X ; s)& y 2 N(Y; t)& Q¤(s; t)] with semirecursive P¤; Q¤ relations on N Yiannis N. Moschovakis: E®ective descriptive set theory II. The basic e®ective notions 2/7 6/20 Recursive Polish spaces Def A topological space (X ; T ) is Polish if there is a d such that (X ; d) is a Polish metric space which induces T Def A recursive Polish space is a set X together with a family R = R(X ) of subsets of N £ X such that for some d : X £ X ! R and r : N !X the following conditions hold: (RP1) (X ; d; r) is a recursive Polish metric space, and (RP2) the frame R of X is the family of semirecursive subsets of N £ X I Every recursive Polish metric space (X ; d; r) determines a recursive Polish space (X ; R(X )) by setting R(X ) = the family of semirecursive subsets of N £ X I If (RP1), (RP2) hold: (d; r) is a compatible pair for X I Every Polish space is recursive in some " 2 N I Every (naturally de¯ned) computable Polish space is recursive Yiannis N. Moschovakis: E®ective descriptive set theory II. The basic e®ective notions 3/7 7/20 The analogies between the classical and the e®ective theory I A pointset is any subset P ⊆ X of a recursive Polish space, formally a pair (P; X ) I A pointclass is any collection ¡ of pointsets, e.g., 0 0 0 §1 = the semirecursive pointsets; ¦1 = fX n P : P 2 §1g; 0 0 0 0 §1 = the open pointsets ¶ §1; ¦1 = all closed pointsets ¶ ¦1 I The arithmetical and analytical pointclasses 0 0 0 N 0 0 0 0 ¦k = :§k ; §k+1 = 9 ¦k ; ¢k = §k \ ¦k 1 N 0 1 0 1 N 1 1 1 1 §1 = 9 ¦1; ¦k = :§k ; §k+1 = 9 ¦k ; ¢k = §k \ ¦k I The ¯nite Borel and projective pointclasses 0 0 0 N 0 0 0 0 ¦k = :§k ; §k+1 = 9 ¦k ; ¢k = §k \ ¦k 1 N 0 1 1 1 N 1 1 1 1 §1 = 9 ¦1; ¦k = :§k ; §k+1 = 9 ¦k ; ¢k = §k \ ¦k I Missing analogy: Hyperarithmetical » Borel Yiannis N. Moschovakis: E®ective descriptive set theory II. The basic e®ective notions 4/7 8/20 From the classical to the e®ective { coding I A coding (in N ) of a set A is any surjection ¼ : C !!A, C ⊆ N I i i i The pointclasses §k ; ¦k ; ¢k are all (naturally) coded, and i i G 2 §k () G 2 §k with a recursive code I 0 The Borel pointclasses: starting with §1 = the open sets, set 0 S A 2 §» () A = i2!(X n Ai ) S 0 with each Ai in ´<» §´ (A ⊆ X ; » > 1) S B = §0 = the Borel sets »<@1 » I ¤ ® (i) = ®(i + 1); (¯)i (j) = ¯(hi; ji) I Borel codes: starting with K1 = f® 2 N : ®(0) = 0g, set for » > 1 S S K = K [f® : ®(0) = 1 & (8i)[(®¤) 2 K ]g; K = K » 1 i ´<» ´ 1·»<@1 » X S ¤ X S X ®(0) = 0 : B = N(X ; ® (i)), ®(0) > 0 : B = (X n B ¤ ) ® i ® i (® )i Yiannis N. Moschovakis: E®ective descriptive set theory II. The basic e®ective notions 5/7 9/20 Hyperarithmetical as e®ective Borel For A ⊆ X : I X Def ® is a K»-code of A : ® 2 K» & A = B® 0 ² A 2 §» () A has a K»-code I X Def ® is a Borel code of A : ® 2 K& A = B® ² A is Borel () A has a Borel code I Def A is HYP () A has a recursive Borel code I Def f : X!Y is HYP if f(x; s): f (x) 2 N(Y; s)g 2 HYP I 0 Def (Louveau 1980) A 2 §» () A has a recursive K»-code ² For X = Y = N, these de¯nitions of HYP agree with the classical ones 0 CK CK ² The pointclasses §» stabilize for » ¸ !1 , and for » < !1 on N they are (essentially) those de¯ned by Mostowski and Kleene ² A ⊆ X is Borel exactly when it is HYP(®) for some ® 2 N ² f : X!Y is Borel (measurable) exactly when it is HYP(®) for some ® 2 N Yiannis N.
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