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Math 117C Notes: Effective Math 117c notes: Effective DST Forte Shinko May 27, 2019 1 Overview of effective DST Effective descriptive set theory is a field at the intersection of computability theory and descriptive set theory. It can often help us prove results in descriptive set theory which are not easily handled with the classical machinery. First of all, recall the arithmetical hierarchy on N: 0 0 0 Σ1 Σ2 Σ3 0 0 0 ∆1 ∆2 ∆3 ··· 0 0 0 Π1 Π2 Π3 The insight of effective DST is relate this hierarchy to the Borel hierarchy: 0 0 0 Σ1 Σ2 Σ3 0 0 0 ∆1 ∆2 ∆3 ··· 0 0 0 Π1 Π2 Π3 These pointclasses are defined as follows (on any topological space): 0 1. A set is Σ1 if it is open. 0 0 2. A set is Πn if its complement is Σn. 0 0 3. A set is Σn+1 if it is the union of countably many Πn sets. 0 0 0 4. A set is ∆n if it is both Σn and Πn. Here is an outline of the corresponding notions between the fields: Computability theory Effective DST Classical DST N Baire space N := N! arithmetical hierarchy first ! levels of Borel hierarchy 0 0 0 0 0 0 Σα; Πα; ∆α Σα; Πα; ∆α computably enumerable set effectively open set open set computable function continuous function hyperarithmetical set Borel set computable ordinal countable ordinal CK !1 !1 analytical hierarchy projective hierarchy 1 1 1 1 1 1 Σn; Πn; ∆n Σn; Πn; ∆n 1 2 Effective open sets ! <! Recall that the Baire space N := N has a basic open set Ns for every s 2 N , defined by Ns := fx 2 N : x sg The basic open sets in N are of the form fng for n 2 N. We will work on spaces of the form X = Nk × N l. One reason is that we'd like to recover as a special case the usual computability theory on N. A better reason is that our subsets of X should be thought of as sets definable in second order arithmetic. Definition 1. Let X = Nk × N l. 0 k <! l 1. A subset A ⊂ X is Σ1 (or effectively open) if there is a c.e. subset B ⊂ N × (N ) such that [ A = fn1g × · · · × fnkg × Ns1 × · · · Nsl (n1;:::;nk;s1;:::;sl)2T (ie. A is an “effective union" of basic open sets). 0 0 2. A subset A ⊂ X is Π1 (or effectively closed) if its complement is Σ1. 0 Recall that a Σ1 subset of a Polish space is defined to be an open subset, so the notation reflects the fact that this is its effective analogue. Remark 1. 1. There are only countably many effectively open sets, since there are only countably many c.e. sets. So there are many open sets which are not effectively open. 2. A subset of N is effectively open iff it is c.e. 0 S <! 3. If X = N , then A ⊂ N is Σ1 iff A = s2B Ns for some c.e. B ⊂ N . We can choose B to be closed under extensions (ie. if s 2 B and s ≺ t then t 2 B), and it is an exercise to show that we can choose 0 B to be computable. So the complement of B is computable tree, and thus a subset of N is Π1 iff A = [T ] for some computable tree on N (where [T ] is the set of infinite branches of T ). k l 0 k+l Proposition 1. Let X = N × N and let A ⊂ X. Then A is Σ1 iff there is a computable relation R ⊂ N such that A(x1; : : : ; xk; n1; : : : ; nk) () 9mR(x1 m; : : : ; xk m; n1; : : : ; nk) S <! Proof. We'll do the case X = N . Let A = s2B Ns for some c.e. B ⊂ N . Let B = im f for some computable f : N ! N<! (this doesn't cover when B is empty, but this case is trivial). Then we have A(x) () 9s 2 B[s ≺ x] () 9n[f(n) ≺ x] () 9m9n[f(n) = x m] The converse is immediate. The analog of a continuous function is a computable function. Definition 2. Let X = Nk × N l. 1. A function f : X ! N is computable if the relation f(x) = n is effectively open, ie. the set f(x; n) 2 X × N : f(x) = ng is effectively open. 2. A function f : X !N is computable if the relation f(x) 2 Ns is effectively open, ie. the set <! f(x; s) 2 X × N : f(x) 2 Nsg is effectively open. 3. A function X ! Nm × N n is computable if each of its projections is computable. Remark 2. In the case of a function f : N ! N, this agrees with the usual notion of a computable function. 2 Proposition 2. 0 1. Σ1 is closed under the following: finite union, finite intersection, bounded quantification, projection along N (ie. 9N), projection along N (ie. 9N ), computable preimage (ie. preimage under a computable function). 0 2. Π1 is closed under the following: finite union, finite intersection, bounded quantification, coprojection along N (ie. 8N), coprojection along N (ie. 8N ), computable preimage. One should think of projection as “effective union". 0 A useful tool is that of a universal Σ1 set. k l 0 0 Proposition 3. Let X = N × N . Then there is a universal Σ1 set for X, that is, a Σ1 subset U ⊂ N × X 0 such that for every Σ1 subset A ⊂ X, there is some n 2 N such that Un = A. Proof. We will do the case X = N . Fix a computable enumeration (We)e of the c.e. sets, then let U(e; x) := 9nWe(x n). We can relativize all of the above notions to an oracle x 2 N . k l 0 0 Definition 3. Let X = N × N and let x 2 N . A subset A ⊂ X is Σ1(x) if there is a Σ1(x) subset B ⊂ Nk × (N<!)l such that [ A = fn1g × · · · × fnkg × Ns1 × · · · Nsl (n1;:::;nk;s1;:::;sl)2B The importance of relativization is that it relates the effective notion of an open set to the classical one. Proposition 4. Let A ⊂ Nk × N l. Then TFAE: 0 1. A is Σ1 (ie. open). 0 2. A is Σ1(x) for some x 2 N . 0 S <! 0 Proof. We will do the case X = N . If A is Σ1, then A = s2B Ns for some B ⊂ N . Then A is Σ1(B). This allows us to prove theorems in descriptive set theory (boldface pointclasses) by using the effective theory (lightface pointclasses). 3 The arithmetical hierarchy We now define the rest of the arithmetical hierarchy. 0 0 0 k l Definition 4. The pointclasses Σn; Πn; ∆n for n ≥ 1 for spaces of the form N × N are defined as follows: 0 1. A set is Σ1 if it is effectively open. 0 0 2. A set is Πn if its complement is Σn. 0 0 3. A set is Σn+1 if it is the projection along N of a Πn set. 0 0 0 4. A set is ∆n if it is both Σn and Πn. A set in one of these pointclasses is called an arithmetical set. The arithmetical hierarchy looks like the following: 0 0 0 Σ1 Σ2 Σn 0 0 0 ∆1 ∆2 ··· ∆n ··· 0 0 0 Π1 Π2 Πn 3 Proposition 5. Let X = Nk × N l. 0 N 1. Σn is closed under bounded quantification, 9 -quantification, and computable preimage. 0 N 2. Πn is closed under bounded quantification, 8 -quantification and computable preimage. 0 3. ∆n is closed under bounded quantification and computable preimage. 0 CK Remark 3. We can define Σα sets for α < !1 by taking “effective unions" at limit stages, giving what is known as the hyperarithmetical hierarchy. Again, we can relativize to any oracle, and this lets us relate hyperarithmetical sets to Borel sets: 0 0 0 Proposition 6. Let Γ be one of Σα; Πα; ∆α, and let Γ be the corresponding boldface pointclass. Then TFAE for a subset A of Nk × N l: 1. A is Γ. 2. A is Γ(x) for some x 2 N . 4 The analytical hierarchy 1 1 1 k l Definition 5. The pointclasses Σn; Πn; ∆n for n ≥ 1 for spaces of the form N × N are defined as follows: 1 0 1. A set is Σ1 if it is the projection along N of a Π1 set. 1 1 2. A set is Πn if its complement is Σn. 1 1 3. A set is Σn+1 if it is the projection along N of a Πn set. 1 1 1 4. A set is ∆n if it is both Σn and Πn. A set in one these pointclasses is called an analytical set. 1 Remark 4. \Analytical set" should not be confused with \analytic set", which means Σ1. The analytical hierarchy extends the hyperarithmetical hierarchy: 0 0 0 1 1 1 Σ1 Σ2 Σα Σ1 Σ2 Σn 0 0 0 1 1 1 ∆1 ∆2 ··· ∆α ··· ∆1 ∆2 ··· ∆n ··· 0 0 0 1 1 1 Π1 Π2 Πα Π1 Π2 Πn Proposition 7. Let X = Nk × N l. 1 N N N 1. Σn is closed under 9 ; 8 ; 9 , computable preimage. 1 N N N 2. Πn is closed under 9 ; 8 ; 8 , computable preimage. 1 N N 3. ∆n is closed under 9 ; 8 , computable preimage. 1 1 We saw above that every arithmetical set is ∆1.
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