A Certain Reducibility on Admissible Sets

Siberian Mathematical Journal, Vol. 50, No. 2, pp. 330–340, 2009 c 2009 Puzarenko V. G. Original Russian Text Copyright 

A CERTAIN REDUCIBILITY ON ADMISSIBLE SETS V. G. Puzarenko

UDC 510.5:510.225

Abstract: We introduce a certain reducibility on admissible sets which preserves definable predicates. Some lattice-theoretic properties are given of the ordered sets of the classes of admissible sets equivalent under this reducibility. Furthermore, we give a transformation that assigns to each admissible set some hereditarily finite set and preserves the following list of descriptive set-theoretic properties (with account taken of the levels of a definable hierarchy): enumerability, quasiprojectibility, uniformization, existence of a universal function, separation, and extension. We introduce the notion of jump of an admissible set which translates the descriptive set-theoretic properties considered above to the corresponding properties lowering levels by 1. Keywords: computably enumerable set, enumeration reducibility, Σ-reducibility, descriptive set-theoretic properties, admissible set, hereditarily finite set, natural ordinal

In [1] the notion of Σ-reducibility on admissible sets was introduced that preserves the Σ-theory. Its main merit, as well as demerit, is the preservation of the structural properties of an admissible set such as the height of an admissible set and the structure of elements. In this paper we study some reducibility that preserves the Σ-theory rather than the particularities. It is this reducibility that will be called Σ-reducibility here. The notion of Σ-reducibility was introduced in [2]. It turned out that to study the most important properties of this reducibility we can consider only the hereditarily finite sets, i.e., the inclusion-least admissible sets. We show that for each admissible set there is an equivalent hereditarily finite set preserving a series of descriptive set-theoretic properties; in particular, the reduction principle and the existence of a universal Σ-function. As a corollary of this transformation, we give a series of lattice-theoretic properties. The main result and its corollaries were announced in [3]. However, the signature estimation is improved here. A plenary talk on the conference “Mal cev Meeting–2004” was also dedicated to these results. The lattice-theoretic and structural properties of this reducibility were studied earlier [4–7]. It turned out that the reducibility coincides on countable admissible sets with the reducibility which was introduced in [8]. It acts on the classes of arbitrary admissible sets in the same way as the reducibility suggested in [9]. We introduce the notion of jump of an admissible set which translates the descriptive set-theoretic properties in the corresponding properties lowering levels by 1. 1. Preliminaries 1.1. Computability and e-reducibility. We use some standard terminology that can be found, e.g., in [10, 11]. We just give the specific notions to be used in this paper. By  we denote equality by definition. f : A → B and f : A  B mean that a map f is injective and surjective respectively. By ω we denote the set of naturals. Let  ·, · be a computable function yielding a one-to-one correspondence between ω × ω and ω. The author was supported by the Russian Foundation for Basic Research (Grants 06–01–04002–NNIOa and 05–01–00481), the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh–4787.2006.1), and the Russian Science Support Foundation. Novosibirsk. Translated from Sibirski˘ı Matematicheski˘ı Zhurnal, Vol. 50, No. 2, pp. 415–429, March–April, 2009. Original article submitted September 1, 2007. 330

c 2009 Springer Science+Business Media, Inc. 0037-4466/09/5002–0330 

As usual, by the join A ⊕ B we mean {2x | x ∈ A} ∪ {2x + 1 | x ∈ B}. By P(X) we denote the set of all subsets of X. We often identify functions with their graphs. Given a partial function ϕ, by δϕ, ρϕ, and Γϕ we will denote the domain, range, and graph of ϕ respectively. As usual, by enumeration reducibility (or, in short, e-reducibility) we mean the reducibility on sets of naturals which is denoted by ≤e and defined as A ≤e B ⇔ ∀t (t ∈ A ⇔ ∃D ( t, D ∈ W & D ⊆ B)) for some computably enumerable set W (here D is a finite subset of naturals which can be identified with a number in the strong table). The relation ≤e is a preorder on P(ω) which naturally induces ordering on the set of e-degrees P(ω) / , where A ≡ B ⇔ A ≤ B & B ≤ A. Given A ⊆ ω, by d (A) we denote an e-degree that ≡e e e e e contains A. The collection of e-degrees forms the upper semilattice Le with bottom under the order induced by ≤e . Notice that de (A)  de (B) = de (A ⊕ B) where a  b is the supremum of e-degrees a and b, the bottom 0 contains exactly all computably enumerable sets. Let L = L, ≤, , 0 be an upper semilattice with bottom. A nonempty collection I ⊆ L is called an ideal of L provided that (1) a ≤ b & b ∈ I ⇒ a ∈ I; (2) a, b ∈ I ⇒ a  b ∈ I. An ideal I is called principal if I is generated by some element b ∈ I; i.e., I = {c ∈ L | c ≤ b} (we will  Otherwise, I is called nonprincipal. denote this ideal by b). Let L be an upper semilattice. Notice that the collection J (L) of all ideals of L forms a lattice under ⊆ and I1 ∗ I2  {x ∈ L | ∃i1 ∈ I1 , ∃i2 ∈ I2 [x ≤ i1  i2 ]}, I1 ∗ I2  I1 ∩ I2 with bottom {0} and top L. An upper semilattice L = L, ≤, , 0 with bottom is called distributive if a, b0 , b1 ∈ L have the following property: if a ≤ b0  b1 then there are c0 , c1 ∈ L such that a = c0  c1 and c0 ≤ b0 , c1 ≤ b1 . Given A ⊆ ω, put K(A)  {x ∈ ω | x ∈ Φx (A)},

J(A)  K(A) ⊕ (ω \ K(A)).

a

Given A ∈ a, we call  de (J(A)) the e-jump of a. Given I ∈ J (Le ), put I∗  {S ⊆ ω | de (S) ∈ I}. 1.2. Elements of the theory of admissible sets. We use the standard terminology that can be found in [12, 13]. We give here just some notation and propositions from [5]. By A, B, C, . . . (possibly, with indices) we denote admissible sets with domains A, B, C, . . . (with the same indices). By an admissible set A we mean a structure of KPU whose domain is well founded under ∈. Therefore, the set of all ordinals Ord A of it is well ordered. By computably enumerable (computable) sets we mean sets that are definable by some Σ-formula (a Σ- and Π-formula simultaneously). Computably enumerable (computable) subsets are called Σ- (Δ-)subsets. For an admissible set A, the collections of all n-ary Σ- and Δ-predicates of A are denoted by Σ(An ) and Δ(An ) respectively. We omit n for n = 1. Hereditarily finite sets form an important class of admissible sets. We can define HF (M ) inductively:  HF0 (M ) = M ; HFn+1 (M ) = HFn (M ) ∪ Pω (HFn (M )); HF (M ) = HFn (M ), n ω then m ΣA m = Sn . Proof. 1. Suppose to contrary that there is a Σ1 -listed admissible set but not hereditarily finite set. Then there are an infinite element a ∈ A and Σ-functions f0 : ω  A, f1 : a  ω. Note that ω ∈ A by Σ-replacement to f1 [13, Theorem 4.6]. Again apply Σ-replacement to f0 . We have A ∈ A, which is a contradiction. 333

4. This follows from the fact that each formula ∀xϕ is equivalent on A to ∀k ∈ ω∃x((f (k) = x) ∧ ϕ), where f is a ΣA n -function enumerating A via ω and k is not free in ϕ.  It follows from Proposition 5.2 and [6, Theorem 3.1] that generally n-P does not imply (n + 1)-P where P is separation or extension. Proposition 2.3. If n ≥ 1 and A is an n-quasiprojectible admissible set then A does not satisfy n-extension. Proof. Let A be n-quasiprojectible. If A is Σn -listed then there is a universal ΣA n -function f (x, y), -functions where and so sg(f (x, x)) has no extension in the class of ΣA n  ∅, if x = ∅, sg(x) = {∅}, if x = ∅. Now, let A be not Σn -listed. There is a ΣA n -function f : R  A for some R ⊂ ω. It has no extension A in the class of Σn -functions; otherwise, h  ω would be the ΣA n -function enumerating A where h is an extension of f .  In [13] a series A of admissible sets projectible into ω ∈ A is given. HYP(N) is such an admissible set where N is the standard model of arithmetic. These admissible sets satisfy (4) of Proposition 2.2. 3. Σ-Reducibility: Definition and Basic Properties Let A and B be admissible sets. Definition 3.1. We say that A is Σ-reducible to B (A Σ B) if ν −1 (Σ(A2 )) ⊆ Σ(B2 ) for some ν : B  A. In this event we say also that A is Σ-reducible to B via ν (ν : A Σ B). Corollary 3.1. Σ is reflexive and transitive. We say that A and B are Σ-equivalent (A ≡Σ B) if A Σ B and B Σ A. [1, Lemma 1] implies Corollary 3.2. If A HYP B then A Σ B. The converse of Corollary 3.2 does not hold, which follows from [1, Proposition 1] and Theorem 3.1. Corollary 3.3. If A Σ B then A ≤Σ B. The converse of Corollary 3.3 does not hold either; e.g., HYP(N) ≤Σ HF(HYP(N)) whereas HYP(N)  Σ HF(HYP(N)) [7].  The classical computability is the least element under the above-introduced level of complexity. Proposition 3.1. HF(∅) Σ A for every admissible set A. Hence, the class of Σ-degrees has bottom under Σ-reducibility. Proposition 3.2. ν : A Σ B iff ν −1 (Σ(An )) ⊆ Σ(Bn ) for all n < ω. Proof. (⇒) is clear for n = 2; for n = 1 it follows from the fact that Σ-predicates are closed under projections and cartesian products. Let n > 2 and C ∈ Σ(An ). Denote by C  (∈ Σ(A2 )) the predicate (x), 1 ≤ i < n, is a Σ{x1 , . . . , xn−1 , xn  | C(x1 , . . . , xn−1 , xn )}. Since the ith projection prn−1 i  n−1  n−1  n−1 , it follows that ν −1 (C) = y  , . . . , y  , y  | ∃y (ν(y1 )) = function on A defined on A 1 1 n−1 2 i=1 pri

  n−1  ν(yi ) ∧ (ν(y1 ) ∈ A ) ∧ C (ν(y1 ), ν(y2 )) is a Σ-predicate on B.  By induction on the quantifier complexity, we infer the following 



B −1 ΠA ⊆ ΠB , and ν −1 (ΔA ) ⊆ ΔB with arity Proposition 3.3. ν : A Σ B iff ν −1 ΣA m ⊆ Σm , ν m m m m preserving for all m ≥ 1. 334

Lemma 3.1. If ν : A Σ B then R0 = {x, n | n ∈ ω ⊆ Ord(B), ν(x) = n} is Δ on B. Proof. This predicate can be defined by Σ-recursion: x, 0 ∈ R0 ⇔ ν(x) = ∅ ⇔ ¬ (ν(x) = ∅); x, n + 1 ∈ R0 ⇔ ∃x (x , n ∈ R0 ∧ ν(x ) + 1 = ν(x)); x, n + 1 ∈ R0 ⇔ ¬ Nat(ν(x)) ∨ (ν(x) = ∅) ∨ ∃x (x , n ∈ R0 ∧ (ν(x ) + 1 = ν(x))); where Nat(a) means “a is a natural ordinal.”  Corollary 3.4. If ν : A Σ B then R1 = {x, n | n ∈ ω ⊆ Ord(B), ν(x) ∈ An } is Δ on B. Proposition 3.4. If A Σ B then Sω (A) ⊆ Sω (B). In particular, Ie (A) ≤ Ie (B). Proof. Let ∅ = S ⊆ P(ω) be computable in A and let Q ∈ Σ(A2 ) be such that S = {{n | Q(a, n)} | a ∈ A}. Then ν −1 (Q) ∈ Σ(B2 ) where ν : A Σ B. It is easy to verify that S = {{n | ∃y(y, n ∈ R0 ∧ ν(b), ν(y) ∈ Q)} | b ∈ B}, where R0 is Δ on B by Lemma 3.1.



The following assumption is immediate from flattening [12]. Furthermore, it ensues from the proof of Proposition 1.2 in [4]. Proposition 3.5. Let M be a structure in some finite signature and let A be an admissible set. M ≤Σ A iff HF(M) Σ A. Theorem 3.1. Let A be an admissible set. Then there is an oriented graph MA without loops, dom(MA ) = A, such that the following hold (n ≥ 1): (1) A ≡Σ HF(MA ); (2) A satisfies n-P iff HF(MA ) also satisfies n-P where P is a basic property. Proof. Let A be an admissible set and let U be a Σ-predicate on A universal for the class of all unary Σ-predicates. By Σ-reflection, there is a ternary Δ0 -predicate U  such that A |= U (x, y) ≡ ∃uU  (x, u, y). By “Pair” and “Triple” we denote the collections of “finite” functions f on A with δf = 2 and = 3 respectively. We define V as follows: ⎧ Pair(b), if a = 0, ⎪ ⎪ ⎪ ⎪ Triple(b) ∧ (b(0) = 0), if a = 1, ⎪ ⎪ ⎪ ⎪ ⎪ Triple(b) ∧ (b(0) = 1), if a = 2, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Triple(b) ∧ (b(1) = a(0)) ∧(b(2) = a(1)), if Pair(a), V (a, b)  ⎪ ⎪ ⎪ a(1) = b, if Triple(a) ∧ (a(0) = 0), ⎪ ⎪ ⎪ ⎪ ⎪ a(2) = b, if Triple(a) ∧ (a(0) = 1), ⎪ ⎪ ⎪ ⎪  ⎪ U (a(1), a(2), b), if Triple(a) ∧ Pair(a(0)) ⎪ ⎪ ⎩ ∧(a(0)(0) = b). Now, let MA be A, V . It is evident that MA is an oriented graph without loops. We prove that the claims are true for this structure. 1. It is easy that V is Δ on A, and so MA ≤Σ A. By Proposition 3.5, HF(MA ) Σ A. To verify A Σ HF(MA ), it suffices to show that U will be Σ on HF(MA ) and Σ-functions a(0), a(1) on A, determined on Pair, are also Σ-functions on HF(MA ): a(0) = b ⇔ (V (0, a) ∧ ∃x(V (a, x) ∧ (V (1, x) ∧ V (x, b)))), a(1) = b ⇔ (V (0, a) ∧ ∃x(V (a, x) ∧ (V (2, x) ∧ V (x, b)))), U (x, y) ⇔ ∃u∃z(V (0, z) ∧ ((z(0) = x) ∧ ((z(1) = u) ∧∃a(V (z, a) ∧ (¬ V (1, a) ∧ (¬ V (2, a) ∧ V (a, y))))))). 335

2. Let P be reduction, uniformization, separation, or extension. If HF(MA ) satisfies n-P , n ≥ 1, then so does A. Conversely, we only consider separation. We first give an auxiliary construction. Lemma 3.2. There is an embedding ı : HF (A) → A that is Σ on HF(MA ) and ı(HF (A)) ∈ Δ(HF(MA )). Proof of Lemma 3.2. It follows from Theorem 1 in [14] that there is a partial Σ-function Term : Ord(HF(MA ))×A