Intrinsic bounds on complexity and definability at limit levels∗ John Chisholm Ekaterina Fokina Department of Mathematics Department of Mathematics Western Illinois University Novosibirsk State University Macomb, IL 61455 630090 Novosibirsk, Russia [email protected] [email protected] Sergey S. Goncharov Academy of Sciences, Siberian Branch Mathematical Institute 630090 Novosibirsk, Russia [email protected] Valentina S. Harizanov Julia F. Knight Department of Mathematics Department of Mathematics George Washington University University of Notre Dame [email protected] [email protected] Sara Miller Department of Mathematics University of Notre Dame [email protected] May 31, 2007 Abstract We show that for every computable limit ordinal α,thereisacom- 0 0 putable structure that is ∆α categorical, but not relatively ∆α categor- A 0 ical, i.e., does not have a formally Σα Scott family. We also show that for every computable limit ordinal α, there is a computable structure with 0 A an additional relation R that is intrinsically Σα on , but not relatively 0 A intrinsically Σ on , i.e., not definable by a computable Σα formula with α A ∗The authors gratefully acknowledge support of the Charles H. Husking endowment at the University of Notre Dame. The last five authors were also partially supported by the NSF binational grant DMS-0554841, and the fourth author by the NSF grant DMS-0704256. In addition, the second author was supported by a grant of the Russian Federation as a 2006-07 research visitor to the University of Notre Dame. 1 finitely many parameters. Earlier results in [7], [10], [8] establish the same facts for computable successor ordinals α. 1 Introduction and Preliminaries The languages we consider are all computable; that is, the set of symbols is computable, and we can effectively determine the type (relation or function) and the arity of a symbol. The structures we consider all have universes that are subsets of ω. When we measure complexity of a structure ,weidentify with its atomic diagram D( ).Thus, is computable if D( A) is computable,A and computable relative to A means computableA in D( ). A A A 0 0 1.1 ∆α-categoricity and relative ∆α-categoricity Definition 1. Let be a computable structure, and let α be a computable ordinal. A 0 1. The structure is ∆α-categorical if for all computable copies of , there is a ∆0 isomorphismA from onto . B A α A B 0 2. The structure is relatively ∆α-categorical if for all copies of (not just the computableA ones), there is an isomorphism from ontoB Athat is ∆0 relative to . A B α B There are syntactical conditions that completely account for relative cate- goricity. 0 0 Definition 2. A formally Σα Scott family for is a Σα set Φ of computable Σ0 formulas ϕ(c, x),withafixed finite tuple c ofA parameters from ,suchthat: α A 1. Every finite tuple a in satisfies some formula ϕ(c, x) in Φ; A 2. For any tuples a and b in and any formula ϕ(c, x) in Φ,ifa and b both satisfy ϕ(c, x), then thereA is an automorphism of mapping a to b. A The following result is in [4] and [6]. Theorem 1.1. For a computable structure and a computable ordinal α,the following are equivalent: A 1. is relatively ∆0 -categorical; A α 2. has a formally Σ0 Scott family. A α 0 For ∆α-categoricity, the syntactical conditions in 2 of Theorem 1.1 are suf- ficient, but not necessary. Goncharov [7] showed the following. Theorem 1.2. There is a structure that is computably categorical but not relatively computably categorical. A 2 Theorem 1.2 has been lifted to successor levels in [8]. Theorem 1.3. For every computable successor ordinal α, there is a structure that is ∆0 -categorical but not relatively ∆0 -categorical. A α α In the present paper, we extend Theorem 1.3 to computable limit ordinals. 0 1.2 Intrinsically Σα relations Definition 3. Let be a computable structure, and let R be an additional relation on . A A 0 1. The relation R is intrinsically Σα on if for every isomorphism F from onto a computable structure, F (R) Ais Σ0 . A α 0 2. The relation R is relatively intrinsically Σα on if for every isomorphism F from onto a copy , F (R) is Σ0 relative toA . A B α B There are syntactical conditions that completely account for relative intrin- 0 sically Σα relations. The following result is in [4] and [6]. Theorem 1.4. The relation R is relatively intrinsically Σ0 on iff R is de- α A finable by a computable Σα formula ϕ(c, x) with a finite tuple c of parameters from . A 0 For a relation to be intrinsically Σα, the syntactical condition in Theorem 1.4 is sufficient but not necessary. Manasse [10] showed the following. Theorem 1.5. There is a computable structure with an additional relation R that is intrinsically c.e. but not relatively intrinsicallyA c.e. In [8], Manasse’s result is lifted to successor levels. Theorem 1.6. For every computable successor ordinal α, there is a computable 0 structure with an additional relation R that is intrinsically Σα but not rela- tively intrinsicallyA Σ0 on . α A In the present paper, we extend Theorem 1.6 to computable limit ordinals. 1.3 Enumerations The results of Goncharov (Theorem 1.2) and Manasse (Theorem 1.5) were both based on an enumeration theorem, which was proved independently by Badaev and Selivanov. To state this theorem, we need some definitions. Definition 4. Let ωω. S ⊆ 1. The family is discrete if for every g ,thereexistsp ω<ω such that g istheuniqueextensionofS p in S. ∈ S ∈ 3 2. The family is effectively discrete if there is a c.e. set Γ ω<ω such that: S ⊆ (a) every g extends some p Γ,and ∈ S ∈ (b) for any g, g0 and p Γ,ifg and g0 both extend p,theng = g0. ∈ S ∈ Definition 5. Let ωω. S ⊆ 1. A function G : ω2 ω is an enumeration of if is the family of → S S functions of the form ga(t)=G(a, t) for a ω–we refer to a as a G- ∈ index for ga. 2. An enumeration G of is Friedberg if every g has a unique G-index. S ∈ S 3. Two Friedberg enumerations G and H are computably equivalent if there is a computable function k such that G(a, t)=H(k(a),t); i.e., k(a) is the H-index for the function with G-index a. Here is the result of Badaev [5] and Selivanov [12]. Theorem 1.7 (Badaev, Selivanov). There is a family ωω such that: S ⊆ 1. is discrete but not effectively discrete, and S 2. has a unique computable Friedberg enumeration, up to computable equiv- alence.S The results in [8], lifting the results of Goncharov and Manasse to successor levels, involved relativizing and coding. Theorem 1.7, relativized but otherwise unchanged, was the basis of the proof. This method fails for limit ordinals. In the present paper, we vary the enumeration theorem. We consider functions g(t), computed using a sequence of oracles the strength of which increases with t. In this section, we consider the limit ordinal α = ω. 1.4 Enumeration theorems for α = ω Definition 6. Let ωω. S ⊆ 1. The family is ω-discrete if there is a Σ0 set Γ ω<ω such that every S ω ⊆ g extends some p Γ,andifg, g0 both extend the same p Γ, ∈ S ∈ ∈ S ∈ then g = g0. 2. An enumeration G is anti-Friedberg if every g has infinitely many G-indices. ∈ S 0 0 3. Enumerations G and H are strongly ∆ω-equivalent if there is a ∆ω per- mutation k of ω such that G(a, t)=H(k(a),t). 4 4. Let f : ω ω be a strictly increasing computable function with values 1.Anω→-f-enumeration of is an enumeration G such that for some ≥ S 0 ∆f(t) fixed e,foralla and t, G(a, t)=ϕe (a, t). 0 It is helpful to fix some notation. Starting with a standard Σω enumeration 0 0 0 <ω of all Σω sets, we obtain a Σω enumeration of the family of all Σω subsets of ω . <ω We identify the elements of ω with their codes. Let Γe be the intersection <ω 0 of ω with the Σω set having index e in the standard enumeration. We call e an index of Γe.WewriteΓe,t for the finite part of Γe enumerated in at most t 0 steps, where the oracle questions are all answered by ∆ for t0 t. f(t0) Let f ωω be a strictly increasing computable function with≤ values 1. We want an∈ enumeration of all partial functions computed in the same way≥ as an ω-f-enumeration; i.e., using the same sequence of oracles. Let 0 ∆f(t) E(e, a, t)= ϕe (a, t) if this value is defined, ( undefined otherwise. We write He for the partial ω-f-enumeration H such that H(a, t)=E(e, a, t), and we refer to e as an index of He. The result below (Theorem 1.8) is a natural analogue of the result of Badaev and Selivanov. In proving this result, we will actually prove a stronger result. Theorem 1.8 (Enumeration Theorem I, for α = ω). Let f ωω be a strictly increasing computable function with values 1. There is∈ a family ωω such that: ≥ S ⊆ 1.