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. is discrete but not ω-discrete, S 2. has a unique anti-Friedberg ω-f-enumeration, up to strong ∆0 -equivalence. S ω Proof. We construct an anti-Friedberg ω-f-enumeration G of a family .We 0 S proceed by induction on levels t, using oracle ∆f(t) to define the value of G(c, t) for all c. We have the following requirements.
<ω R2e: Γe ω does not witness the ω-discreteness of . ⊆ S 0 R2i+1:IfHi is an anti-Friedberg enumeration of ,thenthereisa∆ permu- S ω tation ki of ω such that for all c, G(c, t)=Hi(ki(c),t).
We will call the requirements of the form R2e even and those of the form R2i+1 odd.
For an even requirement R2e,wesetasideaninfinite set
0 1 Ce = c ,c ,... { e e } of G-indices, an initial value ae for the functions with indices in Ce,andan infinite set Be of possible alternative values. The set Ce will provide indices for
5 at most two functions. One function, g, will be constant, with g(t)=ae for all t. If there is a second function, g0, then there is a single t = t0 such that g0(t0) Be, and for all other t, g0(t)=ae. We start by letting G(c, 0) = ae for ∈ all c Ce. We continue by letting G(c, t)=ae for all c Ce untilwecometo ∈ 0 ∈ t such that, using ∆f(t),weseep Γe,oflength t, with constant value ae. We will satisfy the requirement by∈ making a split,≤ at level t or higher. For an odd requirement R2i+1, we watch the enumeration Hi. At level t, assuming that Hi is an ω-f-enumeration of ,wechooseafinite partial per- t S mutation ki , including all c t in the domain and all d t in the range, such t ≤ ≤ that if ki (c)=d, then for all t0
Tests for Hi at level t+1. We require that there is a finite partial permutation k of ω such that:
1. dom(k) includes all c t and ran(k) includes all d t, ≤ ≤
2. if k(c)=d, then for all t0 t, G(c, t0)=Hi(d, t0),andifc Ce,then ≤ ∈ Hi(d, t +1) = ae, ↓ t 3. if there is no e i such that R2e splits at level t,thenk k . ≤ ⊇ i
If at some stage si,weseethatHi has passed all of the level t +1 tests, t+1 witnessed by the function k,thenweletki = k. If there is no such stage
6 t+1 si,thenki will be undefined. In this case, Hi will not be an anti-Friedberg ω-f-enumeration of ,andR2i+1 will be forbidden at all levels >t+1. S Suppose for some e t, R2e is not forbidden at level t +1, and we have foundanappropriatep of≤ length t. Suppose we come to a stage at which for ≤ all i
1. every anti-Friedberg ω-f-enumeration of is strongly ∆0 -equivalent to G, S ω 2. is discrete, S 3. there is a set 0 such that S ⊆ S 0 (a) the set of G-indices for elements of 0 is Σ ,and S ω 0 <ω (b) for every Σ set Γ 2 ,ifeachg 0 extends some p Γ,then ω ⊆ ∈ S ∈ some p Γ has extensions g1,g2 with g1 0 and g2 0. ∈ ∈ S ∈ S − S
7 Remark 1.Supposethat , G,and 0 are as in Enumeration Theorem II. If H is any anti-Friedberg ωS-f-enumerationS of , then the set of H-indices of 0 S functions in 0 is Σ . S ω 0 Proof. Let k be the ∆ω permutation witnessing that G and H are strongly 0 ∆ω-equivalent. This function maps the G-indices of elements of 0 onto the H-indices. S Remark 2. Enumeration Theorem II implies Enumeration Theorem I.
Proof. Let , G,and 0 be as in Enumeration Theorem II. We must show that is not ω-discrete.S SupposeS it is, and let Γ ω<ω witness this. Each element S ⊆ of must extend some element of Γ. In particular, each element of 0 extends S S some element of Γ.Thenwehavep Γ and g1,g2 p such that g1 0 and ∈ ⊇ ∈ S g2 0. This is a contradiction. ∈ S − S 0 0 1.5 ∆ω-categoricity but not relative ∆ω-categoricity We now present our first main result, the analogue of Theorems 1.2 and 1.3, in thecasewhenα = ω. 0 Theorem 1.10. There is a computable structure that is ∆ω-categorical but 0 A not relatively ∆ω-categorical. Proof. Let f(t)=2t +1.Let be as in Theorem 1.8, and let G be an anti- Friedberg ω-f-enumeration of S.Welet S =(A, I, U, V, W, Q, P, <), A where the universe A is the disjoint union of sets I,U,V,W,andI,U,V,W are predicates for the respective sets. We identify the ath element of I with a ω, and think of a as an index for a function in S.TherelationQ maps U onto∈ I so as to partition U into infinite sets Ua, corresponding to a I,itmapsV ∈ onto U so as to partition V into infinite sets Va,t,anditmapsW onto V so as to partition W into sets Wa,t,x.TherelationP acts as a predecessor function on the sets Ua and Va,t, making each of these sets into a copy of ω with the usual predecessor function p,wherep(0) = 0,andp(n+1)= n.Fortheelement playing the role of x in Va,t,therelation< is an ordering on Wa,t,x of type
ωt 2 if G(a, t)=x, ωt · otherwise. ½ The following is a well-known result. A precise statement may be found in [1]. In [2], there is a similar result, with Z replacing ω. These results are based on an idea due to Watnick [13]. 0 Lemma 1.11. Given a ∆2t+1 index for a linear ordering ,wecaneffectively pass to a computable index for an ordering of type ωt . L ·L Using Lemma 1.11, we show the following.
8 Lemma 1.12. The structure has a computable copy. A 0 Proof. Recall that G(a, t) is computed by a uniform procedure using ∆2t+1. 0 We have a uniform procedure for computing from a, t and x a ∆2t+1-index for an ordering that has two elements if G(a, t)=x, and one element otherwise. Applying Lemma 1.11, we get a uniformly computable family of linear orderings a,t,x such that L ωt 2 if G(a, t)=x, a,t,x = · L ∼ ωt otherwise. ½ Using this family of orderings, we get a computable copy of the structure described above. A Lemma 1.13. The structure is ∆0 -categorical. A ω Proof. By Lemma 1.12, we may suppose that the structure is computable. From any computable copy A
=(B,I0,U0,V0,W0,Q0,P0,<0), B we obtain an anti-Friedberg ω-f-enumeration H of as follows. We identify the bth element of B with b, and think of it as an index.S The fact that H and 0 0 G are strongly ∆ω-equivalent gives us a ∆ω function k mapping 1 1 the G- indices of every function in onto the H-indices for the same function.− Using 0 S ∆2, we can match the elements of Ua with those of Uk0 (a),andwecanmatch the elements of Va,t with those of Vk0(a),t.TomatchtheelementsofWa,t,x 0 with those of Wk0(a),t,x,weusetheoracle∆ω. Wehavec.e.setsofcomputable infinitary formulas ψ (x):β<ωt 2 for all t,whereψ (x) says that the set { β · } β of predecessors of x has order type β. We match each element of Wa,t,x with the corresponding element of Wk0(a),t,x satisfying the same formula. We need a further fact about the structure , stated in terms of the standard A “back-and-forth relations α”. ≤ Definition 7. Let and be structures, with tuples a in and b in . A B A B
1. We have ( , a) 1 ( , b) if the existential formulas true of b in are true of a in .A ≤ B B A 2. For countable α>1, ( , a) α ( , b) if for all 1 β<αand all d in , A ≤ B ≤ B there exists c in such that ( , b, d) β ( , a, c). A B ≤ A
By a well-known result of Karp, for any countable ordinal α, ( , a) α ( , b) A ≤ B iff the Σα formulas (of Lω1ω)trueofb in arealsotrueofa in . The back- and-forth relations for well orderings are knownB (see [3]). A t t Proposition 1.14. ω 2 2t+1 ω · ≤
9 Returningtothestructure ,supposea, a0 I,wherea, a0 are indices for A ∈ functions g, g0 that agree on an initial segment of length T . Using the proposition above, it is easy to see that a and a0 satisfy the same Σ2T formulas in . A Lemma 1.15. The structure has no formally Σ0 Scott family. A ω 0 Proof. Suppose there is a formally Σω Scott family Φ, with parameters c.The orbit of c is defined by a computable Σn formula for some n. Therefore, we may suppose that Φ has no parameters. If a, a0 I are indices for functions ∈ that agree on initial segment of length T ,where2T n,then( ,a) n ( ,a0). Using ∆0 , we can associate with each a I aformula≥ ϕ(x) ofA Φ satis≤ fiAed by ω ∈ a.Infact,wecanfind a disjunct of ϕ(x),sayϕa(x), which is computable Σn for some n.ChooseT such that 2T n,andletpa be the finite sequence of length T that is an initial segment≥ of the function with G-index a.The 0 set of these chosen pa is Σω, and it witnesses that is ω-discrete. This is a contradiction. S The above lemmas complete the proof of Theorem 1.10.
0 0 1.6 Intrinsically Σω but not relatively intrinsically Σω re- lations We now present our second main result, the analogue of Theorems 1.5 and 1.6, inthecasewhenα = ω. Theorem 1.16. There is a computable structure with an additional relation R that is intrinsically Σ0 but not relatively intrinsicallyA Σ0 on . ω ω A Proof. Let f(t)=2t +1.Let , 0,andG be as in Theorem 1.9. Then satisfies the conclusion of TheoremS S 1.8, with anti-Friedberg ω-f-enumerationS G.Let =(A, I, U, V, W, Q, P, <) A be the structure obtained from and G as in Theorem 1.10. We effectively identify the elements of I with naturalS numbers, and we think of them as G- indices for functions in .LetR I be the set of G-indices for functions in 0. 0 S ⊆ 0 S Then R is Σω.If is a computable copy of ,wehavea∆ω isomorphism F B 0 A 0 from onto .ThenF (R) is Σω.WehaveshownthatR is intrinsically Σω. A B 0 We will now show that R is not relatively intrinsically Σω. Suppose it is. Then R is defined in be a computable Σω formula. As in the previous proof, A the orbit of any tuple c is defined by a formula that is computable Σn for some n,sowemaysupposethatR is defined by a computable Σω formula with no parameters. Using a ∆0 oracle, we choose for each a R, a disjunct of ϕ(x), ω ∈ say ψa(x),whichissatisfied by a.Ifψa(x) is computable Σm,wechooseT such that 2T m,andweletpa be the restriction to length T of the function with ≥ 0 G-index a.ThesetΓ of these chosen pa is Σ .Eachfunctionin 0 extends ω S some element of Γ. By the properties of , 0,andG in Theorem 1.9, there S S exist pa Γ,withextensionsg 0 and g0 0.Sayψ (x) is computable ∈ ∈ S ∈ S − S a
10 Σm,andpa has length T ,where2T m.Letb be a G-index for g0.Thenb ≥ satisfies the Σm formulas true of a, including ψa(x),sob must be in R.Thisis a contradiction.
2 Intrinsic complexity at arbitrary computable limit ordinals
In this section, we extend the results of the previous section to arbitrary com- putable limit ordinals. We refer to Kleene’s system of ordinal notation ,which is described in Rogers’ classic text [11]. If α is a successor ordinal, thenO from the notation for α,wecanfind the notation for its predecessor. If α is a limit ordi- nal, then from the notation for α,wecaneffectively find a computable sequence of notations for ordinals αn such that αn <αn+1 and limn αn = α. We identify computable ordinals with their unique notations on a fixed path through . We can do some ordinal arithmetic effectively, without leaving our fixed path.O Given the notation a for α,wecaneffectively find the notation for α +1–it is 2a. We can also effectively find the notation for 2α.Weshowthis by computable transfinite recursion. If α is either 0 or a limit ordinal, then 2α = α, so the notation for α is the same as the notation for 2α.Ifα = β +1, b a successor ordinal, then 2α =2β +2,soifb is the notation for β,then22 is the notation for 2α. 0 Definition 8 (∆α oracles). To every computable ordinal α (or its notation 0 on the fixedpaththrough ), we associate a Turing complete ∆α set. We call 0 O 0 (n 1) this set ∆α.Forfinite n 1,wemaylet∆n = − .Forinfinite α,weuse H(a),wherea is our chosen≥ notation for α. ∅
Definition 9. Let α be a computable limit ordinal, and let (αn)n ω be the increasing sequence with limit α, obtained from our notation for α.Let∈ ωω. S ⊆ 1. The family is α-discrete if there is a Σ0 set Γ ω<ω such that every S α ⊆ g extends some p Γ,andifg, g0 both extend p Γ,theng = g0. ∈ S ∈ ∈ S ∈ 0 0 2. Enumerations G and H are strongly ∆α-equivalent if there is a ∆α per- mutation k of ω such that G(a, t)=H(k(a),t). 3. Let f be a computable sequence of notations (on the fixedpaththrough ) for a strictly increasing sequence of ordinals with limit α.Anα-f- enumerationO of is an enumeration G such that for some fixed e,forall S 0 ∆f(t) a and t, G(a, t)=ϕe (a, t).
We will now state the general case of Enumeration Theorem I (Theorem 1.8). The proof is the same as for the special case (Theorem 1.8), so we omit it. Theorem 2.1 (Enumeration Theorem I). Let f be a computable function giving the notations (on the fixedpaththrough ) for a strictly increasing se- quence of ordinals with limit α. There is a familyO ωω such that: S ⊆
11 1. is discrete but not α-discrete, S 0 2. has a unique anti-Friedberg α-f-enumeration,uptostrong∆α equiva- Slence.
Here is the generalization of Enumeration Theorem II (Theorem 1.9). Again, we omit the proof. Theorem 2.2 (Enumeration Theorem II). Let f be a computable function giving the notations for a strictly increasing sequence of ordinals with limit α. There is a family ωω with an α-f-enumeration G, which has the following properties: S ⊆
1. every anti-Friedberg α-f-enumeration of is strongly ∆0 -equivalent to G, S α 2. is discrete, S 3. there is a set 0 such that S ⊆ S 0 (a) the set of G-indices for elements of 0 is Σ ,and S α 0 <ω (b) for every Σ set Γ 2 ,ifeachg 0 extends some p Γ,then α ⊆ ∈ S ∈ some p Γ has extensions g1,g2 with g1 0 and g2 0. ∈ ∈ S ∈ S − S 0 0 2.1 ∆α-categoricity but not relative ∆α-categoricity We now present the general case of our first main result, extending Theorem 1.10 to arbitrary computable limit ordinals. Theorem 2.3. Let α be a computable limit ordinal. There is a structure that 0 0 A is ∆α-categorical but not relatively ∆α-categorical.
Proof. From the notation for αt on the fixed path through ,wecaneffectively O compute the notation for 2αt,andfor2αt +1.Weletf(t)=2αt +1.Let be as in Theorem 2.1, and let G be an anti-Friedberg α-f-enumeration. We letS
=(A, I, U, V, W, Q, P, <), A where, as in the proof of Theorem 1.10, I,U,V,W partition A into infinite sets, Q maps U onto I, V onto U,andW onto V ,andP acts as a predecessor function on every Ua and every Va,t, making these sets into copies of ω.Forthe element playing the role of x in Va,t,therelation< is an ordering on Wa,t,x of type ωαt 2 if G(a, t)=x, · ωαt otherwise. ½
The following lemma is in [1]. A related result, with Z replacing ω,isin[2]. Lemma 2.4. Given a ∆0 index for a linear ordering ,wecaneffectively 2αt+1 L pass to a computable index for an ordering of type ωαt . ·L
12 Lemma 2.5. The structure has a computable copy. A 0 Proof. Recall that G(a, t) is computed by a uniform procedure using ∆2αt+1. 0 Using ∆2αt+1,wecanfind an index for a 2-element ordering if G(a, t)=x, and a 1-element ordering otherwise. Applying Lemma 2.4, we get a uniformly αt computable family of linear orderings a,t,x,havingtypeω 2 if G(a, t)=x L · and ωαt otherwise. Using this family of orderings, we get a computable copy of the structure described above. A Lemma 2.6. The structure is ∆0 -categorical. A α Proof. For any computable copy
=(B,I0,U0,V0,W0,Q0,P0,<0) B of , we get an anti-Friedberg α-f-enumeration H of . We think of elements A S of I0 as indices. For each a I0,wehaveasetU 0 U 0,whichweidentify ∈ a ⊆ with the natural numbers. For each t U 0 ,wehaveasetV 0 V 0,which ∈ a a,t ⊆ we identify with the natural numbers. For each x Va,t,x0 ,wehaveaset ∈ 0 Wa,t,x0 W 0 ordered by <0.Givena and t,andusing∆2αt+1,wecanfind the ⊆ αt αt unique x Va,t0 such that Wa,t,x0 has type ω 2,asopposedtoω .Welet ∈ · 0 H(a, t) be this x.ThefactthatH and G are strongly ∆α-equivalent gives us a 0 ∆α function matching the G-indices in I with the H-indices in I0. Then, using 0 ∆α, we can match the rest of the structures. Lemma 2.7. The structure has no formally Σ0 Scott family. A α 0 Proof. Suppose there is a formally Σα Scott family Φ. In principle, there may be a finite tuple c of parameters. However, the orbit of c is defined by a formula that is computable Σβ for some β<α, so we may suppose that Φ has no αt αt parameters. It is well known that ω 2 2α +1 ω (see [3]). From this, it · ≤ t follows that if a, a0 I are indices for functions in that agree on all t such that ∈ S 0 2αt <β,then( ,a) β ( ,a0). Using an oracle for ∆α, we can associate with each a I a formulaA ≤ of ΦAsatisfied by a.Infact,wecanfind a disjunct of this ∈ formula, say ϕa(x), which is computable Σβ for some β<α.ChooseT such that 2αt >β,andletpa be the initial segment of length T for the function with 0 G-index a. The set of these pa’s is Σα,anditwitnessesthat is α-discrete, which is a contradiction. S This completes the proof of Theorem 2.3.
0 0 2.2 Intrinsically Σα but not relatively intrinsically Σα re- lations We now present the general case of our second main result, extending Theorem 1.16 to arbitrary computable limit ordinals.
13 Theorem 2.8. Let α be a computable limit ordinal. There is a computable 0 structure with an additional relation R that is intrinsically Σα but not rela- A 0 tively intrinsically Σα.
Proof. Let f be as in the proof of Theorem 2.3; i.e., f(t)=2αt +1.Let , 0, and G be as in Theorem 2.2. Let be the structure obtained from andSGSas A S in Theorem 2.3. Let R I consist of the set of G-indices for functions in 0. 0 ⊆ 0 S Then R is Σα.If is a computable copy of ,wehavea∆α isomorphism F from onto .ThenB F (R) is Σ0 . Therefore,AR is intrinsically Σ0 . A B α α Suppose ϕ(c, x) is a computable Σα formula. As in the proof of Theorem 2.3, we can define the orbit of c by a formula that is computable Σβ for some β<α, so we may assume that there are no parameters in the formula ϕ(x).Usingan oracle for ∆0 ,wechooseforeacha R,adisjunctofϕ(x),sayψ (x),whichis α ∈ a satisfied by a.Ifψa(x) is computable Σβ,wetakeT such that 2αT >β,andwe let pa be the restriction of the function g with G-index a to T .LetΓ be the set 0 of these pa’s. Then Γ is Σ . Since every function in 0 extends some element α S of Γ,thereexistpa Γ and g 0 such that g pa.Sayb is a G-index for ∈ ∈ S − S ⊇ g.Thenb must satisfy ψa(x), but it does not, which is a contradiction.
3Conclusion
In this section, we mention some problems that remain open. We also recall some definitions needed to state these problems . Definition 10. Let be a computable structure, and let R be an additional relation on . A A 1 1. The relation R is intrinsically ∆1 on if in every computable copy of , 1 0 A A the image of R is ∆1 (that is, ∆α for some computable ordinal α). 1 2. The relation R is relatively intrinsically ∆1 if for every isomorphism F from onto a copy , F (R) is ∆1 relative to . A B 1 B The following result is due to Soskov [12], and is re-worked in [9]. Theorem 3.1. For an additional relation R on a computable structure ,the following are equivalent. A
1 1. The relation R is intrinsically ∆1. 1 2. The relation R is relatively intrinsically ∆1; in fact, there is a computable 0 ordinal α such that R is relatively intrinsically ∆α. 3. The relation R is definable by some computable infinitary formula ϕ(c, x) with a finite tuple c of parameters from . A
14 Thus, for relations, we have a fairly complete picture. For every computable 0 ordinal α, being intrinsically Σα does not imply being relatively intrinsically 0 Σα. However, intrinsically hyperarithmetical and relatively intrinsically hyper- arithmetical relations are the same. A similar problem for hyperarithmetical categoricity is open. Problem 1. For a computable structure , are the following equivalent? A 1 1. The structure is ∆1-categorical; i.e., for all computable copies of , 1 A B A there is a ∆1 isomorphism. 1 2. The structure is relatively ∆1-categorical. (This implies that is rel- 0 A A atively ∆α-categorical for some computable ordinal α.)
We now recall the definition of effective stability of structures. Definition 11. Let be a computable structure. A 0 1. The structure is ∆α stable if for every computable copy of ,all isomorphisms fromA onto are ∆0 . B A A B α 0 2. The structure is relatively ∆α stable if for every copy of ,all isomorphisms fromA onto are ∆0 relative to . B A A B α B Goncharov’s example [7] of a computable structure that is computably cate- gorical but not relatively computably categorical is rigid. Therefore, the struc- ture is computably stable but not relatively computably stable. In [8], it was asked whether this result lifts to computable successor ordinals. We may also ask the question for computable limit ordinals. Problem 2. For computable ordinals α>1, is there a computable structure 0 0 A that is ∆α stable but not relatively ∆α stable?
References
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1 [9] Goncharov, S. S., V. S. Harizanov, J. F. Knight, and R. Shore, “Π1 relations and paths through ,” Journal of Symbolic Logic, vol. 69(2004), pp. 585— 611. O [10] Manasse, M. S., Techniques and Counterexamples in Almost Categori- cal Recursive Model Theory, Ph.D. Dissertation, University of Wisconsin, Madison, 1982. [11] Rogers, H., Jr., Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York, 1967. [12] Soskov, I. N., “Intrinsically hyperarithmetical sets,” Mathematical Logic Quarterly, vol. 42(1996), pp. 469—480. [13] Watnick, R., “A generalization of Tennenbaum’s theorem on effectively finite recursive linear orderings”, Journal of Symbolic Logic, vol. 49(1984), pp. 563—569.
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