Enumerations in computable structure theory
Sergey Goncharov Academy of Sciences, Siberian Branch Mathematical Institute 630090 Novosibirsk, Russia [email protected] Valentina Harizanov Julia Knight Department of Mathematics Department of Mathematics The George Washington University University of Notre Dame Washington, D.C. 20052, U.S.A. Notre Dame, IN 46556, U.S.A. [email protected] [email protected] Charles McCoy University of Notre Dame Notre Dame, IN 46556, U.S.A. [email protected] Russell Miller Department of Mathematics Queens College—City University of New York Flushing, NY 11367, U.S.A. [email protected] Reed Solomon Department of Mathematics University of Connecticut Storrs, CT 06269, U.S.A. [email protected]∗ February 16, 2005
∗Goncharov, Harizanov, Knight, Miller, and Solomon gratefully acknowledge NSF support under binational grant DMS-0075899. Goncharov was partially supported by the Russian grant NSh-2112.2003.1. Harizanov
1 0 Introduction
Families of sets with special enumeration properties have been used to produce a number of interesting examples in computable structure theory. Selivanov [22] constructed a family of sets that Goncharov [13] used to produce a structure that is computably categorical but not relatively computably categorical. Manasse [18] used Selivanov’s family of sets to produce a computable structure with a relation that is intrinsically computably enumerable (c.e.) but not relatively intrinsically c.e. Goncharov [12] constructed families of sets that he then used to produce computable structures with computable dimension n,forallfinite n [13]. Wehner [25] constructed a family of sets that yields a structure with isomorphic copies in exactly the non-computable Turing degrees. Slaman [23] produced another such structure. Here, we lift the results of Goncharov, Manasse, and Slaman and Wehner to higher lev- els. Using Selivanov’s construction, in relativized form, we show that for each computable 0 successor ordinal α, there is a computable structure that is ∆α categorical, but not relatively 0 ∆α categorical. From this structure, we obtain another computable structure, with a relation 0 0 that is intrinsically ∆α, but not relatively intrinsically ∆α. Using the enumeration results of Goncharov, relativized, we show that for each computable successor ordinal α,andeachfinite 0 n, there is a computable structure with exactly n computable copies, up to ∆α isomorphism. Using the enumeration result of Wehner, also relativized, we show that for each computable successor ordinal α, there is a structure with copies in just the degrees of sets X such that 0 0 ∆α(X) is not ∆α. In particular, for each finite n, there is a structure with copies in just the non-lown degrees. Section 1 has some basic definitions. In Section 2, we state the enumeration results of Selivanov, Goncharov, and Wehner. In Section 3, we say how enumeration properties of a family of sets translate into properties of certain graph structures derived from the family. In Section 4, we prove the basic results of Goncharov, Manasse, and Slaman and Wehner, using the results from Sections 1, 2, and 3. In Section 5, we describe a construction that for a computable successor ordinal α, transforms a graph into a structure ∗ such that 0 G G G has a ∆ copy iff ∗ has a computable copy. We indicate how various special features of α G G translate into features of ∗. This construction requires the existence of a pair of structures G 0 0, 1, which are uniformly relatively ∆ categorical and have nice properties with respect to B B α the standard back-and-forth relations γ for γ<α. We describe the structures in Section 5, but we delay proving that they have the≤ required properties until Section 7. In Section 6, we use the construction taking to ∗ to lift the results from Section 4. In Section 8, we state some open problems. G G
1 Background
We consider only computable languages, and structures with universe contained in a com- putable set of constants. We identify sentences with their Gödel numbers. In measuring complexity of a structure , we identify the structure with its atomic diagram D( ).Thus, A A was partially supported by the UFF grant of the George Washington University. Miller was partially supported by a VIGRE postdoc under NSF grant #9983660 to Cornell University.
2 is a subset of ω, and it makes sense to say that is computable,ortospeakoftheTuring Adegree of . Our main goal in this section is to giveA definitions and state some known results. All of theA material may be found in [4], with examples and proofs. Other relevant references include [11], [14], and [17].
1.1 Notions related to computable categoricity Let be a computable structure. We say that is computably categorical if for all computable A A 0 = , there is a computable isomorphism from onto . Similarly, is ∆α categorical if B ∼ A 0 A B A for all computable ∼= ,thereisa∆α isomorphism. We say that is relatively computably categorical if for allB =A , there is an isomorphism that is computableA relative to ,and 0 B ∼ A 0 B A is relatively ∆α categorical if for all = ,thereisa∆α( ) isomorphism. B ∼ A 0 B There are syntactical conditions that imply ∆α categoricity, and are equivalent to relative 0 ∆α categoricity. The conditions involve the existence of nice “Scott families”. The notion comes from the proof of Scott’s Isomorphism Theorem ([21], [16]), which says that for a countable structure ,thereisanLω ω sentence whose countable models are just the copies A 1 of . Scott derived the “Scott sentence” for from a family of Lω1ω formulas defining the orbitsA of tuples in . A A Scott family Afor is a set Φ of formulas, with a fixed finite tuple of parameters c in , such that A A 1. each tuple in satisfies some ϕ Φ,and A ∈ 2. if a, b are tuples in satisfying the same formula ϕ Φ, then there is an automorphism of taking a to b.A ∈ A According to this definition, a Scott family for may contain formulas that are not satisfied by any tuple in .IfΦ has parameters c,andA = ϕ(a),whereϕ Φ,thenϕ defines the orbit of a in theA expanded structure ( , c). IfA| there are nice isomorphisms∈ from onto its copies, then we expect a nice Scott family.A A formally c.e. Scott family is a c.e. A 0 0 Scott family consisting of finitary existential formulas. A formally Σα Scott family is a Σα Scott family consisting of “computable Σα”formulas. A detailed discussion of computable infinitary formulas is given in [4]. For our purposes here, an intuitive definition, together with one characteristic property, will suffice. Roughly speaking, computable infinitary formulas are Lω1ω formulas in which the infinite disjunctions and conjunctions are over c.e. sets. There is a useful hierarchy of computable infinitary formulas. A computable Σ0 or Π0 formula is a finitary quantifier-free formula. For α>0,a computable Σα formula is a c.e. disjunction of formulas of the form uψ,whereψ is computable ∃ Πβ for some β<α,andacomputable Πα formula is a c.e. conjunction of formulas of the form uψ,whereψ is computable Σβ for some β<α. (To make this precise, we would assign ∀indices to the formulas, based on Kleene’s system of notations for computable ordinals.) The important property of these formulas is given in the following theorem. Theorem 1.1. For a structure ,theset A a : = ϕ(a) { A| } 3 0 0 is Σα( ) if ϕ(x) is computable Σα,andΠα( ) if ϕ(x) is computable Πα.Moreover,thisholds with allA imaginable uniformity, over structuresA and formulas.
It is easy to see that if has a formally c.e. Scott family, then it is relatively computably A 0 categorical, so it is computably categorical. More generally, if has a formally Σα Scott A 0 0 family, then we can see, using Theorem 1.1, that it is relatively ∆α categorical, so it is ∆α categorical. Goncharov [13] showed that, under some added effectiveness conditions (on a single copy), if is computably categorical, then it has a formally c.e. Scott family. Ash [1] A 0 showed that, under some effectiveness conditions (on a single copy), if is ∆α categorical, 0 A then it has a formally Σα Scott family. For the relative notions, the effectiveness conditions disappear. The following result is from [5] and [8].
Theorem 1.2 (Ash-Knight-Manasse-Slaman, Chisholm). A computable structure is 0 0 A relatively ∆α categorical iff it has a formally Σα Scott family. In particular, is relatively computably categorical iff it has a formally c.e. Scott family. A
It would be pleasant if computable categoricity and relative computable categoricity were the same–then we could drop the effectiveness conditions from Goncharov’s result. However, Goncharov [13] showed that this is not the case, using an enumeration result of Selivanov [22]. There are examples with further properties. Cholak, Goncharov, Khoussainov, and Shore [9] gave an example of a structure that is computably categorical, but ceases to be after naming a constant. It follows from Theorem 1.2 that such a structure is not relatively computably categorical. 0 A rigid structure is one with no nontrivial automorphisms. If a rigid structure is ∆α 0 categorical, then it is also ∆α stable; i.e., every isomorphism from onto a computable copy 0 A is ∆α. For a rigid structure , we may replace the notion of a Scott family by that of a defining family, where this is aA set Φ of formulas with just x free, and with a fixed finite tuple of parameters, such that
1. each element of satisfies some formula ϕ(x) Φ,and A ∈ 2. no formula of Φ is satisfied by more than one element of . A If is rigid, and the isomorphisms from to its copies are nice, then we expect a nice definingA family. A defining family Φ is said toA be formally c.e. if it is a c.e. set of finitary 0 0 existential formulas, and it is formally Σα if it is a Σα set of computable Σα formulas. For a rigid computable structure , there is a formally c.e. Scott family iff there is a formally A 0 0 c.e. defining family, and there is a formally Σα Scott family iff there is a formally Σα defining family. The parameters in the Scott family and the defining family will be the same.
0 1.2 Intrinsically and relatively intrinsically Σα relations Let be a computable structure, and let R be a relation on .WesaythatR is intrinsically A A 0 c.e. if in all computable ∼= ,theimageofR is c.e., and R is intrinsically Σα if in all computable = ,theimageofB A R is Σ0 .WesaythatR is relatively intrinsically c.e. if in B ∼ A α
4 0 all ∼= ,theimageofR is c.e. relative to ,andR is relatively intrinsically Σα if in all =B , theA image of R is Σ0 ( ). B B ∼ A α B If R is definable in by a computable Σα formula, with a finite tuple of parameters, then A 0 0 R is relatively intrinsically Σα,soitisintrinsicallyΣα. Ash and Nerode [6] showed that under some effectiveness conditions, on a single copy, if R is intrinsically c.e., then it is defined by a computable Σ1 formula, with a finite tuple of parameters. Barker [7] showed that under 0 some effectiveness conditions, on a single copy, if R is intrinsically Σα,thenitisdefined by a computable Σα formula, with a finite tuple of parameters. For the relative notions, the effectiveness conditions are not needed. The following result is in [5] and [8].
Theorem 1.3 (Ash-Knight-Manasse-Slaman, Chisholm). Let be a computable struc- 0 A ture. Then a relation R on is relatively intrinsically Σα iff it is defined by a computable Σα formula, with a finite tupleA of parameters. In particular, R is relatively intrinsically c.e. iff it is defined by a computable Σ1 formula, with parameters. It would be pleasant if the intrinsically c.e. and relatively intrinsically c.e. relations were the same. However, Manasse [18] produced an example showing that this is not so. His construction also used the family of sets constructed by Selivanov [22].
1.3 Notions related to computable dimension The computable dimension of a structure is the number of computable copies, up to com- 0 A putable isomorphism. Similarly, the ∆α dimension is the number of computable copies, up to 0 ∆α isomorphism. Goncharov [13] showed that there are structures of computable dimension n,forallfinite n. McCoy [20] showed that computable dimension does not relativize.
Theorem 1.4 (McCoy). Suppose is a computable structure. If is not relatively com- A A putably categorical, then for all n>1,thereexist 1,..., n isomorphic to such that for i, B B A j with 1 i We begin with some definitions. For P (ω) an enumeration is a binary relation ν such that S ⊆ = ν(i):i ω ,whereν(i)= x :(i, x) ν . S { ∈ } { ∈ } When we say that the enumeration is computable (c.e., respectively) we mean that the binary relation is computable (c.e., respectively). We note that in some of the literature, ν is called computable when the binary relation is merely c.e. It is easy to see that a family has a computable enumeration just in case the family +,where S S + = A A : A , S { ⊕ ∈ S} has a c.e. enumeration. 5 An enumeration is Friedberg if it is 1 1, in the sense that if i = j,thenν(i) = ν(j). Suppose ν, µ are two enumerations of the− same family .Wewrite6 ν µ if there6 is a computable function f such that for all i, S ≤ ν(i)=µ(f(i)); i.e., we can effectively pass from a ν-index to a µ-index for the same set. We say that ν and µ are computably equivalent if µ ν and ν µ. Notethatifµ and ν are Friedberg enumerations of the same family ,then≤ µ ν implies≤ ν µ. S ≤ ≤ Afamily P (ω) is discrete if for each A ,thereexistsσ 2<ω such that for all B , S ⊆ ∈ S ∈ ∈ S σ χ iff B = A. ⊆ B The family is effectively discrete if there is a c.e. set E 2<ω such that ⊆ (a) for each A ,thereisσ E such that σ χA,and (b) for all σ ∈E Sand all A, B ∈ ,ifσ χ ,χ⊆,thenA = B. ∈ ∈ S ⊆ A B In [22], Selivanov proved the following. Theorem 2.1 (Selivanov). There exists a family P(ω), which has a unique computable Friedberg enumeration, up to computable equivalence,S ⊆ and is discrete but not effectively dis- crete. Actually, Selivanov produced a family of functions f ωω such that the family of sets ∈ Af = x, f(x) : x ω , representing the graphs of the functions, has the properties above. For such{h a family,i any∈ } c.e. enumeration is actually computable. Hence, Selivanov’s family also has a unique c.e. Friedberg enumeration, up to computable equivalence. Goncharov established the following result in [12]. Theorem 2.2 (Goncharov). For every finite n 1, there is a family of sets with just n c.e. Friedberg enumerations, up to computable equivalence.≥ In [19], Marchenkov proved that any family of computable unary functions with two computable Friedberg enumerations, which are not computably equivalent, has infinitely many computable Friedberg enumerations, up to computable equivalence. In [25], Wehner obtained the following result. Theorem 2.3 (Wehner). 1. There is a family P (ω) such that for each noncomputable set X, has an enumer- ation computableS in⊆X,but has no computable enumeration. S S 2. There is a family P (ω) such that for each noncomputable set X, has an enumer- ation c.e. relativeS to⊆X,but has no c.e. enumeration. S S The enumeration results of Selivanov, Goncharov, and Wehner all relativize. In the next section, we describe a general method for turning a family of sets with special enumeration properties into a directed graph structure with related properties. 6 3 Turning a family of sets into a graph Let be a family of sets. For each A ,adaisy graph A consists of one index point a at the center,S with a a,andforeachn∈ SA,apetal of theG form → ∈ a a0 an a. → → ···→ → The petals are disjoint except for the index point, which is common to all. Let ( ) be the G S union of a disjoint family of daisy graphs A,oneforeachA . G ∈ S We put the important facts about this construction into the following technical lemmas. Lemma 3.1. For any family P (ω),both ( ) and ( +) are rigid graphs. S ⊆ G S G S This is clear. Lemma 3.2. If has a unique c.e. Friedberg enumeration, up to computable equivalence, then ( ) is computablyS categorical. Similarly, if has a unique computable Friedberg enu- meration,G S up to computable equivalence, then ( +S) is computably categorical. G S Proof. If ν is a c.e. Friedberg enumeration of ,thenwecanuseν to produce a computable copy of ( ), with a computable function takingS i to the index point for the daisy graph G S ν(i).If is a computable copy of ( ),then yields a c.e. Friedberg enumeration µ of , asG follows.H First, there is a computableG S functionH taking the ith index point of to i.WecanS easily recognize index points in –they are the points a such that = a Ha.Ifa is the ith index point, then we let µ(i) beH the set coded in the daisy graph withH|a as→ its center. If has a unique computable Friedberg enumeration, up to computable equivalence, then for anyS two computable copies of ( ),weeffectivelymatchuptheindexpoints,andwecanthen effectively match up the remainingG S points in the daisies to obtain a computable isomorphism. If has a unique computable Friedberg enumeration, up to computable equivalence, then + hasS a unique c.e. Friedberg enumeration, up to computable equivalence, and we have seen thatS ( +) is computably categorical. G S Lemma 3.3. If has just n c.e. Friedberg enumerations, up to computable equivalence, then ( ) has computableS dimension n.Similarly,if has just n computable Friedberg enumerations,G S up to computable equivalence, then ( +) Shas computable dimension n. G S Proof. Suppose that has just n c.e. Friedberg enumerations, up to computable equivalence. S Let ν1,...,νn be computably nonequivalent c.e. enumerations of .Foreachk,let k be a S H computable copy of ( ) with a computable function taking each i to the index point for νk(i) G S in k.Fork = m, the fact that νk and νm are not computably equivalent means that there is H 6 no computable isomorphism from k to m. Therefore, ( ) has computable dimension at least n.Suppose is a computableH copy ofH ( ),andletνGbeS a computable enumeration of H G S S with a computable function taking i to the index point for the daisy graph of type ν(i) in . G H For some k, ν is computably equivalent to νk, and then we have a computable isomorphism from onto k.Therefore, ( ) has computable dimension at most n. H H G S 7 If has just n computable Friedberg enumerations, up to computable equivalence, then + hasS just n c.e. enumerations, up to computable equivalence, and we have seen that ( +) Shas computable dimension n. G S Lemma 3.4. If is discrete, then every element of ( +) has a finitary existential definition with no parameters.S G S Proof. First, let a betheindexpointforthedaisygraphoftype A A,whereA .Since is discrete, we can fixafinite binary string α such that α χ G,andforany⊕ B∈fromS ,if S ⊆ A S B = A,thenα χB.Thestringα corresponds to a particular collection of odd and even length6 cycles in the6⊆ daisy graph with index point a. From this, we get an existential formula defining a in ( +).Ifc is some other element of the daisy graph containing a,thenforsome n and k, c isG theSkth element of a cycle of length n, which starts and ends with a. Using this, we get an existential definition for c. Lemma 3.5. Suppose has a computable Friedberg enumeration, and is discrete but not effectively discrete. ThenS ( +) does not have a formally c.e. defining family. G S Proof. We may apply the proof of Lemma 3.2 and assume that ( +) is computable. Suppose there is a formally c.e. defining family Φ, hoping for a contradiction.G S By Lemma 3.4, each element of ( +) has a finitary existential definition, and we may assume that there are no parametersG inS the formulas of Φ. We consider the c.e. set D consisting of pairs (ϕ, a) such that ϕ Φ,anda is an index point satisfying ϕ(x) in ( +). For every such pair, the formula ϕ(x) describes∈ the way a sits in a finite subgraph of (G +S),wherethefinite subgraph includes part of the daisy with index point a, and possiblyG partsS of some other daisies, with index points b1,...,bn. By enlarging the finite subgraph, we may suppose that any petal represented in it is completely included, and there are enough petals to give information distinguishing among the sets A, Bi that correspond to the indices a, bi. Thatis,foreachdistinctpairofsets X, Y A, B1,...,Bn , there must be some number k such that k X and k Y ,ork Y and k∈ {X. These differences} are recorded in the graph by the existence∈ of an appropriate6∈ even∈ length6∈ cycle in one daisy graph and an odd length cycle in the other daisy graph. Furthermore, we can find such differences effectively by searching. From the lengths of the petals, we see that α χ and β χ ,whereα, β are distinct sequences of the same finite length. Note ⊆ A i ⊆ Bi i that if c is an index point corresponding to some C ,whereα χC ,thenc also satisfies the formula ϕ(x),soC = A. We have a c.e. set E of∈ fiSnite sequences⊆ α obtained effectively in this way from the pairs (ϕ, a) in D.Therefore, is effectively discrete, a contradiction. S For P (ω), we may also form a graph structure ∞( ) made up of infinitely many S ⊆ G S copies of A for each A .Thestructure ∞( ) is not rigid. Copies of ∞( ) correspond to arbitraryG enumerations∈ S of –not just toG FriedbergS enumerations. G S S 8 Lemma 3.6. Let P (ω).ForanysetX, there is an enumeration of c.e. in X iff there S ⊆ S is a copy of ∞( ) computable in X. Similarly, there is an enumeration of computable in G S + S X iff there is a copy of ∞( ) computable in X. G S Proof. Clearly, if there is a copy of ∞( ) computable in X, then there is an enumeration of c.e. in X; in fact, we get an enumerationG S in which each element of has infinitely many indices.S Now, suppose ν is an enumeration of c.e. in X.WecandefineS another enumeration µ by µ( i, j )=ν(i).Thisµ enumerates everyS set in infinitely many times, and it is also h i S c.e. in X.Fromµ,wegetacopyof ∞( ) computable in X. + G S If there is a copy of ∞( ) computable in X, then there is an enumeration of com- putable in X;infact,thereisanenumerationinwhicheachelementofG S has infinitelyS many indices. If there is an enumeration of computable in X, then there is anS enumeration of + + S S c.e. in X.Wegetacopyof ∞( ) computable in X,asabove. G S 4 Results of Goncharov, Manasse, Slaman, and Wehner In this section, we review the basic results that we plan to lift. Here is the result of Goncharov [13]. Theorem 4.1 (Goncharov). There is a rigid graph structure that is computably categorical but not relatively computably categorical. G Proof. We take the family from Selivanov’s Theorem (Theorem 2.1). By Lemma 3.1, the structure ( +) is rigid. ByS Lemma 3.2, it is computably categorical. By Lemma 3.5, it has no formallyG S c.e. defining family. Therefore, by Theorem 1.2, it is not relatively computably categorical. As a corollary of Theorem 4.1, we obtain Manasse’s result on intrinsically c.e. but not relatively intrinsically c.e. relations. Theorem 4.2 (Manasse). There is a computable structure with a relation R that is intrinsically c.e. but not relatively intrinsically c.e. A Proof. The cardinal sum of disjoint structures 0, 1, in the same relational language, is B B formed by taking the disjoint union of the structures and adding predicates P0 and P1,which hold of the elements of 0 and 1, respectively. Let be the cardinal sum of two disjoint computable copies of theB graphB structure from TheoremA 4.1, and let R be the unique isomorphism. The fact that is computablyG categorical implies that R is intrinsically c.e. Suppose R is relatively intrinsicallyG c.e., hoping for a contradiction. For any copy of ,we take the disjoint union of the universes, and form a copy of . There is an isomorphismH G from onto , computable in .Then is relatively computablyA categorical, a contradiction. G H H G Next, we obtain Goncharov’s result on computable dimension. 9 Theorem 4.3 (Goncharov). For each finite n, there is a rigid graph structure with computable dimension n. G Proof. By Goncharov’s Enumeration Theorem (Theorem 2.2), there is a family of sets with just n c.e. Friedberg enumerations, up to computable equivalence. By Lemma 3.1, (S) is a rigid graph. By Lemma 3.3, it has computable dimension n. G S Here is the result of Slaman and Wehner on degrees of structures. Theorem 4.4 (Slaman, Wehner). There is a structure with copies in just the noncom- putable degrees. A Proof. By Theorem 2.3, there is a family of sets with enumerations c.e. in all noncom- S putable sets, but no c.e. enumeration. By Lemma 3.6, ∞( ) hascopiescomputableinX, for all noncomputable sets X, but no computable copy.G (WeS could also take a family with enumerations computable in all noncomputable sets, but no computable enumeration,S and + form ∞( ).) G S 0 5Codinga∆α structure in a computable one To lift the basic results of Goncharov and Manasse, we first relativize them, producing a 0 directed graph that is ∆α. We then pass to a computable structure ∗,usingapair of structures toG code the arrow relation (from the graph). For a graph G,andapairof G structures 0, 1 for the same relational language, let B B ∗ =(G U,G,U,Q,...), G ∪ where 1. G is the universe of , G 2. G and U are disjoint, 3. Q is a ternary relation assigning to each pair a, b G an infinite set U(a,b),where ∈ x U(a,b) iff Qabx, ∈ 4. the sets U(a,b) form a partition of U, 5. each of the other relations of ∗ (in ...) corresponds to some symbol in the language of G 0, 1, and is the union of its restrictions to the sets U(a,b), B B 6. for each pair a, b G,if (a,b) is the structure (U(a,b),...),then ∈ U 10 0,if = a b, (a,b) = B G| → U ∼ 1, otherwise. ½ B 0 We give conditions on the pair of structures i (i =0, 1) under which a ∆ graph structure B α gives rise to a computable structure ∗.Weneedsomedefinitions. The standard back-and- G G forth relations β on the set of pairs (i, b):b i ,aredefined inductively as follows: ≤ { ∈ B } (i) (i, b) 1 (j, c) if the existential formulas true of c in j are true of b in i, ≤ B B (ii) if β>1, (i, b) β (j, c) if for all c0 in j,andallγ such that 1 γ<β, there exists ≤ B ≤ b0 in i such that (j, c, c0) γ (i, b, b0). B ≤ Remark: By a result of Karp [15], (i, b) β (j, c) iff all Πβ formulas of Lω ω true of b in i ≤ 1 B aretrueofc in j (not just the computable Πβ formulas). B A pair of structures 0, 1 is α-friendly if the structures are computable and the stan- {B B } dard back-and-forth relations β for β<αare c.e., uniformly in β. (To make this precise, we ≤ fix a notation a for α in O andidentifyeachordinalβ<αwithitsuniquenotationb Lemma 5.1. Let α be a computable successor ordinal. Let 0, 1 be such that B B 1. the pair 0, 1 is α-friendly, {B B } 2. 0, 1 satisfy the same Πβ sentences (of Lω ω) for β<α, B B 1 3. each i (i =0, 1) satisfies some computable Πα sentence that is not true in the other. B 0 Then for any ∆α set S, there is a uniformly computable sequence ( n)n ω such that C ∈ 0,ifn S, n = B ∈ C ∼ 1,otherwise. ½ B 0 0 Lemma 5.1 is related to results in [3], where a Πα set (as opposed to a ∆α one) is coded in a computable sequence of structures. The proof of Lemma 5.1 uses the same machinery; namely, Ash’s α-systems. The reader who is not familiar with this machinery will find a thorough discussion in [4]. Proof. Suppose α = β +1. Wegiveauniformeffective procedure for constructing n.Let 0 C C be an infinite computable set of constants, for the universe. We have a ∆β function gn : ω 0, 1 that is eventually constant, with limit value 1 if n S,and0 otherwise. We want → { } ∈ 0, if limk gn(k)=1, n = B C ∼ 1, if limk gn(k)=0. ½ B For simplicity, we suppose that 0, 1 are structures for a finite relational language. Also, for convenience, we suppose thatB theyB have disjoint, computable universes. To put ourselves in a position to apply Ash’s metatheorem, we begin by defining a β- ˆ system (L, U, P, , E, ( γ)γ<β).LetL be the set of all finite partial 1 1 functions from C ≤ − 11 ˆ to 0,or 1.LetU = 0, 1 .Let = .For L,letE( ) be the set of atomic sentences B B { } ∅ ∈ and negations of atomic sentences that makestruein 0,or 1.LetP be the set of finite B B alternating sequences 0u1 1u2 2 ... (ending with an element of U or L) such that ˆ 1. 0 = , 2. uk U,and k L, ∈ ∈ 3. dom( k) includes the first k elements of C, 4. if uk =0,thenran( k) 0,ifuk =1,thenran( k) 1, and in either case, ran( k) includes the first k elements⊆ B of the structure, ⊆ B 5. if uk = uk+1,then k k+1. ⊆ For , 0 L,welet 0 0 if E( ) E( 0),andfor0 <γ<β,welet γ 0 if ∈ ≤ ⊆ ≤ dom( ) dom( 0), and for any extension µ0 of 0 and any δ<γ, there is an extension µ ⊆ of , such that µ0 δ µ. These are the standard back-and-forth relations. We have defined≤ the β-system. We can show that Ash’s four conditions are satisfied: 1. 0 0 implies E( ) E( 0), ⊆ ⊆ 2. γ 0 implies δ 0 if γ>δ, ≤ ≤ 3. γ is transitive and reflexive, ≤ 0 0 1 k 4. if σ u P , γ0 γ1 ... γk 1 ,andβ>γ0 >γ1 >...>γk 1 >γk,thenthere ∈ ≤ 0≤ ≤ − i − exists ∗ such that σ u ∗ P and for i =0, 1,...k,wehave γ ∗. ∈ ≤ i The first three conditions are obvious. For Condition 4, the important thing is that for any γ<βand any extension µ,thereisµ0, with range in the opposite structure, such that µ γ µ0. 0 ≤ Next, we define a ∆β instruction function gn∗ (related to the function gn), such that if σ P ,whereσ has length 2k +1,theng∗(σ) = limk gn(k).Arun of (P, g∗) is an infinite ∈ n n path π = 0u1 1 ... through P in which the terms from U are given by gn∗.Fortherunπ, 1 F − = k k is a 1 1 function from C onto the desired structure i, with inverse F.If n is the structure∪ induced− by F on C,then B C E(π)= kE( k)=D( n). ∪ C By Ash’s metatheorem, there is a run π such that E(π) is c.e. Then the resulting n is computable. Moreover, the uniformity in the metatheorem means that given n,wecanC effectively find a computable index for n. C We need pairs of structures i satisfying the hypotheses of Lemma 5.1. In addition, B 0 each i will be uniformly relatively ∆α categorical ; i.e., given an X-computable index for B 0 = i,wecanfind a ∆α(X) index for an isomorphism from i onto . By the comments C ∼ B B C 0 following Theorem 1.1, to show that a structure is uniformly relatively ∆α categorical, it is 0 B enough to show that it has a formally Σα Scott family Φ with no parameters. We introduce 12 some notation to describe certain structures. If 1, 2 are structures for the same relational C C language, we write 1 2 for the cardinal sum, where this includes unary predicates for the two universes. C |C Proposition 5.2. For each computable successor ordinal α 2,thereexist 0, 1 such that ≥ B B 1. the pair 0, 1 is α-friendly, {B B } 2. 0, 1 satisfy the same Πβ sentences (of Lω ω) for β<α, B B 1 3. each i (i =0, 1) satisfies some computable Πα sentence that is not true in the other, B 0 4. each i is uniformly relatively ∆ categorical. B α We note that if α is a limit ordinal, then structures that satisfy the same Πβ formulas for all β<αalso satisfy the same Πα formulas. Therefore, there is no possibility of extending Proposition 5.2 to limit ordinals. Since proving Proposition 5.2 immediately would disrupt the flow of the argument, we present the structures that satisfy the proposition, but we delay proving that these structures have the required properties until Section 7. The structures we use are all either linear orderings or cardinal sums of linear orderings. We use ω to denote the order type of the natural numbers, and Z to denote the order type of the integers. For any ordering ξ,wewriteξ∗ for the reverse ordering. For any ordinal δ>0,wedefine γ ξδ = Z ω. · γ<δ X 0 We treat Z as a single point, so ξδ is 2 γ ξδ = ω + Z ω + Z ω + + Z ω + . · · ··· · ··· Now, we describe the pairs of structures corresponding to successor ordinals α 2.For ≥ α =2,welet 0, 1 be orderings of type ω and ω∗. For a successor ordinal α>2,wecan write α as eitherB γB+2n +1 or γ +2n +2,wheren ω and γ is either a limit ordinal or zero. Notice that if γ =0,thenn 1.Ifweletβ = γ +∈n,then ≥ 2β +1=2γ +2n +1=γ +2n +1. Similarly, 2β +2=γ +2n +2. So, to consider all successor ordinals α>2,itsuffices to look at 2β +1and 2β +2for all β 1. ≥For 2β +1, we use the cardinal sums β β ξβ (ξβ + Z ) and (ξβ + Z ) ξβ. | | For example, for α =3,whenβ =1,weuse ω (ω + Z) and (ω + Z) ω, | | 13 and for α =5,whenβ =2,weuse 2 2 (ω + Z ω) (ω + Z ω + Z ) and (ω + Z ω + Z ) (ω + Z ω). · | · · | · To meet the conditions of Proposition 5.2, it suffices to prove the following lemma. Lemma 5.3. For al l β 1, ≥ β β 1. ξβ (ξβ + Z ) 2β (ξβ + Z ) ξβ, | ≡ | β β 2. each of ξβ (ξβ + Z ) and (ξβ + Z ) ξβ satisfies a computable Π2β+1 sentence not true in the other, | | β β 3. the pair ξβ (ξβ + Z ), (ξβ + Z ) ξβ is (2β +1)-friendly, { | | } β β 0 4. ξβ (ξβ + Z ), (ξβ + Z ) ξβ are uniformly relatively ∆2β+1 categorical. | | For 2β +2, we use the orders β β Z ω and Z ω∗. · · For example, for α =4,whenβ =1,weuseZ ω and Z ω∗,andforα =6,whenβ =2, 2 2 · · we use Z ω and Z ω∗. To meet the conditions of Proposition 5.2, it suffices to prove the following lemma.· · Lemma 5.4. For al l β 1, ≥ β β 1. Z ω 2β+1 Z ω∗, · ≡ · β β 2. each of Z ω and Z ω∗ satisfies a computable Π2β+2 sentence not true in the other, · · β β 3. the pair Z ω, Z ω∗ is (2β +2)-friendly, { · · } β β 0 4. Z ω, Z ω∗ are uniformly relatively ∆2β+2 categorical. · · We now continue with the general argument, delaying the proofs of Lemmas 5.3 and 5.4 until Section 7. Lemma 5.5. Let α be a computable successor ordinal, and let 0, 1 be as in Proposition B B 5.2. Suppose is a graph structure, and ∗ is constructed from , i in the way that was G G 0 G B described at the beginning of this section. Then has a ∆α copy iff ∗ has a computable copy. 0 G G More generally, for any X ω, has a ∆α(X) copy iff ∗ has an X-computable copy. In addition, we have: ⊆ G G 0 0 0 (a) if has just one ∆ copy, up to ∆ isomorphism, then ∗ is ∆ categorical, G α α G α 0 0 0 (b) if has just n ∆ copies, up to ∆ isomorphism, then ∗ has ∆ dimension n, G α α G α 0 (c) if has no Σα Scott family consisting of finitary existential formulas, then ∗ has G 0 G no formally Σα Scott family. 14 Proof. For (a), suppose that ∗ is computable, let ∗ be a computable copy of ∗,andlet G H G H be the image of under the isomorphism. From the computable Πα sentences distinguishing G the structures i, we get computable Πα and Σα definitions of the relation on G.Now, 0 B 0 0 → is ∆α (the universe is computable and is ∆α). Therefore, there is a ∆α isomorphism f fromH onto .Foreachpair(a, b) in ,wecane→ ffectively determine a computable index for G H G 0 the structure (f(a),f(b)) corresponding to (a,b),and,using∆α, we can determine whether it U U 0 0 is a copy of 0 or 1.Since i is uniformly ∆ categorical, we can effectively find a ∆ index B B B α α for an isomorphism f(a,b) from (a,b) onto (f(a),f(b)). Then the union of f and the functions 0 U U f(a,b) is a ∆ isomorphism from ∗ onto ∗. α G H 0 For (b), let i∗, i Claim 1:Leta, b and a0, b0 be two pairs of tuples from ∗ with the following properties: G a = a0 ; b = b0 ; a and a0 are in G;eachd from b is in U(a ,a ) for some pair a1,a2 a,and | | | | | | | | 1 2 ∈ each d0 in b0 is similarly connected to some pair of elements from a0; if two elements of a or b are equal, then so are the corresponding elements in a0 or b0,andvice versa;andifd b ∈ is in U(a ,a ),wherea1,a2 a,thend0 U(a ,a ) for the corresponding d0 b0 and a0 ,a0 a0, 1 2 ∈ ∈ 10 20 ∈ 1 2 ∈ and vice versa.Fixanyβ<α. Suppose that for all a1,a2 a,ifd isthepartofb in U(a ,a ), ∈ 1 2 and d0 is the corresponding part of b0 in U(a ,a ),wehave(U(a ,a ), d) β (U(a ,a ), d0).Then 10 20 1 2 ≤ 10 20 ( ∗, a, b) β ( ∗, a0, b0). G ≤ G Proof of Claim 1: The proof proceeds by induction on β. The base case, where β =1,is easy to check–we use the fact that i 1 j. Assuming that the claim holds for all γ<β, B ≤ B we prove it for β.Letv0 be a tuple in ∗,whichbreaksintov0 G and v0 U.Itsuffices G 1 ∈ 2 ∈ toprovethatforanygivenγ<β, there is a tuple v,whichbreaksintov1 G and v2 U ∈ ∈ such that ( ∗, a0, b0, v0) γ ( ∗, a, b, v). We assume, without loss of generality, that the tuples G ≤ G 0 a0 and v10 are disjoint and the tuples b and v20 are disjoint. Furthermore, we assume that for any y v20 ,therearex1,x2 from a0, v10 such that y U(x1,x2). We can achieve this property by slightly∈ expanding our tuples. ∈ Let v1 be a tuple of elements of G,disjointfroma,andsuchthat v1 = v0 .Consider | | | 1| 0 0 an arbitrary pair of distinct elements r0,s0 from a0, v10 .Letd be the tuple of elements from b that are in U(r0,s0) (this tuple is empty unless r0,s0 are both from a0), and let w0 be the tuple of elements from v20 that are in U(r0,s0).Letr, s be the elements of a, v1, which correspond to r0 and s0,andletd be the tuple of elements from b that are in U(r,s) (this tuple is empty unless r, s are both from a). If r0,s0 are both from a0,then(U(r,s), d) β (U(r ,s ), d0) by the ≤ 0 0 hypothesis of the claim. Otherwise, d and d0 are empty and, by the properties of our coding structures (since β<α), we have U(r,s) β U(r ,s ). In either case, there is a tuple of elements ≤ 0 0 15 w from U(r,s) such that (U(r ,s ), d0, w0) γ (U(r,s), d, w).Wedeclarethetuplew to be the part 0 0 ≤ of v2 that corresponds to w0,asapartofv20 . We repeat this process for each pair r0,s0 from a0, v10 to build v2. Notice that, once v2 is 0 completed, we have satisfied the hypotheses of this claim with the sequences a0, v10 , b , v20 and a, v1, b, v2,andtheordinalγ. Then, by the induction hypothesis, ( ∗, a0, b0, v0) γ ( ∗, a, b, v), as required. This completes the proof of Claim 1. G ≤ G 0 Now, suppose Φ∗ is a formally Σ Scott family for ∗.Letc denote the set of parameters α G in these formulas and assume c is split into c1 G and c2 U. We can assume that for each ∈ ∈ y from c2,therearex1,x2 from c1 such that y U(x1,x2). To arrive at a contradiction, we 0 ∈ 0 produce a Σα Scott family of finitary existential formulas for .Weuse∆α as an oracle and give an effective list of this Scott family of finitary existentialG formulas. For any tuple a G, 0 ∈ using ∆ ,wecanfind a computable Σα formula ϕ(c, x) Φ∗ such that = ϕ(c, a).Wemay α ∈ G| suppose that ϕ(c, x) has the form ( u) ψ(c, x, u),whereψ is computable Πβ for some β<α. 0 ∃ Using the oracle ∆ ,wecanfind b such that = ψ(c, a, b).Sayb = b1, b2,whereb1 G α G| ∈ and b2 U. Expanding the tuples, if necessary, we may assume that for each y from b2, ∈ 0 the “parents” of y are in c1, a, b1;i.e.,y U(x ,x ) for x1,x2 in c1, a, b1.Using∆ ,wecan ∈ 1 2 α determine, for each pair of points (a1,a2) from a, b1, c1, whether there is an arrow from a1 to a2 in .Letδ(c1, x, u1) be a finitary formula (in the language of ) that describes the graph G G structure on c1, a, b1.Noticethata (as a tuple in )satisfies the finitary existential formula G u1 δ(c1, x, u1). ∃ Claim 2:If =(u1) δ(c1, a0, u1), then there is an automorphism of that fixes c1 and G| ∃ G takes a to a0. ProofofClaim2:Assumethat = δ(c1, a0, b0 ). Suppose there is a tuple b0 from U such that G| 1 2 for b0 = b0 , b0 we have ( ∗, c, a, b) β ( ∗, c, a0, b0). Then, ∗ = ψ(c, a0, b0),andso ∗ = ϕ(c, a0). 1 2 G ≤ G G | G | Therefore, there is an automorphism of ∗ that fixes c and takes a to a0. However, any G automorphism of ∗ induces an automorphism of ,soa and a0 are automorphic in ,as required. G G G 0 0 It remains to show that there is an appropriate tuple b2.Wechooseb2 so that we can 0 0 0 apply Claim 1 to the sequences: c1, c2, a, b1, b2 and c1, c2, a0, b1, b2.Sincec1, a, b1 and c1, a0, b1 both satisfy δ, equality and the graph relation are preserved between these two tuples. → Let u, v be a pair of distinct elements in c1, a, b1,andletu0,v0 be the corresponding pair in c1, a0, b0 .Since = u v iff = u0 v0,wehaveU(u,v) = U(u ,v ).Ifd is the part of b2 1 G| → G| → ∼ 0 0 in U(u,v),thenwecanchoosed0 in U(u ,v ) such that (U(u,v), d) β (U(u ,v ), d0).Letb0 be the 0 0 ≤ 0 0 2 result of combining the chosen tuples d0 in the appropriate way. Now, we can apply Claim 1 to get ( ∗, c, a, b) β ( ∗, c, a0, b0), as required. This completes the proof of Claim 2. G ≤ G 0 Now, we let Φ consist of the formulas ( u) δ(c, x, u), obtained as above. This is a Σα Scott family for , consisting of existential formulas.∃ This contradiction completes the proof of (c). G 16 6 Lifting the basic results Here is our lifting of the result of Goncharov on structures that are computably categorical but not relatively computably categorical. 0 Theorem 6.1. For each computable successor ordinal α, there is a structure that is ∆α 0 categorical but not relatively ∆α categorical. 0 0 Proof. We relativize Theorems 2.1 and 4.1 to ∆α, getting a rigid ∆α graph structure such that: G (1) has just one ∆0 copy, up to ∆0 isomorphism, G α α (2) has no Σ0 Scott family consisting of finitary existential formulas. G α 0 Next, we apply Lemma 5.5 to pass from to a computable structure ∗ that is ∆α categorical, 0 G G 0 with no formally Σα Scott family. By Theorem 1.2, it follows that ∗ is not relatively ∆α categorical. G Here is our lifting of the result of Manasse on relations that are intrinsically c.e. but not relatively intrinsically c.e. Theorem 6.2. For each computable successor ordinal α, there is a computable structure with 0 0 a relation that is intrinsically Σα but not relatively intrinsically Σα. Proof. Let and ∗ be as in the proof of Theorem 6.1. Let be the cardinal sum of two G G A copies of ∗,andletR be the unique isomorphism between the associated copies of .Suppose G G is a computable copy of ,say is the cardinal sum of 1∗ and 2∗,andlet i be the B A B 0 H H H copy of associated with i∗. The structures i are ∆α, and the image of R–the unique G H 0 H 0 isomorphism from 1 onto 2–is Σα. Therefore, R is intrinsically Σα. H H 0 We must show that R is not relatively intrinsically Σα on ∗. Supposing that it is, we 0 G arrive at a contradiction by showing that ∗ is relatively ∆ categorical. Let ∗ be a copy of G α H ∗,andlet be the associated copy of . We may suppose that ∗ and ∗ are disjoint, and formG the cardinalH sum = . By our assumption,G the image of GR, the uniqueH isomorphism 0 B ∼ A from onto ,isΣα( ∗).Now,weextendR to an isomorphism f from ∗ onto ∗,still 0 G H H G H ∆ ( ∗). α H For each pair a, b in ,wecanfind the R-images a0,b0 in .Let (a,b) be the structure G H U0 (U(a,b),...),andlet (a ,b ) be the corresponding part of ∗.Using∆ ,wecandetermine V 0 0 H α whether = a b,sowecandeterminewhich i is isomorphic to (a,b).Wecanfind an G| → B U ∗-computable index for the corresponding structure (a0,b0).Since i is uniformly relatively H0 0 V B ∆α categorical, we can find a ∆α( ∗) index for an isomorphism f(a,b) from (a,b) onto (a0,b0). H 0 U V The union of R with these f(a,b) is a ∆α( ∗) isomorphism from ∗ onto ∗. Therefore, ∗ is 0 H G H G relatively ∆α categorical, a contradiction. Here is our lifting of the result of Goncharov on structures with finite computable dimen- sion. 17 Theorem 6.3. For each computable successor ordinal α and each finite n, there is a com- 0 putable structure with ∆α dimension n. 0 Proof. First, we relativize Theorems 2.2 and 4.3 to ∆α, getting a rigid graph structure with 0 0 G just n ∆α copies, up to ∆α isomorphism. Then we apply Lemma 5.5 to pass from to a 0 G computable structure ∗ with ∆ dimension n. G α Here is our lifting of the result of Slaman and Wehner. Theorem 6.4. For each computable successor ordinal α, there is a structure with copies in 0 0 just the Turing degrees of sets X such that ∆α(X) is not ∆α. In particular, for each finite n, there is a structure with copies in just the non-lown degrees. 0 Proof. We can relativize Theorems 2.3 and 4.4 to ∆α, getting a graph structure (not rigid) such that the degrees of copies of are just the degrees of sets that are not ∆G0 .Next,we G α applyLemma5.5topassfrom to a structure ∗, where the degrees of copies of ∗ are G 0 G 0 G just the degrees of sets X such that ∆α(X) is not ∆α.Ifα = n +1,wheren is finite, then 0 0 the degrees of copies of ∗ are the degrees of sets X such that ∆n+1(X) is not ∆n+1; i.e., (n) (n) G X T . 6≤ ∅ 7 Pairs of structures In this section, we prove Proposition 5.2. Recall that we broke the proof into three parts. We need to verify that the orderings ω and ω∗ work for the case α =2, and we need to prove Lemmas 5.3 and 5.4 (which are restated below as Lemmas 7.1 and 7.2). The analysis of various order types draws heavily on the work of Ash [2]. First, we consider the orderings ω and ω∗. The orderings can be distinguished by finitary Π2 sentences saying that there is no first, or last, element. Since both orderings are infinite, we have ω 1 ω∗ and ω∗ 1 ω. ≤ ≤ Each ordering is rigid, with a c.e. defining family consisting of finitary Σ2 formulas ϕn(x) saying that there are exactly n elements to the left, or right, of x. Similarly, any tuple of elements x in ω or ω∗ can be defined by a conjunction of such formulas. These properties 0 imply that ω and ω∗ each have a formally Σ2 Scott family without parameters. To see that ω, ω∗ is 2-friendly, fix computable copies of ω and ω∗ in which we can determine the size{ of the} interval (x, y) for any x L0 = A0 + a1 + A1 + ...+ an + An and L1 = B0 + b1 + B1 + ...+ bn + Bn, 18 we have Ai γ Bi,foralli =0,...,n. From these two facts and the existence of our nice ≤ copies of ω and ω∗, it is clear that we can enumerate the 1 relation between tuples in these models. ≤ In the rest of this section, we will prove Lemmas 5.3 and 5.4. Recall that for any ordinal δ>0, γ 2 γ ξδ = Z ω = ω + Z ω + Z ω + + Z ω + . · · · ··· · ··· γ<δ X We will also use the ordering ηδ = ξδ + ξδ∗. Whenever we mention ξδ or ηδ, we assume that δ>0. We repeat the statements of the lemmas, so the reader can avoid flippingbackandforthbetweenthissectionandSection5. Lemma 7.1. For al l β 1, ≥ β β 1. ξβ (ξβ + Z ) 2β (ξβ + Z ) ξβ, | ≡ | β β 2. each of ξβ (ξβ + Z ) and (ξβ + Z ) ξβ satisfies a computable Π2β+1 sentence not true in the other, | | β β 3. the pair ξβ (ξβ + Z ), (ξβ + Z ) ξβ is (2β +1)-friendly, { | | } β β 0 4. ξβ (ξβ + Z ), (ξβ + Z ) ξβ are uniformly relatively ∆2β+1 categorical. | | Lemma 7.2. For al l β 1, ≥ β β 1. Z ω 2β+1 Z ω∗, · ≡ · β β 2. each of Z ω and Z ω∗ satisfies a computable Π2β+2 sentence not true in the other, · · β β 3. Z ω, Z ω∗ is (2β +2)-friendly, { · · } β β 0 4. Z ω, Z ω∗ are uniformly relatively ∆2β+2 categorical. · · c For the rest of this section, we use Πγ to abbreviate the expression “computable Πγ”. We begin by working toward part 2 of Lemmas 7.1 and 7.2. We need to see how complicated certain statements are in the various structures we have introduced. We assume that all points x, y we discuss come from a structure of the form δ δ δ δ Z , Z ω, Z ω∗, ξδ, ξδ + Z , ηδ, · · or the reverse ordering of one of these structures. α c Lemma 7.3. For points x y, the statement “x, y are in the same copy of Z ”isΣ2α and ≤ α c the statement “there are exactly n copies of Z between x and y”isΣ2α+2. 19 Proof. We proceed by induction on α.Forα =1,“x, y are in the same Z”isgivenby ( [x, y] = n), n>0 | | _ where [x, y] = n is the standard finitary Σ2 statement saying that the closed interval between | | c x and y has size n. The disjunction is clearly Σ2, as required. We abbreviate this formula by x y. ∼Z To say “there are exactly n copies of Z between x and y”, we say that there exist x0 (xi Z xi+1)& ( z) xi z xi+1 (xi Z z z Z xi+1) . i x Zδ y (x Zδ,n y). ∼ ∨ n>0 ∼ _ Here, we are using the obvious generalizations of the abbreviations given above in the α =1 case. This formula has the form Σc Σc , and hence is Σc , as required. We abbreviate 2δ ∨ 2δ+2 2α this formula by x α y. ∼Z To say that “there are exactly n copies of Zα between x and y”wesaythereexist x0 (xi Zα xi+1)& ( z) xi z xi+1 (xi Zα z z Zα xi+1) . i (x δ y). ∼Z δ<α _ c c c Since x δ y is Σ ,and2δ<α,thisformulaisΣ = Σ , as required. Saying “x, y have ∼Z 2δ α 2α exactly n copies of Zα between them” is exactly as in the successor ordinal case. β β β Lemma 7.4. For Z ω and Z ω∗, the statements “there is a least copy of Z ”and“there ·β · c is a greatest copy of Z ”arebothΣ2β+2. Proof. “There is a least copy of Zβ”is ( y)( x 20 Lemma7.4proves2inLemma7.2.Wedelayverifying2forLemma7.1untilwehave analyzed the complexity of some more statements. To check 1 from Lemmas 7.1 and 7.2, we need to understand the back-and-forth relations on our structures, which means that we need to understand how the structures can be partitioned by a finite number of points. We mention two useful facts, both of which were pointed out by Ash [2]. Facts:Foranyγ>0 and n ω,wehave: ∈ γ 1. Z = ξγ∗ +1+ξγ, γ 2. ξγ + Z n + ξγ∗ = ηγ (n +1). · · The first equality follows by a simple examination of the linear orderings involved. The γ second equality is obtained by applying the first equality to rewrite Z n as (ξγ∗ +1+ξγ) n and then rearranging the parentheses. Notice that before rearranging the· parentheses, we can· absorb the 1’s into either ξγ or ξγ∗ . We next examine the proper initial and final segments of the orderings in which we are interested. β For Z , the only proper initial segment is ξβ∗ , and the only proper end segment is ξβ. • For ξβ, the proper initial segments are either finite, or ηγ (n +1)for some γ<βand • n ω. The only proper final segment is ξ . · ∈ β β β For Z ω, the proper initial segments have the form Z n + ξβ∗ .Wecanhaven =0,in • · · β which case the initial segment is just ξβ∗ . The only proper final segment is ξβ + Z ω, · which is, of course, ξβ+1. β β Since Z ω∗ is (Z ω)∗, the only proper initial segment is ξβ∗+1,andtheproperfinal • · · β segments have the form ξβ + Z n.Again,wecanhaven =0, in which case the end · segment is ξβ. We use this information to tell us how various orderings can be partitioned by a finite number of points. For the first example, we provide a detailed explanation, and we leave the similar explanations for the rest to the reader. If Zβ is partitioned by a finite number of points, then we have β Z = ξβ∗ +1+σ1 +1+σ2 +1+ +1+σk 1 +1+ξβ ··· − if and only if each σi is either finite, or ηγ (n +1)for some γ<βand n ω.Toobtain · β ∈ this equivalence, notice that ξβ∗ is the only proper initial segment of Z .Theremainingfinal segment is ξβ,soσ1 must be an initial segment of ξβ. Hence, we have the required form for such an initial segment. After partitioning off σ1, the remaining end segment is again ξβ. Therefore, σ2 must have the prescribed form. Continuing this process, we see that each σi must be an initial segment of ξβ,andhencewegettheequivalence. 21 For ξβ, ξβ = σ0 +1+σ1 +1+ +1+σk 1 +1+ξβ ··· − β if and only if each σi is either finite or ηγ (n +1)for some γ<βand n ω.ForZ ω, · ∈ · β β Z ω = Z m + ξβ∗ +1+σ1 +1+ +1+σk 1 +1+ξβ+1 · · ··· − if and only if each σi is either finite or ηγ (n +1)for some γ β and n ω.Noticethat in this case, in contrast to the earlier cases,· we can have γ = β.≤ Also, as explained∈ above, we β can have m =0in this case. For Z ω∗, · β β Z ω∗ = ξβ∗+1 +1+σ1 +1+ +1+σk 1 +1+ξβ + Z m · ··· − · if and only if each σi is either finite or ηγ (n +1)for some γ β and n ω.Again,wecan have γ = β in this case, and we can also have· m =0, as explained≤ above.∈ For η n, β · η n = σ0 +1+σ1 +1+ +1+σk β · ··· if and only if each σi is either finite, or η (m +1) for some γ<βand m ω,orη (m +1) γ · ∈ β · for some m β β Lemma 7.5. Let x be an element of ξβ + Z . The properties “x is in the Z summand” and c “x is in the ξβ summand” are both Σ2β+1. Proof. To say that x is the Zβ summand, we say ( y ( y>x)(x β y). ∃ 6∼Z The analysis that this is the correct statement is similar to the argument for being in the Zβ summand. (Although it will not be important for our discussion, these two formulas actually c show that the properties in the lemma are ∆2β+1.) β β Lemma 7.5 shows that (2) from Lemma 7.1 holds for ξβ (ξβ + Z ) and (ξβ + Z ) ξβ. c | | Consider the Σ2β+1 sentence ( x, y)(x ( y0 1. If β>γ>0 and n, m 1,thenη m<2γ+1 η n. ≥ β · γ · β 2. If β>0,thenξβ + Z <2β+1 ξβ. 3. If m>n 1 and γ>0,thenη m<2γ+1 η n. ≥ γ · γ · We have already verified the fact that there is a strict inequality in 2 of this lemma. Lemma 7.7 follows by an analysis similar to the ones given below for other order types. Notice that 2 in Lemma 7.7 tells us that β β ξβ (ξβ + Z ) 2β (ξβ + Z ) ξβ. | ≡ | Therefore, 1 from Lemma 7.1 holds. 23 Lemma 7.8. For al l β>0, ξβ+1 <2β+1 ξβ. Proof. First, notice that the statement ( x, y)(x We are still left with having to verify that the final intervals ξβ+1 and ξβ match up correctly. However, notice that we have reduced the level of the back-and-forth relation required between them. Continuing this process of matching intervals other than the final interval, we eventually reach the 1 relation, which is satisfied between ξβ and ξβ+1 since both are infinite. ≤ A similar argument gives the following slightly stronger result. Lemma 7.9. For al l β 1 and all n ω, ≥ ∈ β ξβ+1 <2β+1 ξβ + Z n. · β β We are now ready to verify 1 from Lemma 7.2 for the structures Z ω and Z ω∗. · · β β Lemma 7.10. Z ω 2β+1 Z ω∗. · ≤ · β Proof. Suppose we are given a partition of Z ω∗ by finitely many points. Since we know what · this partition looks like, we know that we can partition Zβ ω to match the intervals exactly, except for the initial and end intervals. To see that the end intervals· match up, we need to see β that ξβ + Z n 2β ξβ+1. This inequality follows from Lemma 7.9. To verify that the initial · ≤ β intervals match up, we need to see that ξβ∗+1 2β Z n + ξβ∗ .ByLemma7.9weknowthat β ≤ · β β ξβ+1 2β ξβ + Z n. However, it is also the case that (ξβ + Z n)∗ = Z n + ξβ∗ .Therefore, ≤ · · · theinitialintervalsmatchupcorrectlysinceforanyδ,ifL0 δ L1 then L∗ δ L∗. ≤ 0 ≤ 1 24 β β Lemma 7.11. Z ω∗ 2β+1 Z ω. · ≤ · Proof. This follows from Lemma 7.10, together with the fact that for all δ,ifL0 δ L1,then ≤ L∗ δ L∗. 0 ≤ 1 We next turn our attention to 4 from Lemmas 7.1 and 7.2, and verify the appropriate categoricity results. Lemma 7.12 shows that 4 holds for Zβ ω, and an almost identical β · argumentshowsthatitholdsforZ ω∗. · β 0 Lemma 7.12. Z ω has a formally Σ2β+2 Scott family with no parameters. · Proof. Let x =(x0,...,xn) be a tuple such that x0 < th β 1. We say x0 is in the l copy of Z for the appropriate l.Byourworkabove,this c statement is Σ2β+2. β β 2. If xi and xi+1 are not in the same copy of Z and there are exactly n copies of Z c between xi and xi+1,thenwesayxi β xi+1. This statement is Σ . ∼Z ,n 2β+2 β 3. If xi and xi+1 are in the same Z ,thenfixtheminimumαi β such that xi and αi ≤ xi+1 are in the same Z .Noticethatαi = δi +1for some δi. Add a clause saying α δi xi Z i xi+1 xi Zδi ,n xi+1 for the appropriate number of copies of Z between xi and ∼ ∧ ∼ c c xi+1. This statement is Σ2αi , and hence is Σ2β+2. c We have explained why this formula is Σ2β+2, and it is clear that any two tuples satisfying this formula are automorphic. Furthermore, every tuple must satisfy a formula of this form. Therefore, the Scott family consists of all formulas obtained in this way, corresponding to different tuples. β 0 Lemma 7.13. Both ξβ and ξβ + Z have formally Σ2β+1 Scott families with no parameters. β Proof. We prove that ξβ + Z has an appropriate Scott family. (The proof for ξβ is almost β identical, except that we do not need to say whether points lie in the ξβ summand or the Z summand, and we do not need to include the clause below, which concerns pairs of points in the Zβ summand.) β Let x be a tuple of elements in ξβ + Z such that x0 β 1. For each xi,wesaywhetheritisintheξβ summand or the Z summand. th γ 2. For each xi in the ξβ summand, we say it is in the n copy of Z for the appropriate γ<βand n 1. ≥ β 3. For each pair of points xi,xi+1 that are both in Z , we determine the interval between xi and xi+1 exactly as in step 3 in the proof of Lemma 7.12. 25 th γ 4. For each pair of points xi,xi+1 that are both in ξβ and are both in the n copy of Z , we determine the interval between xi and xi+1 as in step 3 in the proof of Lemma 7.12, exceptthatweworkinZγ instead of Zβ. β Notice that if xi,xi+1 are either not both in ξβ,ornotbothinZ ,orarebothinξβ but not in the same copy of Zγ, then we do not need to add any more information about the interval between them. c Finally, we check that the formula is Σ2β+1.Wehavealreadyverified this property for c steps 1 and 2. For steps 3 and 4, the formula is Σ2αi for some αi β. Therefore, the entire c ≤ formula is Σ2β+1. It remains to verify 3 from Lemmas 7.1 and 7.2. For 3 in Lemma 7.2, we need to show β β that there are computable copies of Z ω and Z ω∗, which have the property that for all γ<2β +2, the set of pairs (a, b) with · · β β a Z ω, b Z ω∗, a = b , ∈ · ∈ · | | | | and β β (Z ω, a) γ (Z ω∗, b) · ≤ · β β is uniformly c.e. in γ. Recall that (Z ω, a) γ (Z ω∗, b) if and only if each partitioned β · ≤ β · interval in Z ω is γ the corresponding interval in Z ω∗. This reduces our work to describing · ≤ β· β the γ relations between all possible subintervals of Z ω and Z ω∗, and to constructing nice computable≤ copies of these structures, in which we can· compute the· initial and final segments determined by each point, and we can compute the bounded interval determined by any pair of points. β β We have already seen the relations γ on all the subintervals of Z ω and Z ω∗.These are: ≤ · · β (i) ξβ+1 <2β+1 ξβ + Z m for all m, β · (ii) ξβ∗+1 <2β+1 Z m + ξβ∗ for all m, · (iii) η (n +1)<1 m for all m and n, γ · (iv) n<1 m if and only if n>m, (v) η m<2γ+1 η n for all m>n 1,and γ · γ · ≥ (vi) ηα m<2γ+1 ηγ n for all m, n 1 and α>γ. The first· two statements· follow from≥ Lemma 7.9, the third and fourth statements follow from considering the size of the orderings, and the last two statements were proved by Ash andstatedinLemma7.7. β β It remains to show that there are nice computable copies of Z ω and Z ω∗.Supposewe · · could construct a nice copy of Zβ in which we understood the relationship between any pair β β β of points. We could use this copy of Z to build uniform copies of Z ω and Z ω∗,inthe · · sense that for each point we would know which copy of Zβ that point sits in. In particular, we would know the initial and final segments determined by each point, and the intervals β β between any pair of points. These copies of Z ω and Z ω∗ would establish 3 in Lemma 7.2. · · Therefore, it sufficestoshowthereisacomputablecopyofZβ in which we can determine the interval between any pair of points, in terms of the orderings given above. That is, for any 26 two points x 27 segment of the Zβ summand. If the first partition point is in Zβ,then β ξβ + Z = ξβ + ξβ∗ +1+σ1 +1+ +1+σk +1+ξβ ··· if and only if each σi is either finite, or ηγ (n+1) for some γ<βand n ω.Theξβ +ξβ∗ term · β ∈ consists of all of the ξβ summand, and the initial segment of the Z summand. As above, it can be rewritten as ηβ. We have already specified all the back-and-forth relations γ between the subintervals β ≤ occurring in ξβ and ξβ + Z ,forallγ<2β +1.Asabove,itsuffices to show that we can β construct computable copies of ξβ and ξβ + Z , in which we know for each point whether it β is in ξβ or Z , and we know the interval determined by any pair of points. However, given anicecopyofZβ as described above, it is straightforward to put these copies together in a β uniform way to form nice copies of ξβ and ξβ + Z . 8Problems 0 Problem 1. For a computable limit ordinal α, is there a computable structure that is ∆α 0 categorical but not relatively ∆α categorical? Problem 2. For a computable limit ordinal α, is there a computable structure with a 0 0 A relation R that is intrinsically Σα but not relatively intrinsically Σα? Problem 3. If is ∆1 categorical, must it be relatively ∆1 categorical? A 1 1 Soskov [24] showed that for a computable (or hyperarithmetical) structure and a relation R on ,ifR is invariant under automorphisms of , and hyperarithmetical, thenA it is definable A A 1 1 by a computable infinitary formula. Hence, intrinsically ∆1 and relatively intrinsically ∆1 relations are the same. 0 Problem 4. For a computable limit ordinal α and finite n, is there a structure with ∆α dimension n? Problem 5. Is it true that for any computable successor ordinal α, there is a rigid computable 0 0 structure that is ∆α categorical but not relatively ∆α categorical? References [1] C. J. Ash, “Categoricity in hyperarithmetical degrees”, Annals of Pure and Applied Logic, vol. 34 (1987), pp. 1—14. [2] C. J. Ash, “A construction for recursive linear orderings”, Journal of Symbolic Logic,vol. 56 (1991), pp. 673—683. [3] C. J. Ash and J. F. Knight, “Pairs of recursive structures”, Annals of Pure and Applied Logic, vol. 46 (1990), pp. 211—234. 28 [4]C.J.AshandJ.F.Knight,Computable Structures and the Hyperarithmetical Hierarchy, Elsevier, Amsterdam, 2000. [5] C. Ash, J. Knight, M. Manasse, and T. Slaman, “Generic copies of countable structures”, Annals of Pure and Applied Logic, vol. 42 (1989), pp. 195—205. [6] C. J. Ash and A. Nerode, “Intrinsically recursive relations”, in Aspects of Effective Al- gebra (J. N. Crossley, editor), Upside Down A Book Co., Steel’s Creek, Australia, 1981, pp. 26—41. 0 [7] E. Barker, “Intrinsically Σα relations”, Annals of Pure and Applied Logic, vol. 39 (1988), pp. 105—130. [8] J. Chisholm, “Effective model theory vs. recursive model theory”, Journal of Symbolic Logic, vol. 55 (1990), pp. 1168—1191. [9] P. Cholak, S. Goncharov, B. Khoussainov, and R. A. Shore, “Computably categorical structures and expansions by constants”, Journal of Symbolic Logic, vol. 64 (1999), pp. 13—37. [10]Yu.L.Ershov,TheTheoryofNumberings, Nauka, Moscow, 1977 (in Russian). [11] Yu. L. Ershov and S. S. Goncharov, Constructive Models, Siberian School of Algebra and Logic, Kluwer Academic/Plenum Publishers, 2000 (English translation). [12] S. S. Goncharov, “Computable single-valued numerations”, Algebra and Logic,vol.19 (1980), pp. 507—551 (Russian); pp. 325—356 (English translation). [13] S. S. Goncharov, “The quantity of nonautoequivalent constructivizations”, Algebra and Logic, vol. 16 (1977), pp. 257—282 (Russian); pp. 169—185 (English translation). [14] V. S. Harizanov, “Pure computable model theory”, Handbook of Recursive Mathematics, vol. 1 (Yu. L. Ershov, S. S. Goncharov, A. Nerode, and J. B. Remmel, editors, V. W. Marek, assoc. editor), Elsevier, Amsterdam, 1998, pp. 3—114. [15] C. R. Karp, Languages with Expressions of Infinite Length, Ph.D. Dissertation, University of Southern California, 1959. [16] H. J. Keisler, Model Theory for Infinitary Logic,North-Holland,Amsterdam,1971. [17] B. Khoussainov and R. A. Shore, “Effective model theory: The number of models and their complexity”, Models and Computability, Invited Papers from Logic Colloquium ’97 (S. B. Cooper and J. K. Truss, editors), London Mathematical Society Lecture Notes Series 259, Cambridge University Press, Cambridge, 1999, pp. 193—239. [18] M. S. Manasse, Techniques and Counterexamples in Almost Categorical Recursive Model Theory, Ph.D. Dissertation, University of Wisconsin, Madison, 1982. 29 [19] S. S. Marchenkov, “On computable enumerations of families of general recursive func- tions”, Algebra and Logic, vol. 11 (1972), pp. 588—607 (Russian); pp. 326—336 (English translation). [20] C. F. D. McCoy, “Finite computable dimension does not relativize”, Archive for Mathe- matical Logic, vol. 41 (2002), pp. 309—320. [21] D. Scott, “Logic with denumerably long formulas and finite strings of quantifiers”, The Theory of Models (J.W.Addison,L.Henkin,andA.Tarski,editors),North-Holland, 1965, pp. 329—341. [22] V. L. Selivanov, “Enumerations of families of general recursive functions”, Algebra and Logic, vol. 15 (1976), pp. 205—226 (Russian), pp. 128—141 (English translation). [23] T. A. Slaman, “Relative to any nonrecursive set”, Proceedings of the American Mathe- matical Society, vol. 126 (1998), pp. 2117—2122. [24] I. N. Soskov, “Intrinsically hyperarithmetical sets”, Mathematical Logic Quarterly,vol. 42 (1996), pp. 469—480. [25] S. Wehner, “Enumerations, countable structures and Turing degrees”, Proceedings of the American Mathematical Society, vol. 126 (1998), pp. 2131—2139. 30