DEGREES OF CATEGORICITY AND THE HYPERARITHMETIC HIERARCHY
BARBARA F. CSIMA, JOHANNA N. Y. FRANKLIN, AND RICHARD A. SHORE
Abstract. We study arithmetic and hyperarithmetic degrees of categoricity. We extend a result of Fokina, Kalimullin, and R. Miller to show that for every computable ordinal α, 0(α) is the degree of categoricity of some computable structure . We show additionally that for α a computable successor ordinal,A every de- gree 2-c.e. in and above 0(α) is a degree of categoricity. We further prove that every degree of categoricity is hyperarithmetic and show 1 that the index set of structures with degrees of categoricity is Π1 complete.
1. Introduction Though classically two structures are considered the same if they are isomorphic, in computable model theory we must distinguish between particular presentations of a structure, since they may have different computable properties. We say a structure is computably categorical, if between any two computable presentations that are isomorphic, there exists a computable isomorphism. For example, the structure (η, <), the countable dense linear order without endpoints, is easily seen to be computably categorical since the usual back-and-forth isomorphism can be constructed computably between computable copies. However, there are many computable structures that are not computably cate- gorical. A well-known example is the structure (ω, <). It is easy to construct a computable copy, b, of (ω, <) such that any isomorphism between it and the standard copy,N , computes the halting set. It is also easy to see that in order to constructN an isomorphism between two copies of (ω, <), all we need to do is identify the least element of each
Date: May 8, 2012. B. Csima was partially supported by Canadian NSERC Discovery Grant 312501. R. Shore was partially supported by NSF Grant DMS-0852811 and John Templeton Foundation Grant 13408. This project was started by B. Csima and R. Shore while the former was visiting MIT, and we thank them for their hospitality. B. Csima would also like to thank the Max Planck Institute for Mathematics in Bonn, Germany for a productive visit. 1 2 CSIMA, FRANKLIN, AND SHORE
0 order, then the next element of each order, and so on. These are Σ1 questions, so 00 can certainly build an isomorphism. This leads us to recall the relativized definition of computable categoricity. Definition 1.1. A computable structure is d-computably categorical C for a Turing degree d if, for every computable ∼= , there is a d- computable isomorphism from to . A C C A We have observed above that the structure (ω, <) is not only 00- computably categorical, but that if c is such that (ω, <) is c-computably categorical, then c 00 (since, in particular, c computes an isomor- ≥ phism between and b). Thus, for the structure (ω, <), the degree 00 exactly describesN theN difficulty of computing isomorphisms between copies of the structure. This natural idea of a “degree of categoricity” for a computable struc- ture was first introduced by Fokina, Kalimullin and Miller in [2]. Definition 1.2. [2] A Turing degree d is said to be the degree of cat- egoricity of a computable structure if d is the least degree such that is d-computably categorical. TheC degree d is a degree of categoricity ifC it is the degree of categoricity of some computable structure. So far, every known degree of categoricity d has the following addi- tional property: There is a structure with computable copies 1 and C C 2 such that not only does have degree of categoricity d, but every C C isomorphism f : 1 ∼= 2 computes d. If a degree of categoricity has this property, weC say itC is a strong degree of categoricity. In [2], Fokina, Kalimullin, and Miller proved that for all m < ω, 0(m) (and, in fact, any degree 2-c.e. in and above 0(m)) is a strong degree of categoricity, as is 0(ω). They prove the results first for m = 1 and then use relativization and Marker’s extensions [5] to transfer them to all m < ω. In the case for ω, they paste together the structures for the 0(m). In this paper, we give direct constructions by effective trans- finite recursion that extend their results through the hyperarithmetic hierarchy. That is, in Theorem 3.1 we show that for every computable ordinal α, 0(α) is a strong degree of categoricity, and in Theorem 3.2 that, given the additional assumption that α is a successor ordinal, so are all degrees d that are 2-c.e. in and above 0(α). We then go on to show in Theorem 4.1 that any degree of categoricity must be hyper- arithmetic. We use this to show that the index set of structures with 1 degrees of categoricity, which at first glance seems no better than Σ2, 1 (α) is actually Π1. We then use our result that degrees of the form 0 1 are all degrees of categoricity to show that this index set is actually Π1 complete in Theorem 4.2. DEGREES OF CATEGORICITY 3
1.1. Notation and conventions. As many of the results in this paper use structures defined in Hirschfeldt and White [4], we will adopt many of their conventions. In particular, we will use Ash and Knight’s [1] terminology in discussing the hyperarithmetic hierarchy. For general references see Harizanov [3] for computable structure theory, Soare [8] for computability theory and Sacks [7] for hyperarithmetic theory. Definition 1.3. A system of notations for ordinals is comprised of a subset O of the natural numbers, a function O that maps each element || of O to an ordinal, and a strict partial order (1) O contains 1, and 1 O = 0. (2) If a O is a notation| | for the ordinal α, then 2a O and a ∈ a ∈ 2 O = α + 1. For b O, we let b Definition 1.5. AΣ0 (Π0) index for a computable predicate (f, n) is a triple Σ, 0, e ( Π, 0, e ) such that e is an index for .R If α is h i h i R a computable ordinal, a Σα index for a predicate (f, n) is a triple Σ, a, e such that a is a notation for α and e is an indexR for a c.e. set h i 4 CSIMA, FRANKLIN, AND SHORE A1 E1 Figure 1. 1 and 1. A E of Πβk indices for predicates k(f, n, x) such that βk < α for all k < ω and Q _ (f, n) ( x) k(f, n, x). R ⇔ ∃ Q k<ω AΠα index for a predicate (f, n) is a triple Π, a, e such that a R h i is a notation for α and e is an index for a c.e. set of Σβk indices for predicates k(f, n, x) such that βk < α for all k < ω and Q ^ (f, n) ( x) k(f, n, x). R ⇔ ∀ Q k<ω We say that a predicate is Σα (Πα) if it has a Σα (Πα) index and ∆α if it is both Σα and Πα. 2. Hirschfeldt and White’s “back-and-forth trees” The structures in this paper will be built using Hirschfeldt and White’s “ back-and-forth trees.” These are computable subtrees of ω<ω with height ω that will never have infinite paths. We use the construction as in≤ [4]; however, instead of fixing a path O0 O, we ⊂ define structures b and b for each b O, and other structures at A2 E2 ∈ limit ordinals. The structures a and a will be constructed by ef- fective transfinite recursion withA a caseE division as to whether a = 1, b e a = 22 (is the successor of a successor), or a = 23·5 (is the successor of a a a limit). Auxiliary structures ∞ and k for k < ω will be constructed for a = 3 5e. L L · 1 consists of a single node, and 1 consists of a single root node withA infinitely many children, none ofE which have children themselves. 2b Now consider a = 2 , i.e., the successor of a successor ordinal. a A will consist of a single root node with infinitely many copies of b E2 DEGREES OF CATEGORICITY 5 a A Ea b b 2b 2b b 2 2 2b b 2b 2b E E E2 E E E A2 A A 2b Figure 2. a and a for a = 2 . A E e 3 5 3 5e · · L∞ Lk ϕe(0) ϕ (1) ϕe(0) ϕ (1) A A e Aϕe(2) A A e Aϕe(k) Eϕe(k+1) Eϕe(k+2) e e Figure 3. 3·5 and 3·5 . L∞ Lk attached to it, and a will consist of a single root node with infinitely E many copies of 2b and infinitely many copies of 2b attached to it. A 3·5Ee Finally, consider an ordinal of the form a = 2 . To define a and 3·5e A a, we first define auxiliary trees k for each k < ω and an auxiliary e e treeE 3·5 . For each k < ω, we letL 3·5 consist of a single root node L∞ Lk with exactly one copy of ϕ (n) attached to it for every n k and A e ≤ exactly one copy of ϕe(n) attached to it for every n > k. We define 3·5e E ∞ to consist of a single root node with exactly one copy of ϕe(n) attachedL to it for every n < ω. (Note: This is where we makeA use of e our assumption that if 3 5 O then each ϕe(n) is a notation for a successor ordinal.) · ∈ Now we can define a and a. We let a consist of a root node with A 3·5Ee A infinitely many copies of attached to it for every n < ω and a Ln E 6 CSIMA, FRANKLIN, AND SHORE a A Ea e 3 5 e 3 5e e e e e e e e 0· 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 L 0· 1· 1· · · 0· 0· 1· 1· L L L L∞ L∞ L L L L 3·5e Figure 4. a and a for a = 2 . A E 3·5e consist of a root node with infinitely many copies of ∞ and infinitely 3·5e L many copies of n for each n < ω attached to it. L 3·5e 3·5e We say that the back-and-forth trees ∞ and n for n < ω have e L L rank 3 5 and that the back-and-forth trees a and a have rank a. The· above definitions are given by computableA transfiniteE recursion. In fact, we have given a procedure that, for any natural number a, assumes that a O with all fundamental sequences containing only successors and begins∈ to build the desired structure. In any case, a computable tree is constructed, and if a O, then it is of the desired form. ∈ 0 Note that for a = a , we may have, for example, a = a0 even 0 6 E 6∼ E though a O = a O. Nonetheless, in cases where it is not likely to lead | | | | α α to confusion, we will use α, α, k and ∞ to denote copies of a, a a E A L L E a, k and ∞, respectively, for some a O with a O = α. A WeL will makeL use of the following results∈ from Hirschfeldt| | and White [4]. Note that we have slightly reworded their results. Hirschfeldt and White fix a particular path through O, noting that their results work equally well for any path through O, and word their results in terms of computable ordinals α. For us, it will be more convenient to speak of the notations, a O. ∈ Proposition 2.1 (Proposition 3.2, [4]). Let (n) be a Σα predicate. P (1) If α is a successor ordinal, then for every notation a for α, there is a sequence of trees n, uniformly computable from the T notation a and a Σα index for , such that for all n, P ( a if (n) and n = E P T ∼ a otherwise. A DEGREES OF CATEGORICITY 7 (Note: If α Furthermore, for computable , the complexity of “ = ϕB(r)” is the T T | natural complexity of , and an index for ϕB(r) can be found uniformly from the back-and-forthB index of . B We can use the above lemmas to see to what extent 0(α) can be used to compute isomorphisms between back-and-forth trees. Corollary 2.5. Let be any computable tree, and let be a back- T S and-forth limb of with rank( ) Theorem 3.1. For any a O, there is a computable structure a with (strong) degree of categoricity∈ H(a). S Proof. We build the structures a for a O by transfinite recursion. These structures will consist ofS infinitely∈ many disjoint copies of the different kinds of back-and-forth trees described in Section 2, though DEGREES OF CATEGORICITY 9 they will not be trees themselves. To do this, we will describe the back- and-forth trees we want to use and then assign elements of ω to the various parts of these trees by defining a set of edges that will produce these trees. Note that in fact for any a ω the procedure can be used ∈ to build a computable structure a, where the structure has the desired form for each a O. S ∈ For each a O, we will build a “standard” copy of a as well as ∈ S a “hard” but still computable copy ba such that any isomorphism be- S tween a and ba will compute H(a). S S First, we consider the case a = 2. Let 2 consist of infinitely many S 0 disjoint copies of 1 and 1, and fix an approximation Ks s∈ω to 0 . A E { } We define the set of edges of the standard copy 2 to be S ( 2n, 0 , 2n, k ) k > 0 , { h i h i | } so the substructure consisting of the elements in an “even” column is isomorphic to 1 (with 2n, 0 as the root node) and the elements in the “odd” columnsE areh not connectedi to any other elements and are thus, when considered as singletons, substructures that are isomorphic to 1. Now we define the set of edges of the hard copy, b2, to be A S ( 2n, 0 , 2n, t ) n Kt . { h i h i | ∈ } In b2, the root nodes of copies of 1 are of the form 2n, 0 for n K, S E h i ∈ and their child nodes are of the form 2n, t for those t where n Kt. Elements coding pairs of any other formh willi not be connected to∈ any other elements and, when considered as singletons, will be substruc- tures isomorphic to 1. Let f : b2 ∼= 2. Then n K f( 2n, 0 ) = 2m, 0 forA some m. S S ∈ ⇐⇒ h i h i 0 Now we consider two arbitrary computable copies of 2. Since 0 can S answer all Σ1 and Π1 questions, it can determine which elements of each are 1 components, which are roots in 1 components, and which are A E children in 1 components, so it is powerful enough to compute an isomorphismE between these copies. Now we consider the case of 2a where a = 2b for some b O. In this ∈ case, let 2a consist of infinitely many disjoint copies of a and a. S A a E We now verify that 2a has degree of categoricity H(2 ). In the S following discussion, we assume that a O > ω; the case where a O is finite is similar but some indices are off| | by one. | | Since H(2a) is Σ0 , using Proposition 2.1 (1), we can build a com- |a|O putable copy b2a of 2a such that for every n, n, 0 is a parent node S S a h i of a tree isomorphic to a if n H(2 ) and a parent node of a tree E a ∈ isomorphic to a if n H(2 ). We have a standard copy of 2a where A 6∈ S 10 CSIMA, FRANKLIN, AND SHORE 2n, 0 is a parent node of a tree isomorphic to a and 2n + 1, 0 is a h i E h i parent node of a tree isomorphic to a for every n. Let f : b2a ∼= 2a . Then n H(2a) if and only if f( n, 0A) = 2k, 0 for some k.S S ∈ h i h i Conversely, let be an arbitrary computable copy of 2a . We de- a B S scribe an H(2 )-computable isomorphism f : 2a . It is a Σ1 question whether a vertex in has an edge goingB to→ it. S Hence, H(2a) B can certainly compute the root nodes, ri, of all the connected com- ponents of . Each connected component is isomorphic to either a B a E or a, which both have rank a. By Corollary 2.5, H(2 ) uniformly computesA the back-and-forth index of the limb extending from each root node. Thus H(2a) can first define a bijection between the root nodes ri in and the root nodes i, 0 in 2a that is back-and-forth index preserving.B Then, by Corollaryh 2.6,i HS(2a) can extend this to an isomorphism = 2a . B ∼ S e Now we consider the case where a = 3 5 . Let a consist of infinitely · S many disjoint copies of ϕ (k) and ϕ (k) for all k ω. A e E e ∈ To make the hard copy, ba, we proceed as follows. Using Proposition 2.1 (1), let u, v, n+1, 0 beS the parent node of a tree that is isomorphic h i to ϕe(n) if u O ϕe(n) and v H(u) and isomorphic to ϕe(n) other- wise.E Let u, v,≤0, 0 be the parent∈ node of a tree that is isomorphicA to h i ϕe(0) if ( n)[u O ϕe(n)] and isomorphic to ϕe(0) otherwise. To make theE standard∃ copy,≤ we let k, n, 0 be the parentA node of a tree that is h i isomorphic to ϕ (n) if k is even and ϕ (n) if k is odd. E e A e Let f : ba = a; we wish to use f to compute H(a). First, compute S ∼ S f( u, v, 0, 0 ). If f( u, v, 0, 0 ) is not of the form 2y, 0, 0 , then u Proof. We build a graph with degree of categoricity d. Let D d be (α) G 0 ∈ 2-c.e. in and above 0 , and let B and C be Σα sets such that C B ⊂(α) and D = B C. Let Bk k∈ω be an enumeration of B relative to 0 . − { } n n n n n For each n ω, we will make use of vertices labeled ek , ak , bk , ck , dk for k ω, which∈ will all belong to a single connected component of ∈ n the graph. We attach an (n + 4)-cycle to the point e0 . We now use Proposition 2.1 (1) to build two computable copies and b of the graph. Since the description of the component for nGis theG same for each n, we drop the superscript to ease notation. In both copies, we have edges (ek, ek+1), (ek, ak), (ek, bk), (ek, ck), (ek, dk), (ak, bk), (bk, ck), (ck, dk), and (dk, ak) for all k (Figure 5). In both copies, a0 will be a parent node of a tree isomorphic to α and c0 will be a parent node of E a tree isomorphic to α if n C and α otherwise. In , let b0 be a E ∈ A G parent node of a tree isomorphic to α if n B and α otherwise, and E ∈ A let d0 be a parent node of a tree isomorphic to α if n C and α E ∈ A otherwise (Figure 6). In b, we reverse the roles of b0 and d0 (Figure 7). G Now for k > 0, in both copies, we will make ak a parent node of a tree isomorphic to α and ck a parent node of a tree isomorphic to α. In , E A G let bk be a parent node of a tree isomorphic to α if n (Bk+1 Bk) C E ∈ − ∩ and α otherwise, and let dk be a parent node of a tree isomorphic to A α (Figure 8). In b, we reverse the roles of bk and dk (Figure 9). A G Finally, we will add 3-cycles to and b that will guarantee that any isomorphism between these structuresG willG compute H(α). We begin by adding a basic 3-cycle to each graph. Then we attach a copy of α (built the “standard” way) to each of the three elements of the 3-cycleS 12 CSIMA, FRANKLIN, AND SHORE a0 o d0 a1 o d1 ]:: A O W/ G O : ÔÔ // :: Ô / : ÔÔ / :: ÔÔ // : Ô / b0 :: Ô / c0 b1 / / c1 fMM : ÔÔ q8 `A / > MM :: ÔÔ qq AA / }} MM : Ô qq A // } MM : Ô qqq AA / }} MMM:: ÔÔqq AA/ }} M Ôqq } ... 8 e0 / e1 / | (n + 4)-cycle Figure 5. Basic structure of the nth connected compo- nent of and b G G α if n C α hPP E otherwise∈ E PPP α PPP nnA6 PPP nnn PPP nnn PPP nnn PP nnn a0 o d0 O b0 / c0 nn PPP nnn PPP nnn PPP nnn PPP vnnn PP( α if n B α if n C Eα otherwise∈ Eα otherwise∈ A A Figure 6. G α if n B α hPP E otherwise∈ E PPP α PPP nnA6 PPP nnn PPP nnn PPP nnn PP nnn a0 o d0 O b0 / c0 nn PPP nnn PPP nnn PPP nnn PPP vnnn PP( α if n C α if n C Eα otherwise∈ Eα otherwise∈ A A Figure 7. b G in and a copy of bα (built the “hard” way) to each of the three G S elements of the 3-cycle in b. These 3-cycles must be matched up by G DEGREES OF CATEGORICITY 13 α α E aC = A CC {{ CC {{ CC {{ C {{ ak o dk O b / ck k C {{ CC {{ CC {{ CC }{{ C! α α E A Figure 8. for k > 0 if n (Bk+1 Bk) C G ∈ − ∩ α α E aC = E CC {{ CC {{ CC {{ C {{ ak o dk O b / ck k C {{ CC {{ CC {{ CC }{{ C! α α A A Figure 9. b for k > 0 if n (Bk+1 Bk) C G ∈ − ∩ any isomorphism between these structures, which means that such an isomorphism must be able to map a “standard” copy of α to a “hard” copy. S Note that in any computable copy of the structure, for each n, the points isomorphic to e0, e1, e2,... are computable, as are the 4-cycles emanating from them. Each member of such a 4-cycle is a parent node of a tree isomorphic to either α or α, and an isomorphism of two copies of the structure matchesE theseA up correctly. Let p be an isomorphism between and b. This means that p must be able to compute H(α), since itG can computeG an isomorphism between the 3-cycles with copies of α and bα attached to them. S S Now if p(a0) = a0, then either n C (in which case n D), or ∈ 6∈ n B (in which case n D as well). Therefore, if p(a0) = a0, then 6∈ 6∈ n D. If p(a0) = a0, then n B. This means that n D if and only if 6∈n C. Since 6 p computes H∈(α) and n B, p computes∈ the k such 6∈ ∈ that n Bk+1 Bk. Then n D if and only if p(ak) = ak. ∈ − ∈ 14 CSIMA, FRANKLIN, AND SHORE α α E aC = A CC {{ CC {{ CC {{ C {{ a0 o d0 O b / c0 0 C {{ CC {{ CC {{ CC }{{ C! α α A A Figure 10. and b if n B G G 6∈ Conversely, we claim that, given arbitrary computable copies of the structures, D can compute an isomorphism between them. Note that the limbs of α and α all have rank α 1, so by Corollary 2.5, H(α) can computeE the back-and-forthA index− of limbs of the trees attached to any of ak, bk, ck, dk in either copy. Moreover, α has α as a proper E A substructure; i.e., the any back-and-forth index of a limb of α also A occurs as the back-and-forth index of a limb of α, but not conversely. E Thus the fact that a tree attached to some node is α (and not α) is c.e. in H(α). E A If n D, then there are two possibilities: either n is not in B (and therefore,6∈ not in C) or n is in both B and C. In the first case, exactly one of the nodes a0, b0, c0, and d0 (in fact, a0) is a parent node of a tree isomorphic to α (Figure 10). In the second case, all of these nodes are E parent nodes of a tree isomorphic to α (Figure 11). Now, by Corollary E 2.5, we can use H(α) to find at least one α parent node in each copy E and match these up, applying Corollary 2.5 to match up limbs of the α with the same back-and-forth index, and Corollary 2.6 to (uniformly)E extend these to isomorphisms of the limbs. The rest of the four nodes can then be safely matched up, using the same procedure to construct the isomorphism, assuming they are all copies of α—if at some point A it turns out that one is an α and not an α, then they all are. If E A n D, then exactly two of the nodes a0, b0, c0 and d0 are parent nodes ∈ of a tree isomorphic to α (Figures 12 and 13); H(α) can find these in each copy and match themE up. For k > 0, exactly one or two of the nodes ak, bk, ck, dk are parent nodes of a tree isomorphic to α. The only way that there are two is if E n (Bk+1 Bk) C. Now H(α) and hence D can compute whether ∈ − ∩ n Bk+1 Bk, and if n Bk+1 Bk, then n C if and only if n D. ∈ − ∈ − ∈ 6∈ Therefore, D can compute whether there are one or two copies of α E DEGREES OF CATEGORICITY 15 α α E `A > E AA }} AA }} AA } A }} a0 o d0 O b / c0 0 A }} AA }} AA }} AA ~}} A α α E E Figure 11. Copy 1 and Copy 2 if n C ∈ α α E aC = A CC {{ CC {{ CC {{ C {{ a0 o d0 O b / c0 0 C {{ CC {{ CC {{ CC }{{ C! α α E A Figure 12. Copy 1 if n B and n C ∈ 6∈ α α E aC = E CC {{ CC {{ CC {{ C {{ a0 o d0 O b / c0 0 C {{ CC {{ CC {{ CC }{{ C! α α A A Figure 13. Copy 2 if n B and n C ∈ 6∈ with parent nodes among ak, bk, ck, dk, and H(α) can find these in each copy and match them up. 16 CSIMA, FRANKLIN, AND SHORE We conclude by observing that since D T H(α), D can compute an ≥ isomorphism between the copies of α attached to the 3-cycle in each copy. S 4. General properties of degrees of categoricity Fokina, Kalimullin, and R. Miller showed that any strong degree of categoricity is hyperarithmetical. The Effective Perfect Set Theorem is the main ingredient in their proof [2]. Here, we strengthen their result and show that every degree of categoricity is hyperarithmetic. We then go on to show that the index set of structures that have a degree of 1 categoricity is actually Π1 complete. Theorem 4.1. If d is a degree of categoricity, then d HYP . ∈ Proof. Let d HYP , and let be any computable structure. We will 6∈ A show that d is not a degree of categoricity for . Let 0, 1, 2,... be A A A A 0 a list of all computable copies of . Note that f f : 0 = 1 is Π2, 1 A { | A ∼ A } so it is certainly Σ1. Hence by Kreisel’s Basis Theorem [7], there exists an isomorphism f1 : 0 = 1 such that d h f1. Suppose we are given A ∼ A 6≤ isomorphisms fi : 0 = i for 1 i n such that d h f1 fn. A ∼ A ≤ ≤ 6≤ ⊕ · · · ⊕ Then by Kreisel’s Basis Theorem relativized to f1 fn, there exists ⊕· · ·⊕ an isomorphism fn+1 : 0 = n+1 such that d h f1 fn+1. Now A ∼ A 6≤ ⊕ · · · ⊕ let a, b be an exact pair for the sequence f1, f1 f2, f1 f2 f3, .... Since ⊕ ⊕ ⊕ fn is an isomorphism between 0 and n, both a and b can compute an isomorphism between any twoA arbitraryA computable copies of . Therefore, if d is a degree of categoricity for , then d a and d Ab. A ≤ ≤ However, since a, b is an exact pair, d T f1 fn for some n, ≤ ⊕ · · · ⊕ giving us a contradiction. Now we consider the complexity of the index set of structures that have a degree of categoricity. Let 0, 1, 2,... be a list of the partial computable structures. A A A Theorem 4.2. The index set = e e has a degree of categoricity 1 DC { | A } is Π1 complete. Proof. We begin by giving the natural formula ψ(e) that expresses that the structure e has a degree of categoricity, that is, the formula A ( D) ( i)[( F )(F : e = i) ( Fˆ T D)(Fˆ : e = i)] & ∃ { ∀ ∃ A ∼ A → ∃ ≤ A ∼ A ( C)[( i)[( F )(F : e = i) ( Fˆ T C)(Fˆ : e = i)] D T C] . ∀ ∀ ∃ A ∼ A → ∃ ≤ A ∼ A → ≤ } DEGREES OF CATEGORICITY 17 1 We note that ψ(e) is Σ2. However, by Theorem 4.1, we know that any degree of categoricity is hyperarithmetic. Thus many of the existential quantifiers in ψ can be bounded by HYP without changing the meaning of the formula. That is, we have the following formula ψˆ(e) expressing that e has a degree of categoricity. A ( D HYP ) ( i)[( F )(F : e = i) ( Fˆ T D)(Fˆ : e = i)] & ∃ ∈ { ∀ ∃ A ∼ A → ∃ ≤ A ∼ A ( C)[( i)[( F HYP )(F : e = i) ( Fˆ T C)(Fˆ : e = i)] D T C] ∀ ∀ ∃ ∈ A ∼ A → ∃ ≤ A ∼ A → ≤ } 1 We observe that this formula is Π1: quantifiers like D HYP can be 1 ˆ ∃ ∈ written as Π1 formulas, and quantifiers like F T D are arithmetic. Furthermore, we note that we can write F ∃HYP≤ in the second half of the conjunction instead of simply F ∃because∈ we require D to be in HYP and because the first half of the∃ conjunction states that there is an isomorphism from e to i that is computable from D. Therefore, A A ψˆ(e) is equivalent to ψ(e). Now we only need to show that the set 1 DC is Π1 complete. For each a ω, we will, uniformly in a, define a computable structure ∈ a. If a O, then a will have a degree of categoricity. If a O then R ∈ R 6∈ a will have distinguishable computable substructures with degrees of R CK categoricity H(b) for b’s in O of rank unbounded in ω1 . Thus a itself not have a degree of categoricity by Theorem 4.1. R We begin with a folklore reduction that can be found in [9], Prop. 5.4.1: There is a recursive function f such that if a O then f(a) is an CK∈ index for a recursive linear ordering of type α < ω1 and if a / O then CK ∈ f(a) is an index for a recursive linear ordering of type ω1 (1 + η) (the Harrison linear ordering). We then apply a standard translation g of a recursive linear order into a recursive notation system g(e) as can be found in [6] (Theorem 11.8.XX). This translation takes≤ an index e for a recursive well ordering to a recursive notation system in O of limit ordinal length (actually ω times the length of the original ordering) CK and one for a linear order of type ω1 (1 + η) to a recursive notation system (although not a well founded one) with a well founded initial CK segment that is a path in O of type ω1 . Let a be a disjoint labeled union of the b for each b in the notation R S system g(f(a)), i.e. there is a (b + 3)-cycle with a copy of b attached to a node in the cycle. Note that since the notation systemS is recursive and each b is uniformly computable (in b), a is computable. Note S R that we can build “standard” and “hard” copies of a by using (for R any b O) the standard copies of b in the standard copy of a ∈ S R and the hard copies bb in the hard copy ba of a. Since for each S R R 18 CSIMA, FRANKLIN, AND SHORE b in the system, the copy of b in a is labeled by a (b + 3)-cycle, it is easy to see that for any Sb ORin the system, any isomorphism ∈ f : ba a computes H(b). If a O, then the notation system is an initialR → segment R of O of length some∈ limit notation c O. By the uniformities of all our constructions, it is easy to see that∈ H(c) can always compute an isomorphism between any two copies of a. On R the other hand, the uniform replacement of the b by hard and easy ones produces copies such that any isomorphismS uniformly computes H(b) for every b Theorem 4.1 tells us that we can limit our search for a counterexam- ple to the hyperarithmetic case and provides some evidence that these degrees are the same, but no more. Along similar lines, but with a view towards the structures, one might ask the following. Question 5.4. Does there exist a structure which has a degree of cat- egoricity that is not a strong degree of categoricity for that structure? References [1] C.J. Ash and J. Knight. Computable Structures and the Hyperarithmetical Hi- erarchy. Number 144 in Studies in Logic and the Foundations of Mathematics. North-Holland, 2000. [2] Ekaterina B. Fokina, Iskander Kalimullin, and Russell Miller. Degrees of cate- goricity of computable structures. Arch. Math. Logic, 49(1):51–67, 2010. [3] Valentina S. Harizanov. Pure computable model theory. In Handbook of recur- sive mathematics, Vol. 1, volume 138 of Stud. Logic Found. Math., pages 3–114. North-Holland, Amsterdam, 1998. [4] Denis R. Hirschfeldt and Walker M. White. Realizing levels of the hyperarith- metic hierarchy as degree spectra of relations on computable structures. Notre Dame J. Formal Logic, 43(1):51–64 (2003), 2002. [5] David Marker. Non Σn axiomatizable almost strongly minimal theories. J. Sym- bolic Logic, 54(3):921–927, 1989. [6] Hartley Rogers, Jr. Theory of recursive functions and effective computability. McGraw-Hill Book Co., New York, 1967. [7] Gerald E. Sacks. Higher Recursion Theory. Springer-Verlag, 1990. [8] Robert I. Soare. Recursively Enumerable Sets and Degrees. Perspectives in Mathematical Logic. Springer-Verlag, 1987. [9] Walker McMillan White. Characterizations for computable structures. PhD the- sis, Cornell University, 2000. Department of Pure Mathematics, University of Waterloo, 200 Uni- versity Avenue West, Waterloo, Ontario, Canada N2L 3G1 E-mail address: [email protected] Department of Mathematics, 196 Auditorium Road, University of Connecticut, U-3009, Storrs, CT 06269-3009, USA E-mail address: [email protected] 310 Malott Hall, Cornell University, Ithaca, NY 14853-4201, USA E-mail address: [email protected]