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Degrees of Categoricity and the Hyperarithmetic Hierarchy

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Degrees of Categoricity and the Hyperarithmetic Hierarchy 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 (!; <). It is easy to construct a computable copy, b, of (!; <) 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 (!; <), 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 (!; <) is not only 00- computably categorical, but that if c is such that (!; <) is c-computably categorical, then c 00 (since, in particular, c computes an isomor- ≥ phism between and b). Thus, for the structure (!; <), 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 < !, 0(m) (and, in fact, any degree 2-c.e. in and above 0(m)) is a strong degree of categoricity, as is 0(!). They prove the results first for m = 1 and then use relativization and Marker's extensions [5] to transfer them to all m < !. In the case for !, 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 jj of O to an ordinal, and a strict partial order <O on O. In particular, (1) O contains 1, and 1 O = 0. (2) If a O is a notationj j for the ordinal α, then 2a O and a 2 a 2 2 O = α + 1. For b O, we let b <O 2 if either b <O a or bj =j a. 2 (3) Given a limit ordinal λ, we say that the eth partial computable function 'e defines a fundamental sequence for λ if it is total, 'e(k) O 'e(k + 1) for all k, and λ is the least upper bound of ≤ e e the ordinals 'e(k) O. In this case, 3 5 O and 3 5 O = λ. j j e · 2 j · j For b O we will set b <O 3 5 if there exists k such that 2 · b <O 'e(k). For technical convenience, Hirschfeldt and White also require in [4] that the fundamental sequences contain only successor ordinals and have 1 as a first element. We assume without loss of generality that all members of O have this property. Definition 1.4. For a O we define H(a) by effective transfinite recursion as follows: 2 (1) H(1) = , (2) H(2a) =;H(a)0, and e e (3) H(3 5 ) = u; v u <O 3 5 & v H(u) = u; v n(u O · fh i j · 2 g fh i j 9 ≤ 'e(n)& v H(u)) : 2 g It is a result of Spector, see [1], that for a computable ordinal α (i.e., one with a notation in O), the Turing degree of H(a) when a is a notation for α does not depend on the choice of a. It is denoted by 0(α). Now we define the hyperarithmetic hierarchy in terms of computable infinitary formulas. 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 < ! and Q _ (f; n) ( x) k(f; n; x): R , 9 Q k<! 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 < ! and Q ^ (f; n) ( x) k(f; n; x): R , 8 Q k<! 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 !<! with height ! 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 2 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 1 and k for k < ! 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.
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