Abstract Our main result is that there exist structures which cannot be computably recovered from their tree of tuples. This implies that there are structures with no computable copies which nevertheless cannot code any information in a natural/functorial way.

Abstract We define the notion of a completely determined Borel code in reverse mathematics, and consider the principle $CD - PB$ , which states that every completely determined Borel set has the property of Baire. We show that this principle is strictly weaker than $AT{R_0}$ . Any ω-model of $CD - PB$ must be closed under hyperarithmetic reduction, but $CD - PB$ is not a theory of hyperarithmetic analysis. We show that whenever $M \subseteq {2^\omega }$ is the second-order part of an ω-model of $CD - PB$ , then for every $Z \in M$ , there is a $G \in M$ such that G is ${\rm{\Delta }}_1^1$ -generic relative to Z.

Abstract We examine recursive monotonic functions on the Lindenbaum algebra of $EA$ . We prove that no such function sends every consistent φ to a sentence with deductive strength strictly between φ and $\left( {\varphi \wedge Con\left( \varphi \right)} \right)$ . We generalize this result to iterates of consistency into the effective transfinite. We then prove that for any recursive monotonic function f, if there is an iterate of $Con$ that bounds f everywhere, then f must be somewhere equal to an iterate of $Con$ .

Abstract We introduce the notion of infinitary interpretation of structures. In general, an interpretation between structures induces a continuous homomorphism between their automorphism groups, and furthermore, it induces a functor between the categories of copies of each structure. We show that for the case of infinitary interpretation the reversals are also true: every Baire-measurable homomorphism between the automorphism groups of two countable structures is induced by an infinitary interpretation, and every Baire-measurable functor between the set of copies of two countable structures is induced by an infinitary interpretation. Furthermore, we show that the complexities are maintained in the sense that if the functor is ${\bf{\Delta }}_\alpha ^0$ , then the interpretation that induces it is ${\rm{\Delta }}_\alpha ^{in}$ up to ${\bf{\Delta }}_\alpha ^0$ equivalence.

Abstract Using methods from computable analysis, we establish a new connection between two seemingly distant areas of logic: computable structure theory and invariant descriptive set theory. We extend several fundamental results of computable structure theory to the more general setting of topological group actions. As we will see, the usual action of ${S_\infty }$ on the space of structures in a given language is effective in a certain algorithmic sense that we need, and ${S_\infty }$ itself carries a natural computability structure (to be defined). Among other results, we give a sufficient condition for an orbit under effective ${\cal G}$ -action of a computable Polish ${\cal G}$ to split into infinitely many disjoint effective orbits. Our results are not only more general than the respective results in computable structure theory, but they also tend to have proofs different from (and sometimes simpler than) the previously known proofs of the respective prototype results.

Abstract We extend the usual language of second order arithmetic to one in which we can discuss an ultrafilter over of the sets of a given model. The semantics are based on fixing a subclass of the sets in a structure for the basic language that corresponds to the intended ultrafilter. In this language we state axioms that express the notion that the subclass is an ultrafilter and additional ones that say it is idempotent or Ramsey. The axioms for idempotent ultrafilters prove, for example, Hindman’s theorem and its generalizations such as the Galvin--Glazer theorem and iterated versions of these theorems (IHT and IGG). We prove that adding these axioms to IHT produce conservative extensions of ACA0+IHT, ${\rm{ACA}}_{\rm{0}}^ +$ , ATR0, ${\rm{\Pi }}_2^1$ -CA0, and ${\rm{\Pi }}_2^1$ -CA0for all sentences of second order arithmetic and for full Z2for the class of ${\rm{\Pi }}_4^1$ sentences. We also generalize and strengthen a metamathematical result of Wang (1984) to show, for example, that any ${\rm{\Pi }}_2^1$ theorem ∀X∃YΘ(X,Y) provable in ACA0or ${\rm{ACA}}_{\rm{0}}^ +$ there aree,k∈ ℕ such that ACA0or ${\rm{ACA}}_{\rm{0}}^ +$ proves that ∀X(Θ(X, Φe(J(k)(X))) where Φeis theeth Turing reduction andJ(k)is thekth iterate of the Turing or Arithmetic jump, respectively. (A similar result is derived for ${\rm{\Pi }}_3^1$ theorems of ${\rm{\Pi }}_1^1$ -CA0and the hyperjump.)

Abstract Our main result is the equivalence of two notions of reducibility between structures. One is a syntactical notion which is an effective version of interpretability as in model theory, and the other one is a computational notion which is a strengthening of the well-known Medvedev reducibility. We extend our result to effective bi-interpretability and also to effective reductions between classes of structures.

Abstract We study Turing degrees a for which there is a countable structure ${\cal A}$ whose degree spectrum is the collection {x : x ≰ a}. In particular, for degrees a from the interval [0′, 0″], such a structure exists if a′ = 0″, and there are no such structures if a″ > 0‴.

Abstract In this paper, we investigate connections between structures present in every generic extension of the universe V and computability theory. We introduce the notion of generic Muchnik reducibility that can be used to compare the complexity of uncountable structures; we establish basic properties of this reducibility, and study it in the context of generic presentability, the existence of a copy of the structure in every extension by a given forcing. We show that every forcing notion making ω2 countable generically presents some countable structure with no copy in the ground model; and that every structure generically presentable by a forcing notion that does not make ω2 countable has a copy in the ground model. We also show that any countable structure ${\cal A}$ that is generically presentable by a forcing notion not collapsing ω1 has a countable copy in V, as does any structure ${\cal B}$ generically Muchnik reducible to a structure ${\cal A}$ of cardinality ℵ1. The former positive result yields a new proof of Harrington’s result that counterexamples to Vaught’s conjecture have models of power ℵ1 with Scott rank arbitrarily high below ω2. Finally, we show that a rigid structure with copies in all generic extensions by a given forcing has a copy already in the ground model.

Abstract We say that a theory T is intermediate under effective reducibility if the isomorphism problems among its computable models is neither hyperarithmetic nor on top under effective reducibility. We prove that if an infinitary sentence T is uniformly effectively dense, a property we define in the paper, then no extension of it is intermediate, at least when relativized to every oracle in a cone. As an application we show that no infinitary sentence whose models are all linear orderings is intermediate under effective reducibility relative to every oracle in a cone.

Abstract Many classes of structures have natural functions and relations on them: concatenation of linear orders, direct product of groups, disjoint union of equivalence structures, and so on. Here, we study the (un)decidability of the theory of several natural classes of structures with appropriate functions and relations. For some of these classes of structures, the resulting theory is decidable; for some of these classes of structures, the resulting theory is bi-interpretable with second-order arithmetic.

Abstract We describe a variation of Ash’s η-system and give a new proof of Ash’s metatheorem. As an application, we prove a generalization of Ash and Knight’s theorem on pairs of structures.

Abstract Assuming that 0# exists, we prove that there is a structure that can effectively interpret its own jump. In particular, we get a structure such that where is the set of Turing degrees which compute a copy of More interesting than the result itself is its unexpected complexity. We prove that higher-order arithmetic, which is the union of full “nth-order arithmetic for all n, cannot prove the existence of such a structure.

Abstract We introduce the notions of copyable and diagonalizable classes of structures. We then show how these notions are connected to two other notions that had already been studied for some particular classes of structures, namely the listability property and the low property. The main result of this paper is the characterizations of the classes of structures with the low property, that is, the classes whose low members all have computable copies. We characterize these classes as the ones whose structural jumps are listable.

Abstract We study the complexity of the isomorphism relation on classes of computable structures. We use the notion of FF-reducibility introduced in [9] to show completeness of the isomorphism relation on many familiar classes in the context of all equivalence relations on hyperarithmetical subsets of ω.

Given a class of structures K and n ∈ ω, we study the dichotomy between there being countably many n-back-and-forth equivalence classes and there being continuum many. In the latter case we show that, relative to some oracle, every set can be weakly coded in the (n − 1)st jump of some structure in K. In the former case we show that there is a countable set of infinitary Πn relations that captures all of the Πn information about the structures in K. In most cases where there are countably many n-back-and-forth equivalence classes, there is a computable description of them. We will show how to use this computable description to get a complete set of computably infinitary Πn formulas. This will allow us to completely characterize the relatively intrinsically Σ 0 n+1 relations in the computable structures of K, and to prove that no Turing degree can be coded by the (n − 1)st jump of any structure in K unless that degree is already below 0 (n−1).

Abstract Fraïssé studied countable structures through analysis of the age of , i.e., the set of all finitely generated substructures of . We investigate the effectiveness of his analysis, considering effectively presented lists of finitely generated structures and asking when such a list is the age of a computable structure. We focus particularly on the Fraïssé limit. We also show that degree spectra of relations on a sufficiently nice Fraïssé limit are always upward closed unless the relation is definable by a quantifier-free formula. We give some sufficient or necessary conditions for a Fraïssé limit to be spectrally universal. As an application, we prove that the computable atomless Boolean algebra is spectrally universal.

Abstract We study the computability-theoretic complexity and proof-theoretic strength of the following statements: (1) “If is a well-ordering, then so is ”, and (2) “If is a well-ordering, then so is φ(α, )”, where ∝ is a fixed computable ordinal and φ represents the two-placed Veblen function. For the former statement, we show that ω iterations of the Turing jump are necessary in the proof and that the statement is equivalent to over RCA0. To prove the latter statement we need to use ωα iterations of the Turing jump, and we show that the statement is equivalent to . Our proofs are purely computability-theoretic. We also give a new proof of a result of Friedman: the statement “if is a well-ordering, then so is φ( , 0)” is equivalent to ATR0 over RCA0.

Abstract We construct a computable ℵ0-categorical structure whose first order theory is computably equivalent to the true first order theory of arithmetic.

We study the classes of Büchi and Rabin automatic structures. For Büchi (Rabin) automatic structures their domains consist of infinite strings (trees), and the basic relations, including the equality relation, and graphs of operations are recognized by Büchi (Rabin) automata. A Büchi (Rabin) automatic structure is injective if different infinite strings (trees) represent different elements of the structure. The first part of the paper is devoted to understanding the automata-theoretic content of the well-known Löwenheim-Skolem theorem in model theory. We provide automata-theoretic versions of Löwenheim-Skolem theorem for Rabin and Büchi automatic structures. In the second part, we address the following two well-known open problems in the theory of automatic structures: Does every Büchi automatic structure have an injective Büchi presentation? Does every Rabin automatic structure have an injective Rabin presentation? We provide examples of Büchi structures without injective Büchi and Rabin presentations. To answer these questions we introduce Borel structures and usesome of the basic properties of Borel sets and isomorphisms. Finally, in the last part of the paper we study the isomorphism problem for Büchi automatic structures.

Abstract Two linear orderings areequimorphicif each can be embedded into the other. We prove that every hyperarithmetic linear ordering is equimorphic to a recursive one. On the way to our main result we prove that a linear ordering has Hausdorff rank less than if and only if it is equimorphic to a recursive one. As a corollary of our proof we prove that, given a recursive ordinal α, the partial ordering of equimorphism types of linear orderings of Hausdorff rank at most α ordered by embeddablity is recursively presentable.

Abstract. We show that if d ∈ GH1 then (≤ d) has the complementation property, i.e., for all a < d there is some b < d such that a ∧ b = 0 and a ∨ b = d.

Abstract We prove that every countable jump upper semilattice can be embedded in , where a jump upper semilattice (jusl) is an upper semilattice endowed with a strictly increasing and monotone unary operator that we call jump, and is the jusl of Turing degrees. As a corollary we get that the existential theory of 〈D, ≤T, ∨, ′〉 is decidable. We also prove that this result is not true about jusls with 0, by proving that not every quantifier free 1-type of jusl with 0 is realized in . On the other hand, we show that every quantifier free 1-type of jump partial ordering (jpo) with 0 is realized in . Moreover, we show that if every quantifier free type,p(x1,…,xn), of jpo with 0, which contains the formulax1≤ 0(m)& … &xn≤ 0(m)for somem, is realized in , then every quantifier free type of jpo with 0 is realized in . We also study the question of whether every jusl with the c.p.p. and size is embeddable in . We show that for the answer is no, and that forκ= ℵ1it is independent of ZFC. (It is true if MA(κ) holds.)