We prove that the degree structures of the d.c.e. and the 3-c.e. Turing degrees are not elementarily equivalent, thus refuting a conjecture of Downey. More specifically, we show that the following statement fails in the former but holds in the latter structure: There are degrees f > e > d > 0 such that any degree u ≤ f is either comparable with both e and d, or incomparable with both.

We give several characterizations of weakly minimal abelian structures. In two special cases, dual in a sense to be made explicit below, we give precise structure theorems:

• 1. when the only finite 0-definable subgroup is {0}, or equivalently 0 is the only algebraic element (the co-strongly minimal case);

• 2. when the theory of the structure is strongly minimal.

In the first case, we identify the abelian structure as a “near-subspace” A of a vector space V over a division ring D with its induced structure, with possibly some collection of distinguished subgroups of A of finite index in A and (up to acl(∅)) no further structure. In the second, the structure is that of V/A for a vector space and near-subspace as above, with the only further possible structure some collection of distinguished points. Here a near-subspace of V is a subgroup A such that for any nonzero d∈ D, the index of A∩ dA in A is finite. We also show that any weakly minimal abelian structure is a reduct of a weakly minimal module.

As is well-known, the Bernays-Schönfinkel-Ramsey class of all prenex ∃^*∀^*-sentences which are valid in classical first-order logic is decidable. This paper paves the way to an analogous result which the authors deem to hold when the only available predicate symbols are ∈ and =, no constants or function symbols are present, and one moves inside a (rather generic) Set Theory whose axioms yield the well-foundedness of membership and the existence of infinite sets. Here semi-decidability of the satisfiability problem for the BSR class is proved by following a purely semantic approach, the remaining part of the decidability result being postponed to a forthcoming paper.

We examine fields in which model theoretic algebraic closure coincides with relative field theoretic algebraic closure. These are perfect fields with nice model theoretic behaviour. For example, they are exactly the fields in which algebraic independence is an abstract independence relation in the sense of Kim and Pillay. Classes of examples are perfect PAC fields, model complete large fields and henselian valued fields of characteristic 0.

We give several characterizations of Schnorr trivial sets, including a new lowness notion for Schnorr triviality based on truth-table reducibility. These characterizations allow us to see not only that some natural classes of sets, including maximal sets, are composed entirely of Schnorr trivials, but also that the Schnorr trivial sets form an ideal in the truth-table degrees but not the weak truth-table degrees. This answers a question of Downey, Griffiths and LaForte.

We show that there is a first order sentence φ(x; a, b, l) such that for every computable partial order 𝒫 and Δ⁰₂-degree u > 0_e, there are Δ⁰₂-enumeration degrees a ≤ u, b, and l such that 𝒫 ≅ {x : φ(x; a, b, l)}. Allowing 𝒫 to be a suitably defined standard model of arithmetic gives a parameterized interpretation of true arithmetic in the Δ⁰₂-enumeration degrees. Finally we show that there is a first order sentence that correctly identifies a subset of the standard models, which gives a parameterless interpretation of true arithmetic in the Δ⁰₂-enumeration degrees.

A careful exposition of Zilber's quasiminimal excellent classes and their categoricity is given, leading to two new results: the L_ω1,ω(Q)-definability assumption may be dropped, and each class is determined by its model of dimension ℵ₀.

Computability logic (CL) is a recently launched program for redeveloping logic as a formal theory of computability, as opposed to the formal theory of truth that logic has more traditionally been. Formulas in it represent computational problems, “truth” means existence of an algorithmic solution, and proofs encode such solutions. Within the line of research devoted to finding axiomatizations for ever more expressive fragments of CL, the present paper introduces a new deductive system CL12 and proves its soundness and completeness with respect to the semantics of CL. Conservatively extending classical predicate calculus and offering considerable additional expressive and deductive power, CL12 presents a reasonable, computationally meaningful, constructive alternative to classical logic as a basis for applied theories. To obtain a model example of such theories, this paper rebuilds the traditional, classical-logic-based Peano arithmetic into a computability-logic-based counterpart. Among the purposes of the present contribution is to provide a starting point for what, as the author wishes to hope, might become a new line of research with a potential of interesting findings—an exploration of the presumably quite unusual metatheory of CL-based arithmetic and other CL-based applied systems.

If one wants to compute with infinite objects like real numbers or data streams, continuity is a necessary requirement: better and better (finite) approximations of the input are transformed into better and better (finite) approximations of the output. In case the objects are constructively generated, they can be represented by a finite description of the generating procedure. By effectively transforming such descriptions for the generation of the input (respectively, their codes) into (the code of) a description for the generation of the output another type of computable operation is obtained. Such operations are also called effective. The relationship of both classes of operations has always been a question of great interest. In this paper the setting is extended to the case of multifunctions. Various ways of coding (indexing) sets are discussed and their relationship is investigated. Moreover, effective versions of several continuity notions for multifunctions are introduced. For each of these notions an indexing system for sets is exhibited so that the multifunctions that are effective with respect to this indexing system are exactly the multifunction which are effectively continuous with respect to the continuity notion under consideration. Mostly, in addition to being effective the multifunctions need also possess certain witnessing functions. Important special cases are discussed where such witnessing functions always exist.

It is shown that under GCH every poset preserves its cofinality in any cofinality preserving extension. On the other hand, starting with ω measurable cardinals, a model with a partial ordered set which can change its cofinality in a cofinality preserving extension is constructed.

We investigate the pair-splitting number 𝔰_pair which is a variation of splitting number, pair-reaping number 𝔯_pair which is a variation of reaping number and cardinal invariants of ideals on ω. We also study cardinal invariants of F_σ ideals and their upper bounds and lower bounds. As an application, we answer a question of S. Solecki by showing that the ideal of finitely chromatic graphs is not locally Katětov-minimal among ideals not satisfying Fatou's lemma.

For automatic and recursive graphs, we investigate the following problems:

• (A) existence of a Hamiltonian path and existence of an infinite path in a tree

• (B) existence of an Euler path, bounding the number of ends, and bounding the number of infinite branches in a tree

• (C) existence of an infinite clique and an infinite version of set cover

The complexity of these problems is determined for automatic graphs and, supplementing results from the literature, for recursive graphs. Our results show that these problems

• (A) are equally complex for automatic and for recursive graphs (Σ¹₁-complete),

• (B) are moderately less complex for automatic than for recursive graphs (complete for different levels of the arithmetic hierarchy),

• (C) are much simpler for automatic than for recursive graphs (decidable and Σ₁¹-complete, resp.).

We present Woodin's proof that if there exists a measurable Woodin cardinal δ, then there is a forcing extension satisfying all Σ²₂ sentences φ such that CH + φ holds in a forcing extension of V by a partial order in V_δ. We also use some of the techniques from this proof to show that if there exists a stationary limit of stationary limits of Woodin cardinals, then in a homogeneous forcing extension there is an elementary embedding j : V → M with critical point ω₁^V such that M is countably closed in the forcing extension.

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

We prove that the (non-intuitionistic) law of the double negation shift has a bounded functional interpretation with bar recursive functionals of finite type. As an application, we show that full numerical comprehension is compatible with the uniformities introduced by the characteristic principles of the bounded functional interpretation for the classical case.

Let L be a first-order language and Φ and Ψ two Σ¹₁ L-sentences that cannot be satisfied simultaneously in any finite L-structure. Then obviously the following principle Chain_L,Φ,Ψ(n,m) holds: For any chain of finite L-structures C₁, …, C_m with the universe [n] one of the following conditions must fail:

• 1. C₁ ⊨ Φ,

• 2. C_i ≅ C_i+1, for i = 1, …, m-1,

• 3. C_m ⊨ Ψ.

For each fixed L and parameters n, m the principle Chain_L,Φ,Ψ(n,m) can be encoded into a propositional DNF formula of size polynomial in n, m.

For any language L containing only constants and unary predicates we show that there is a constant c_L such that the following holds: If a constant depth Frege system in DeMorgan language proves Chain_L,Φ,Ψ(n, c_L· n) by a size s proof then the class of finite L-structures with universe [n] satisfying Φ can be separated from the class of those L-structures on [n] satisfying Ψ by a depth 3 formula of size 2^log(s)O(1) and with bottom fan-in log(s)^O(1).