We introduce the computable FS-jump, an analog of the classical Friedman–Stanley jump in the context of equivalence relations on the natural numbers. We prove that the computable FS-jump is proper with respect to computable reducibility. We then study the effect of the computable FS-jump on computably enumerable equivalence relations (ceers).

In [20] Krajíček and Pudlák discovered connections between problems in computational complexity and the lengths of first-order proofs of finite consistency statements. Later Pudlák [25] studied more statements that connect provability with computational complexity and conjectured that they are true. All these conjectures are at least as strong as $\mathsf {P}\neq \mathsf {NP}$ [23–25].One of the problems concerning these conjectures is to find out how tightly they are connected with statements about computational complexity classes. Results of this kind had been proved in [20, 22].In this paper, we generalize and strengthen these results. Another question that we address concerns the dependence between these conjectures. We construct two oracles that enable us to answer questions about relativized separations asked in [19, 25] (i.e., for the pairs of conjectures mentioned in the questions, we construct oracles such that one conjecture from the pair is true in the relativized world and the other is false and vice versa). We also show several new connections between the studied conjectures. In particular, we show that the relation between the finite reflection principle and proof systems for existentially quantified Boolean formulas is similar to the one for finite consistency statements and proof systems for non-quantified propositional tautologies.

We initiate the study of computable presentations of real and complex C*-algebras under the program of effective metric structure theory. With the group situation as a model, we develop corresponding notions of recursive presentations and word problems for C*-algebras, and show some analogous results hold in this setting. Famously, every finitely generated group with a computable presentation is computably categorical, but we provide a counterexample in the case of C*-algebras. On the other hand, we show every finite-dimensional C*-algebra is computably categorical.

Lindström’s theorem obviously fails as a characterization of first-order logic without identity ($\mathcal {L}_{\omega \omega }^{-} $). In this note, we provide a fix: we show that $\mathcal {L}_{\omega \omega }^{-} $ is a maximal abstract logic satisfying a weak form of the isomorphism property (suitable for identity-free languages and studied in [11]), the Löwenheim–Skolem property, and compactness. Furthermore, we show that compactness can be replaced by being recursively enumerable for validity under certain conditions. In the proofs, we use a form of strong upwards Löwenheim–Skolem theorem not available in the framework with identity.

Let T be an infinite pseudo-tree. In [2], we showed that the cellularity of the pseudo-tree algebra Treealg(T) was the maximum of four cardinals cT, lT, ϕT, and μT: roughly, cT is the “tallness” of T: lT is the “width” of T, ϕ is the number of “points of finite branching” in T: and μ is the number of “sections of no branching” in T. Here we ask: which inequalities among these four cardinals may be satisfied, in some sense, by a pseudo-tree? We show that the possible inequalities among cT, lT, ϕT, and μT attained by pseudo-trees T are closely related to the existence of generalized Suslin trees.

From the work of Kleene, Post and Davis it is well-known that the arithmetic sets can be characterized as those sets recursive in ∅(n) for some natural number n, where ∅(0) = ∅ and . Actually the arithmetic sets which can be expressed in prenex form with n alternating quantifiers (applied to recursive predicates) are recursive in ∅(n). Hence, starting with ∅, the “jump operation”, which takes a set A into the set , serves to increase the “complexity” of sets in a uniform way.

As far as extending the ω-sequence of degrees of unsolvability (i.e. the degrees represented by the ∅(n)) into the 2nd number class, there is an immediate problem. One knows from Spector [17], corollary 2, p. 585, that there is no l.u.b. for the ω-sequence. So, at ω one must pick a degree in some other “natural” way. Unfortunately, what has seemed “natural” to some mathematicians has not seemed natural to others. Kleene and Davis (cf. [10] and [3] respectively) extended this arithmetic hierarchy of degrees of unsolvability by making use of natural number notations for a certain segment of ordinals of the 1st and 2nd number classes, the “constructive” ordinals. Using the Church-Kleene system S3, one can “sum up” previously obtained sets at the limit notations in a way that is certainly natural from a notational point of view.

The program of reverse mathematics has usually been to find which parts of set theory, often used as a base for other mathematics, are actually necessary for some particular mathematical theory. In recent years, Slaman, Groszek, et al, have given the approach a new twist. The priority arguments of recursion theory do not naturally or necessarily lead to a foundation involving any set theory; rather, Peano Arithmetic (PA) in the language of arithmetic suffices. From this point, the appropriate subsystems to consider are fragments of PA with limited induction. A theorem in this area would then have the form that certain induction axioms are independent of, necessary for, or even equivalent to a theorem about the Turing degrees. (See, for examples, [C], [GS], [M], [MS], and [SW].)

As go the integers so go the ordinals. One motivation of α-recursion theory (recursion on admissible ordinals) is to generalize classical recursion theory. Since induction in arithmetic is meant to capture the well-foundedness of ω, the corresponding axiom in set theory is foundation. So reverse mathematics, even in the context of a set theory (admissibility), can be changed by the influence of reverse recursion theory. We ask not which set existence axioms, but which foundation axioms, are necessary for the theorems of α-recursion theory.

When working in the theory KP – Foundation Schema (hereinafter called KP−), one should really not call it α-recursion theory, which refers implicitly to the full set of axioms KP. Just as the name β-recursion theory refers to what would be α-recursion theory only it includes also inadmissible ordinals, we call the subject of study here γ-recursion theory. This answers a question by Sacks and S. Friedman, “What is γ-recursion theory?”

Given a Polish group G, let $E(G)$ be the right coset equivalence relation $G^{\omega }/c(G)$, where $c(G)$ is the group of all convergent sequences in G. The connected component of the identity of a Polish group G is denoted by $G_0$.

Let $G,H$ be locally compact abelian Polish groups. If $E(G)\leq _B E(H)$, then there is a continuous homomorphism $S:G_0\rightarrow H_0$ such that $\ker (S)$ is non-archimedean. The converse is also true when G is connected and compact.

For $n\in {\mathbb {N}}^+$, the partially ordered set $P(\omega )/\mbox {Fin}$ can be embedded into Borel equivalence relations between $E({\mathbb {R}}^n)$ and $E({\mathbb {T}}^n)$.

We study function fields over p-adically closed fields in the first-order language of fields. Using ideas of Duret [D], we show that the field of constants is definable, and that the genus is an elementary property.