Gödel’s completeness theorem for classical first-order logic is one of the most basic theorems of logic. Central to any foundational course in logic, it connects the notion of valid formula, i.e., a formula satisfied in all models, to the notion of provable formula.

We survey a few standard formulations and proofs of the completeness theorem before focusing on the formal description of a slight modification of Henkin’s proof within intuitionistic second-order arithmetic.

It is usual, in the context of the completeness of intuitionistic logic with respect to various semantics, such as Kripke or Beth semantics, to follow the Curry–Howard correspondence and to interpret the proofs of completeness as programs which turn proofs of validity for these semantics into proofs of derivability.

We apply this approach to Henkin’s proof to phrase it as a program which transforms any proof of validity with respect to Tarski semantics into a proof of derivability.

By doing so, we hope to shed some “effective” light on the relation between Tarski semantics and syntax: proofs of validity are syntactic objects with which we can compute.

We investigate the notion of ideal (equivalently: filter) Schauder basis of a Banach space. We do so by providing bunch of new examples of such bases that are not the standard ones, especially within classical Banach spaces ($\ell _p$, $c_0$, and James’ space). Those examples lead to distinguishing and characterizing ideals (equivalently: filters) in terms of Schauder bases. We investigate the relationship between possibly basic sequences and ideals (equivalently: filters) on the set of natural numbers.

Working within the context of countable, superstable theories, we give many equivalents of a theory having NOTOP. In particular, NOTOP is equivalent to V-DI, the assertion that any type V-dominated by an independent triple is isolated over the triple. If T has NOTOP, then every model N is atomic over an independent tree of countable, elementary substructures, and hence is determined up to back-and-forth equivalence over such a tree. We also verify Shelah’s assertion from Chapter XII of [9] that NOTOP implies PMOP (without using NDOP).

For cardinals $\mathfrak {a}$ and $\mathfrak {b}$, we write $\mathfrak {a}=^\ast \mathfrak {b}$ if there are sets A and B of cardinalities $\mathfrak {a}$ and $\mathfrak {b}$, respectively, such that there are partial surjections from A onto B and from B onto A. $=^\ast $-equivalence classes are called surjective cardinals. In this article, we show that $\mathsf {ZF}+\mathsf {DC}_\kappa $, where $\kappa $ is a fixed aleph, cannot prove that surjective cardinals form a cardinal algebra, which gives a negative solution to a question proposed by Truss [J. Truss, Ann. Pure Appl. Logic 27, 165–207 (1984)]. Nevertheless, we show that surjective cardinals form a “surjective cardinal algebra”, whose postulates are almost the same as those of a cardinal algebra, except that the refinement postulate is replaced by the finite refinement postulate. This yields a smoother proof of the cancellation law for surjective cardinals, which states that $m\cdot \mathfrak {a}=^\ast m\cdot \mathfrak {b}$ implies $\mathfrak {a}=^\ast \mathfrak {b}$ for all cardinals $\mathfrak {a},\mathfrak {b}$ and all nonzero natural numbers m.

We investigate some variants of the splitting, reaping, and independence numbers defined using asymptotic density. Specifically, we give a proof of Con($\mathfrak {i}<\mathfrak {s}_{1/2}$), Con($\mathfrak {r}_{1/2}<\mathfrak {b}$), and Con($\mathfrak {i}_*<2^{\aleph _0}$). This answers two questions raised in [5]. Besides, we prove the consistency of $\mathfrak {s}_{1/2}^{\infty } < $ non$(\mathcal {E})$ and cov$(\mathcal {E}) < \mathfrak {r}_{1/2}^{\infty }$, where $\mathcal {E}$ is the $\sigma $-ideal generated by closed sets of measure zero.

We prove that the structure $(\mathbb {Z},<,+,R)$ is distal for all congruence-periodic sparse predicates $R\subseteq \mathbb {N}$. We do so by constructing a strong honest definition for every formula $\phi (x;y)$ with $\lvert {x}\rvert =1$, providing a rare example of concrete distal decompositions.

In this work, we consider the ideals $m^0(\mathcal {I})$ and $\ell ^0(\mathcal {I})$, ideals generated by the $\mathcal {I}$-positive Miller trees and $\mathcal {I}$-positive Laver trees, respectively. We investigate in which cases these ideals have cofinality larger than $\mathfrak {c}$ and we calculate some cardinal invariants closely related to these ideals.

We show, assuming PD, that every complete finitely axiomatized second-order theory with a countable model is categorical, but that there is, assuming again PD, a complete recursively axiomatized second-order theory with a countable model which is non-categorical. We show that the existence of even very large (e.g., supercompact) cardinals does not imply the categoricity of all finitely axiomatizable complete second-order theories. More exactly, we show that a non-categorical complete finitely axiomatized second-order theory can always be obtained by (set) forcing. We also show that the categoricity of all finite complete second-order theories with a model of a certain singular cardinality $\kappa $ of uncountable cofinality can be forced over any model of set theory. Previously, Solovay had proved, assuming $V=L$, that every complete finitely axiomatized second-order theory (with or without a countable model) is categorical, and that in a generic extension of L there is a complete finitely axiomatized second-order theory with a countable model which is non-categorical.

Let $\mathsf {KP}$ denote Kripke–Platek Set Theory and let $\mathsf {M}$ be the weak set theory obtained from $\mathsf {ZF}$ by removing the collection scheme, restricting separation to $\Delta _0$-formulae and adding an axiom asserting that every set is contained in a transitive set ($\mathsf {TCo}$). A result due to Kaufmann [9] shows that every countable model, $\mathcal {M}$, of $\mathsf {KP}+\Pi _n\textsf {-Collection}$ has a proper $\Sigma _{n+1}$-elementary end extension. We show that for all $n \geq 1$, there exists an $L_\alpha $ (where $L_\alpha $ is the $\alpha ^{\textrm {th}}$ approximation of the constructible universe L) that satisfies $\textsf {Separation}$, $\textsf {Powerset}$ and $\Pi _n\textsf {-Collection}$, but that has no $\Sigma _{n+1}$-elementary end extension satisfying either $\Pi _n\textsf {-Collection}$ or $\Pi _{n+3}\textsf {-Foundation}$. Thus showing that there are limits to the amount of the theory of $\mathcal {M}$ that can be transferred to the end extensions that are guaranteed by Kaufmann’s theorem. Using admissible covers and the Barwise Compactness theorem, we show that if $\mathcal {M}$ is a countable model $\mathsf {KP}+\Pi _n\textsf {-Collection}+\Sigma _{n+1}\textsf {-Foundation}$ and T is a recursive theory that holds in $\mathcal {M}$, then there exists a proper $\Sigma _n$-elementary end extension of $\mathcal {M}$ that satisfies T. We use this result to show that the theory $\mathsf {M}+\Pi _n\textsf {-Collection}+\Pi _{n+1}\textsf {-Foundation}$ proves $\Sigma _{n+1}\textsf {-Separation}$.

We expand the study of generic stability in three different directions. Generic stability is best understood as a property of types in $NIP$ theories in classical logic. In this article, we make attempts to generalize our understanding to Keisler measures instead of types, arbitrary theories instead of $NIP$ theories, and continuous logic instead of classical logic. For this purpose, we study randomization of first-order structures/theories and modes of convergence of types/measures.

We prove that every $\Sigma ^0_2$ Gale-Stewart game can be won via a winning strategy $\tau $ which is $\Delta _1$-definable over $L_{\delta }$, the $\delta $th stage of Gödel’s constructible universe, where $\delta = \delta _{\sigma ^1_1}$, strengthening a theorem of Solovay from the 1970s. Moreover, the bound is sharp in the sense that there is a $\Sigma ^0_2$ game with no strategy $\tau $ which is witnessed to be winning by an element of $L_{\delta }$.

We prove a rigidity result for maps between Čech–Stone remainders under fairly mild forcing axioms.

Let $\mathcal {M}=(M,<,+, \dots )$ be a weakly o-minimal non-valuational structure expanding an ordered group. We show that the full first-order theory $\operatorname {\mathrm {Th}}(\mathcal {M})$ has definable Skolem functions if and only if isolated types in $S_{n}^{\mathcal M}(A)$ are dense for each $ A\subseteq M $ and $ n\in \mathbb {N} $. Using this, we prove that no strictly weakly o-minimal non-valuational expansion of an ordered group has definable Skolem functions, thereby answering Conjecture 1.7 of Eleftheriou et al. (On definable Skolem functions in weakly o-minimal non-valuational structures. J. Symb. Logic, vol. 82 (2017), no. 4).