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).

This paper offers a substantial improvement in the revision-theoretic approach to conditionals in theories of transparent truth. The main modifications are (i) a new limit rule; (ii) a modification of the extension to the continuum-valued case; and (iii) the suggestion of a variation on how universal quantification is handled, leading to more satisfactory laws of restricted quantification.

We study universal-existential fragments of first-order theories of fields, in particular of function fields and of equicharacteristic henselian valued fields. For example, we discuss to what extent the theory of a field k determines the universal-existential theories of the rational function field over k and of the field of Laurent series over k, and we find various many-one reductions between such fragments.

Dedicated to the memory of Alexander Prestel (1941–2024)

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.

Two salient notions of sameness of theories are synonymy, aka definitional equivalence, and bi-interpretability. Of these two definitional equivalence is the strictest notion. In which cases can we infer synonymy from bi-interpretability? We study this question for the case of sequential theories. Our result is as follows. Suppose that two sequential theories are bi-interpretable and that the interpretations involved in the bi-interpretation are one-dimensional and identity preserving. Then, the theories are synonymous.

The crucial ingredient of our proof is a version of the Schröder–Bernstein theorem under very weak conditions. We think this last result has some independent interest.

We provide an example to show that this result is optimal. There are two finitely axiomatized sequential theories that are bi-interpretable but not synonymous, where precisely one of the interpretations involved in the bi-interpretation is not identity preserving.

Gentzen’s sequent calculi were a part of his consistency program, the ultimate aim of which was a proof of the consistency of analysis. Among Gentzen’s series of shorthand notes there was one titled WKR in which various sequent calculi and cut elimination procedures are examined. Nothing of this series has survived, but there is instead a late summary Gentzen wrote of it in 1944. In this article, these calculi and reductions are described in the context of Gentzen’s consistency program, followed by an English translation of his manuscript that is written in what is known as the unified German shorthand (einheitliche Kurzschrift).

We prove that P-points (even strong P-points) and Gruff ultrafilters exist in any forcing extension obtained by adding fewer than $\aleph _{\omega } $-many random reals to a model of CH.These results improve and correct previous theorems that can be found in the literature.

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.

Explanations, and in particular explanations which provide the reasons why their conclusion is true, are a central object in a range of fields. On the one hand, there is a long and illustrious philosophical tradition, which starts from Aristotle, and passes through scholars such as Leibniz, Bolzano and Frege, that give pride of place to this type of explanation, and is rich with brilliant and profound intuitions. Recently, Poggiolesi [25] has formalized ideas coming from this tradition using logical tools of proof theory. On the other hand, recent work has focused on Boolean circuits that compile some common machine learning classifiers and have the same input-output behavior. In this framework, Darwiche and Hirth [7] have proposed a theory for unveiling the reasons behind the decisions made by Boolean classifiers, and they have studied their theoretical implications. In this paper, we uncover the deep links behind these two trends, demonstrating that the proof-theoretic tools introduced by Poggiolesi provide reasons for decisions, in the sense of Darwiche and Hirth [7]. We discuss the conceptual as well as the technical significance of this result.

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 initiate the study of the spectrum of sets that can be realized as the vanishing levels $V(\mathbf T)$ of a normal $\kappa $-tree $\mathbf T$. This is an invariant in the sense that if $\mathbf T$ and $\mathbf T'$ are club-isomorphic, then $V(\mathbf T)\mathbin {\bigtriangleup } V(\mathbf T')$ is nonstationary. Additional features of this invariant imply that the spectrum is closed under finite unions and intersections. The set $V(\mathbf T)$ must be stationary for a homogeneous normal $\kappa $-Aronszajn tree $\mathbf T$, and if there exists a special $\kappa $-Aronszajn tree, then there exists one $\mathbf T$ that is homogeneous and satisfies that $V(\mathbf T)$ covers a club in $\kappa $. It is consistent (from large cardinals) that there is an $\aleph _2$-Souslin tree, and yet $V(\mathbf T)$ is co-stationary for every $\aleph _2$-tree $\mathbf T$. Both $V(\mathbf T)=\emptyset $ and $V(\mathbf T)=\kappa $ (modulo nonstationary) are shown to be feasible using $\kappa $-Souslin trees, even at some large cardinal close to a weakly compact. It is also possible to have a family of $2^\kappa $ many $\kappa $-Souslin trees for which the corresponding family of vanishing levels forms an antichain in the Boolean algebra of the powerset of $\kappa $, modulo the nonstationary ideal.

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 $\Sigma _{n+1}\textsf {-Separation}$.

This paper makes a twofold contribution to the study of expressivity. First, we introduce and study the novel concept of conditional expressivity. Taking a universal logic perspective, we characterize conditional expressivity both syntactically and semantically. We show that our concept of conditional expressivity is related to, but different from, the concept of explicit definability in Beth’s definability theorem. Second, we use the concept to explore inferential relations between collective deontic admissibility statements for different groups. Negative results on conditional expressivity are stronger than standard (unconditional) inexpressivity results: we show that the well-known inexpressivity results from epistemic logic on distributed knowledge and on common knowledge only concern unconditional expressivity. By contrast, we prove negative results on conditional expressivity in the deontic logic of collective agency. In particular, we consider the full formal language of the deontic logic of collective agency, define a natural class of sublanguages of the full language, and prove that a collective deontic admissibility statement about a particular group is conditionally expressible in a sublanguage from the class if and only if that sublanguage includes a collective deontic admissibility statement about a supergroup of that group. Our negative results on conditional expressivity may serve as a proof of concept for future studies.

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.