In a relational language consisting of a single relation $R$, we investigate pseudofiniteness of certain Hrushovski constructions obtained via predimension functions. It is notable that the arity of the relation $R$ plays a crucial role in this context. When $R$ is ternary, by extending the methods recently developed by Brody and Laskowski, we interpret $\langle {\mathbb{Q}}^{+},<\rangle$ in the $\langle {\mathcal{K}}^{+},{\le }^{\ast }\rangle$-generic and prove that this structure is not pseudofinite. This provides a negative answer to the question posed in an earlier work by Evans and Wong. This result, in fact, unfolds another aspect of complexity of this structure, along with undecidability and the strict order property proved in the mentioned earlier works. On the other hand, when $R$ is binary, it can be shown that the $\langle {\mathcal{K}}^{+},{\le }^{\ast }\rangle$-generic is decidable and pseudofinite.

We show that propositional intuitionistic logic is the maximal (with respect to expressive power) abstract logic satisfying a certain form of compactness, the Tarski union property (TUP), and preservation under asimulations.

Let $Con(\mathbf{T})\upharpoonright x$ denote the finite consistency statement “there are no proofs of contradiction in $\mathbf{T}$ with $\le x$ symbols.” For a large class of natural theories $\mathbf{T}$, Pudlák has shown that the lengths of the shortest proofs of $Con(\mathbf{T})\upharpoonright n$ in the theory $\mathbf{T}$ itself are bounded by a polynomial in $n$. At the same time he conjectures that $\mathbf{T}$ does not have polynomial proofs of the finite consistency statements $Con(\mathbf{T}+Con(\mathbf{T}))\upharpoonright n$. In contrast, we show that Peano arithmetic ($\mathbf{PA}$) has polynomial proofs of $Con(\mathbf{PA}+{Con}^{\ast }(\mathbf{PA}))\upharpoonright n$, where ${Con}^{\ast }(\mathbf{PA})$ is the slow consistency statement for Peano arithmetic, introduced by S.-D. Friedman, Rathjen, and Weiermann. We also obtain a new proof of the result that the usual consistency statement $Con(\mathbf{PA})$ is equivalent to ${\epsilon }_0$ iterations of slow consistency. Our argument is proof-theoretic, whereas previous investigations of slow consistency relied on nonstandard models of arithmetic.

Reverse mathematics (RM hereafter) is a program in the foundations of mathematics founded by Friedman and developed extensively by Simpson and others. The aim of RM is to find the minimal axioms needed to prove a theorem of ordinary, that is, non-set-theoretic, mathematics. As suggested by the title, this paper deals with two (relatively rare) RM-phenomena, namely, splittings and disjunctions. As to splittings, there are some examples in RM of theorems $A$, $B$, $C$ such that $A\leftrightarrow (B\wedge C)$, that is, $A$ can be split into two independent (fairly natural) parts $B$ and $C$. As to disjunctions, there are (very few) examples in RM of theorems $D$, $E$, $F$ such that $D\leftrightarrow \left(E\vee F\right)$, that is, $D$ can be written as the disjunction of two independent (fairly natural) parts $E$ and $F$. By contrast, we show in this paper that there is a plethora of (natural) splittings and disjunctions in Kohlenbach’s higher-order RM.

In recent years, the logic of questions and dependencies has been investigated in the closely related frameworks of inquisitive logic and dependence logic. These investigations have assumed classical logic as the background logic of statements, and added formulas expressing questions and dependencies to this classical core. In this paper, we broaden the scope of these investigations by studying questions and dependency in the context of intuitionistic logic. We propose an intuitionistic team semantics, where teams are embedded within intuitionistic Kripke models. The associated logic is a conservative extension of intuitionistic logic with questions and dependence formulas. We establish a number of results about this logic, including a normal form result, a completeness result, and translations to classical inquisitive logic and modal dependence logic.

We propose a canonization scheme for smooth equivalence relations on ${\mathbb{R}}^{\omega }$ modulo restriction to ${\mathsf{E}}_0$-large infinite products. It shows that, given a pair of Borel smooth equivalence relations $\mathsf{E}$, $\mathsf{F}$ on ${\mathbb{R}}^{\omega }$, there is an infinite ${\mathsf{E}}_0$-large perfect product $P\subseteq {\mathbb{R}}^{\omega }$ such that either $\mathsf{F}\subseteq \mathsf{E}$ on $P$, or, for some $\ell <\omega$, the following is true for all $x,y\in P$: $x\mathsf{E}y$ implies $x\left(\ell \right)=y\left(\ell \right)$, and $x\upharpoonright (\omega \setminus \{\ell \})=y\upharpoonright (\omega \setminus \{\ell \})$ implies $x\mathsf{F}y$.

In recent years, some authors have proposed quantitative measures of the coherence of sets of propositions. Such probabilistic measures of coherence (PMCs) are, in general terms, functions that take as their argument a set of propositions (along with some probability distribution) and yield as their value a number that is supposed to represent the degree of coherence of the set. In this paper, I introduce a minimal constraint on PMC theories, the weak stability principle, and show that any correct, coherent, and complete PMC cannot satisfy it. As a matter of fact, the argument offered in this paper can be applied to any coherence theory that uses a priori procedures. I briefly explore some consequences of this fact.

This paper studies two dual notions in module theory—namely, radicals and socles—from the standpoint of reverse mathematics. We first consider radicals of $\mathbb{Z}$-modules, where the radical of a $\mathbb{Z}$-module $M$ is defined as the intersection of $pM=\{px:x\in M\}$ with $p$ taken from all primes. It shows that ${\mathrm{ACA}}_0$ is equivalent to the existence of radicals of $\mathbb{Z}$-modules over ${\mathrm{RCA}}_0$. We then study socles of modules over commutative rings with identity. The socle of an $R$-module $M$ is the largest semisimple submodule of $M$. We show that the existence of socles of modules over a commutative ring with identity is equivalent to ${\mathrm{ACA}}_0$ over ${\mathrm{RCA}}_0$. Vector spaces are semisimple modules over fields. In general, semisimple modules possess nice properties of vector spaces. Lastly, we study characterizations of semisimple modules over commutative rings using techniques of reverse mathematics.

Turing progressions arise by iteratedly adding consistency statements to a base theory. Different notions of consistency give rise to different Turing progressions. In this paper we present a logic that generates exactly all relations that hold between these different Turing progressions given a particular set of natural consistency notions. Thus, the presented logic is proven to be arithmetically sound and complete for a natural interpretation, named the formalized Turing progressions (FTP) interpretation.