Fix $2<n<\omega$. Let ${\mathsf{CA}}_n$ denote the class of cylindric algebras of dimension $n$, and let ${\mathbf{RCA}}_n$ denote the variety of representable ${\mathsf{CA}}_n$’s. Let $L_n$ denote first-order logic restricted to the first $n$ variables. Roughly, ${\mathsf{CA}}_n$, an instance of Boolean algebras with operators, is the algebraic counterpart of the syntax of $L_n$, namely, its proof theory, while ${\mathbf{RCA}}_n$ algebraically and geometrically represents the Tarskian semantics of $L_n$. Unlike Boolean algebras having a Stone representation theorem, ${\mathbf{RCA}}_n⊊{\mathsf{CA}}_n$. Using combinatorial game theory, we show that the existence of certain finite relation algebras $\mathsf{RA}$, which are algebras whose domain consists of binary relations, implies that the celebrated Henkin omitting types theorem fails in a very strong sense for $L_n$. Using special cases of such finite $\mathsf{RA}$’s, we recover the classical nonfinite axiomatizability results of Monk, Maddux, and Biro on ${\mathbf{RCA}}_n$ and we re-prove Hirsch and Hodkinson’s result that the class of completely representable ${\mathsf{CA}}_n$’s is not first-order definable. We show that if $T$ is an $L_n$ countable theory that admits elimination of quantifiers, $\lambda$ is a cardinal $<2^{{\aleph }_0}$, and $\mathbf{F}=\langle {\Gamma }_i:i<\lambda \rangle$ is a family of complete nonprincipal types, then $\mathbf{F}$ can be omitted in an ordinary countable model of $T$.

We show that for any $\omega$-r.e. degree $\mathit{d}$ and $n$-r.e. degree $\mathit{b}$ with $\mathit{d}<\mathit{b}$, there is an $(\omega +1)$-r.e. degree $\mathit{a}$ strictly between $\mathit{d}$ and $\mathit{b}$. We also show that there is a maximal incomplete $(\omega +1)$-r.e. degree. As a corollary, ${\mathcal{D}}_{\omega }$ is not a ${\Sigma }_1$-elementary substructure of ${\mathcal{D}}_{\omega +1}$.

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 the study of the topological notions of dimension and paracompactness, inside Kohlenbach’s higher-order RM. 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 $C$, and the aforementioned topological notions give rise to a number of splittings involving highly natural $A$, $B$, $C$. Nonetheless, the higher-order picture is markedly different from the second-one: in terms of comprehension axioms, the proof in higher-order RM of, for example, the paracompactness of the unit interval requires full second-order arithmetic, while the second-order/countable version of paracompactness of the unit interval is provable in the base theory ${\mathsf{RCA}}_0$. We obtain similarly “exceptional” results for the Urysohn identity, the Lindelöf lemma, and partitions of unity. We show that our results exhibit a certain robustness, in that they do not depend on the exact definition of cover, even in the absence of the axiom of choice.

Linnebo and Shapiro have recently given an analysis of potential infinity using modal logic. A key technical component of their account is to show that under a suitable translation ${}^{◊}$ of nonmodal language into modal language, nonmodal sentences ${\varphi }_1,\dots ,{\varphi }_n$ entail $\psi$ just in case ${\varphi }_1^{◊},\dots ,{\varphi }_n^{◊}$ entail ${\psi }^{◊}$ in the modal logic S4.2. Linnebo and Shapiro establish this result in nonfree logic. In this note I argue that their analysis of potential infinity should be carried out in a free logic. I then extend their key theorems to the setting of negative free logic.

Let $M_n^{♯}\left(\mathbb{R}\right)$ denote the minimal active iterable extender model which has $n$ Woodin cardinals and contains all reals, if it exists, in which case we denote by $M_n\left(\mathbb{R}\right)$ the class-sized model obtained by iterating the topmost measure of $M_n\left(\mathbb{R}\right)$ class-many times. We characterize the sets of reals which are ${\Sigma }_1$-definable from $\mathbb{R}$ over $M_n\left(\mathbb{R}\right)$, under the assumption that projective games on reals are determined:

1. for even $n$, ${\Sigma }_1^{M_n\left(\mathbb{R}\right)}={⅁}^{\mathbb{R}}{\Pi }_{n+1}^1$;

2. for odd $n$, ${\Sigma }_1^{M_n\left(\mathbb{R}\right)}={⅁}^{\mathbb{R}}{\Sigma }_{n+1}^1$.

This generalizes a theorem of Martin and Steel for $L\left(\mathbb{R}\right)$, that is, the case $n=0$. As consequences of the proof, we see that determinacy of all projective games with moves in $\mathbb{R}$ is equivalent to the statement that $M_n^{♯}\left(\mathbb{R}\right)$ exists for all $n\in \mathbb{N}$, and that determinacy of all projective games of length ${\omega }^2$ with moves in $\mathbb{N}$ is equivalent to the statement that $M_n^{♯}\left(\mathbb{R}\right)$ exists and satisfies $\mathsf{AD}$ for all $n\in \mathbb{N}$.

Barendregt gave a sound semantics of the simple type assignment system $\lambda \to$ by generalizing Tait’s proof of the strong normalization theorem. In this paper, we aim to extend the semantics so that the completeness theorem holds.

The four-valued semantics of Belnap–Dunn logic, consisting of the truth values True, False, Neither, and Both, gives rise to several nonclassical logics depending on which feature of propositions we wish to preserve: truth, nonfalsity, or exact truth (truth and nonfalsity). Interpreting equality of truth values in this semantics as material equivalence of propositions, we can moreover see the equational consequence relation of this four-element algebra as a logic of material equivalence. In this paper, we axiomatize all combinations of these four-valued logics, for example, the logic of truth and exact truth or the logic of truth and material equivalence. These combined systems are consequence relations which allow us to express implications involving more than one of these features of propositions.

The study of existentially closed closure algebras begins with Lipparini’s 1982 paper. After presenting new nonelementary axioms for algebraically closed and existentially closed closure algebras and showing that these nonelementary classes are different, this paper shows that the classes of finitely generic and infinitely generic closure algebras are closed under finite products and bounded Boolean powers, extends part of Hausdorff’s theory of reducible sets to existentially closed closure algebras, and shows that finitely generic and infinitely generic closure algebras are elementarily inequivalent. Special properties of algebraically closed (a.c.), existentially closed (e.c.), finitely generic (f.g.), and infinitely generic (i.g.) closure algebras are established along the way.