A partial ordering $\mathbb{P}$ is $n$-Ramsey if, for every coloring of $n$-element chains from $\mathbb{P}$ in finitely many colors, $\mathbb{P}$ has a homogeneous subordering isomorphic to $\mathbb{P}$. In their paper on Ramsey properties of the complete binary tree, Chubb, Hirst, and McNicholl ask about Ramsey properties of other partial orderings. They also ask whether there is some Ramsey property for pairs equivalent to ${\mathit{ACA}}_0$ over ${\mathit{RCA}}_0$.

A characterization theorem for finite-level partial orderings with Ramsey properties has been proven by the second author. We show, over ${\mathit{RCA}}_0$, that one direction of the equivalence given by this theorem is equivalent to ${\mathit{ACA}}_0$ (for $n\ge 3$), and the other is provable in ${\mathit{ATR}}_0$.

We answer Chubb, Hirst, and McNicholl’s second question by showing that there is a primitive recursive partial ordering $\mathbb{P}$ such that, over ${\mathit{RCA}}_0$, “$\mathbb{P}$ is 2-Ramsey” is equivalent to ${\mathit{ACA}}_0$.

We use methods of reverse mathematics to analyze the proof theoretic strength of a theorem involving the notion of coloring number. Classically, the coloring number of a graph $G=(V,E)$ is the least cardinal $\kappa$ such that there is a well-ordering of $V$ for which below any vertex in $V$ there are fewer than $\kappa$ many vertices connected to it by $E$. We will study a theorem due to Komjáth and Milner, stating that if a graph is the union of $n$ forests, then the coloring number of the graph is at most $2n$. We focus on the case when $n=1$.

For a computable structure $\mathcal{M}$, the categoricity spectrum is the set of all Turing degrees capable of computing isomorphisms among arbitrary computable copies of $\mathcal{M}$. If the spectrum has a least degree, this degree is called the degree of categoricity of $\mathcal{M}$. In this paper we investigate spectra of categoricity for computable rigid structures. In particular, we give examples of rigid structures without degrees of categoricity.

Let $T$ be a theory in a countable fragment of ${\mathcal{L}}_{{\omega }_1,\omega }$ whose extensions in countable fragments have only countably many types. Sacks proves a bounding theorem that generates models of high Scott rank. For this theorem, a tree hierarchy is developed for $T$ that enumerates these extensions.

In this paper, we effectively construct a predecessor function for formulas defining types in this tree hierarchy as follows. Let $T_{\gamma }\subseteq T_{\delta }$ with $T_{\gamma }$- and $T_{\delta }$-theories on level $\gamma$ and $\delta$, respectively. Then if $⌜p⌝\left(T_{\delta }\right)$ is a formula that defines a type for $T_{\delta }$, our predecessor function provides a formula for defining its subtype in $T_{\gamma }$.

By constructing this predecessor function, we weaken an assumption for Sacks’s result.

This paper presents a refinement of a result by Conidis, who proved that there is a real $X$ of effective packing dimension $0<\alpha <1$ which cannot compute any real of effective packing dimension $1$. The original construction was carried out below $\varnothing \mathrm{\text{'}\text{'}}$, and this paper’s result is an improvement in the effectiveness of the argument, constructing such an $X$ by a limit-computable approximation to get $X{\le }_T\varnothing \text{'}$.

It is observed that a consistent congruential modal logic is not guaranteed to have a consistent extension in which the Box operator becomes a truth-functional connective for one of the four one-place (two-valued) truth functions.

We explore the consequences, for logical system-building, of taking seriously (i) the aim of having irredundant rules of inference, and (ii) a preference for proofs of stronger results over proofs of weaker ones. This leads one to reconsider the structural rules of REFLEXIVITY, THINNING, and CUT.

REFLEXIVITY survives in the minimally necessary form $\phi :\phi$. Proofs have to get started.

CUT is subject to a CUT-elimination theorem, to the effect that one can always make do without applications of CUT. So CUT is redundant, and should not be a rule of the system.

CUT-elimination, however, in the context of the usual forms of logical rules, requires the presence, in the system, of THINNING. But THINNING, it turns out, is not really necessary. Provided only that one liberalizes the statement of certain logical rules in an appropriate way, one can make do without CUT or THINNING. From the methodological point of view of this study, the logical rules ought to be framed in this newly liberalized form. These liberalized logical rules determine the system of core logic.

Given any intuitionistic Gentzen proof of $\Delta :\phi$, one can determine from it a Core proof of some subsequent of $\Delta :\phi$. Given any classical Gentzen proof of $\Delta :\phi$, one can determine from it a classical Core proof of some subsequent of $\Delta :\phi$. In both cases the Core proof is of a result at least as strong as that of the Gentzen proof; and the only structural rule used is $\phi :\phi$.

In this paper we discuss automorphism groups of saturated models and boundedly saturated models of $\mathsf{PA}$. We show that there are saturated models of $\mathsf{PA}$ of the same cardinality with nonisomorphic automorphism groups. We then show that every saturated model of $\mathsf{PA}$ has short saturated elementary cuts with nonisomorphic automorphism groups.

We generalize the double negation construction of Boolean algebras in Heyting algebras to a double negation construction of the same in Visser algebras (also known as basic algebras). This result allows us to generalize Glivenko’s theorem from intuitionistic propositional logic and Heyting algebras to Visser’s basic propositional logic and Visser algebras.