We prove a number of results concerning the variety of first-order theories and isomorphism types of pairs of the form $(N,M)$, where $N$ is a countable recursively saturated model of Peano Arithmetic and $M$ is its cofinal submodel. We identify two new isomorphism invariants for such pairs. In the strongest result we obtain continuum many theories of such pairs with the fixed greatest common initial segment of $N$ and $M$ and fixed lattice of interstructures $K$, such that $M\prec K\prec N$.

For any extension $T$ of $I{\Sigma }_1$ having a cut-elimination property extending that of $I{\Sigma }_1$, the number of different proofs that can be obtained by cut elimination from a single $T$-proof cannot be bound by a function which is provably total in $T$.

This paper develops a formal system, consisting of a language and semantics, called serial logic (SL). In rough outline, SL permits quantification over, and reference to, some finite number of things in an order, in an ordinary everyday sense of the word “order,” and superplural quantification over things thus ordered. Before we discuss SL itself, some mention should be made of an issue in philosophical logic which provides the background to the development of SL, and with respect to which I wish to contend that the system permits progress.

We prove that each countable rooted K4-frame is a d-morphic image of a subspace of the space $\mathbb{Q}$ of rational numbers. From this we derive that each modal logic over K4 axiomatizable by variable-free formulas is the d-logic of a subspace of $\mathbb{Q}$. It follows that subspaces of $\mathbb{Q}$ give rise to continuum many d-logics over K4, continuum many of which are neither finitely axiomatizable nor decidable. In addition, we exhibit several families of modal logics finitely axiomatizable by variable-free formulas over K4 that d-define interesting classes of topological spaces. Each of these logics has the finite model property and is decidable. Finally, we introduce quasi-scattered and semi-scattered spaces as generalizations of scattered spaces, develop their basic properties, axiomatize their corresponding modal logics, and show that they also arise as the d-logics of some subspaces of $\mathbb{Q}$.

V’yugin has shown that there are a computable shift-invariant measure on $2^{\mathbb{N}}$ and a simple function $f$ such that there is no computable bound on the rate of convergence of the ergodic averages $A_{n}f$. Here it is shown that in fact one can construct an example with the property that there is no computable bound on the complexity of the limit; that is, there is no computable bound on how complex a simple function needs to be to approximate the limit to within a given $\epsilon$.

We give upper and lower bounds for the strength of ordinal definable determinacy in a small admissible set. The upper bound is roughly a premouse with a measurable cardinal $\kappa$ of Mitchell order ${\kappa }^{++}$ and $\omega$ successors. The lower bound are models of ZFC with sequences of measurable cardinals, extending the work of Lewis, below a regular limit of measurable cardinals.

We study the reverse mathematics of pigeonhole principles for finite powers of the ordinal $\omega$. Four natural formulations are presented, and their relative strengths are compared. In the analysis of the pigeonhole principle for ${\omega }^2$, we uncover two weak variants of Ramsey’s theorem for pairs.

Given an ideal $I$, let ${\mathbb{P}}_I$ denote the forcing with $I$-positive sets. We consider models of forcing axioms $MA(\Gamma )$ which also have a normal ideal $I$ with completeness ${\omega }_2$ such that ${\mathbb{P}}_I\in \Gamma$. Using a bit more than a superhuge cardinal, we produce a model of PFA (proper forcing axiom) which has many ideals on ${\omega }_2$ whose associated forcings are proper; a similar phenomenon is also observed in the standard model of $M{A}^{+{\omega }_1}\left(\sigma \text{-closed}\right)$ obtained from a supercompact cardinal. Our model of PFA also exhibits weaker versions of ideal properties, which were shown by Foreman and Magidor to be inconsistent with PFA.

Along the way, we also show (1) the diagonal reflection principle for internally club sets ($\mathit{DRP}\left(I{C}_{{\omega }_1}\right)$) introduced by the author in earlier work is equivalent to a natural weakening of “there is an ideal $I$ such that ${\mathbb{P}}_I$ is proper”; and (2) for many natural classes $\Gamma$ of posets, $M{A}^{+{\omega }_1}(\Gamma )$ is equivalent to an apparently stronger version which we call $M{A}^{+Diag}(\Gamma )$.

We expand the notion of resplendency to theories of the kind $T+\mathrm{p\uparrow }$, where $T$ is a first-order theory and $\mathrm{p\uparrow }$ expresses that the type $p$ is omitted; both $T$ and $p$ are in languages extending the base language. We investigate two different formulations and prove necessary and sufficient conditions for countable recursively saturated models of PA.