In many simple integral domains, such as $\mathbb{Z}$ or $\mathbb{Z}\left[i\right]$, there is a straightforward procedure to determine if an element is prime by simply reducing to a direct check of finitely many potential divisors. Despite the fact that such a naive approach does not immediately translate to integral domains like $\mathbb{Z}\left[x\right]$ or the ring of integers in an algebraic number field, there still exist computational procedures that work to determine the prime elements in these cases. In contrast, we will show how to computably extend $\mathbb{Z}$ in such a way that we can control the ordinary integer primes in any ${\Pi }_2^0$ way, all while maintaining unique factorization. As a corollary, we establish the existence of a computable unique factorization domain (UFD) such that the set of primes is ${\Pi }_2^0$-complete in every computable presentation.

We prove that $(\mathbb{Z},+,0)$ has no proper superstable expansions of finite Lascar rank. Nevertheless, this structure equipped with a predicate defining powers of a given natural number is superstable of Lascar rank $\omega$. Additionally, our methods yield other superstable expansions such as $(\mathbb{Z},+,0)$ equipped with the set of factorial elements.

I consider the phenomenon of conflation—treating distinct things as one—and develop logical tools for modeling it. These tools involve a purely consequence-theoretic treatment, independent of any proof or model theory, as well as a four-valued valuational treatment.

We give two new characterizations of $K$-triviality. We show that if for all $Y$ such that $\Omega$ is $Y$-random, $\Omega$ is $\left(Y\oplus A\right)$-random, then $A$ is $K$-trivial. The other direction was proved by Stephan and Yu, giving us the first titular characterization of $K$-triviality and answering a question of Yu. We also prove that if $A$ is $K$-trivial, then for all $Y$ such that $\Omega$ is $Y$-random, $\left(Y\oplus A\right){\equiv }_{LR}Y$. This answers a question of Merkle and Yu. The other direction is immediate, so we have the second characterization of $K$-triviality.

The proof of the first characterization uses a new cupping result. We prove that if $A{\nleqq }_{LR}B$, then for every set $X$ there is a $B$-random set $Y$ such that $X$ is computable from $Y\oplus A$.

Problem 2 at the 56th International Mathematical Olympiad (2015) asks for all triples $(a,b,c)$ of positive integers for which $ab-c$, $bc-a$, and $ca-b$ are all powers of $2$. We show that this problem requires only a primitive form of arithmetic, going back to the Pythagoreans, which is the arithmetic of the even and the odd.

We show that in the Gromov space of isometry classes of pointed proper metric spaces, the equivalence relations defined by existence of coarse quasi-isometries or being at finite Gromov–Hausdorff distance cannot be reduced to the equivalence relation defined by any Polish action.

We present an elementary three-pass algorithm for computing addition in Ostrowski numeration systems. When $a$ is quadratic, addition in the Ostrowski numeration system based on $a$ is recognizable by a finite automaton. We deduce that a subset of $X\subseteq {\mathbb{N}}^n$ is definable in $(\mathbb{N},+,V_a)$, where $V_a$ is the function that maps a natural number $x$ to the smallest denominator of a convergent of $a$ that appears in the Ostrowski representation based on $a$ of $x$ with a nonzero coefficient if and only if the set of Ostrowski representations of elements of $X$ is recognizable by a finite automaton. The decidability of the theory of $(\mathbb{N},+,V_a)$ follows.

This article studies seriously actualistic quantified modal logics. A key component of the language is an abstraction operator by means of which predicates can be created out of complex formulas. This facilitates proof of a uniform substitution theorem: if a sentence is logically true, then any sentence that results from substituting a (perhaps complex) predicate abstract for each occurrence of a simple predicate abstract is also logically true. This solves a problem identified by Kripke early in the modern semantic study of quantified modal logic. A tableau proof system is presented and proved sound and complete with respect to logical truth. The main focus is on seriously actualistic T (SAT), an extension of T, but the results established hold also for systems based on other propositional modal logics (e.g., K, B, S4, and S5). Following Menzel it is shown that the formal language studied also supports an actualistic account of truth simpliciter.