A semimeasure is a generalization of a probability measure obtained by relaxing the additivity requirement to superadditivity. We introduce and study several randomness notions for left-c.e. semimeasures, a natural class of effectively approximable semimeasures induced by Turing functionals. Among the randomness notions we consider, the generalization of weak $2$-randomness to left-c.e. semimeasures is the most compelling, as it best reflects Martin-Löf randomness with respect to a computable measure. Additionally, we analyze a question of Shen, a positive answer to which would also have yielded a reasonable randomness notion for left-c.e. semimeasures. Unfortunately, though, we find a negative answer, except for some special cases.

In this paper we first show that Robin Smith’s ecthetic system $\mathit{SE}$ for Aristotle’s assertoric syllogistic is not complete, despite what is claimed by Smith. $\mathit{SE}$ is then not adequate to establish that ecthesis allows one to dispense with indirect or per impossibile deductions in Aristotle’s assertoric logic. As an alternative to $\mathit{SE}$, we then present a stronger system $\mathit{EC}$ which is adequate for this purpose. $\mathit{EC}$ is a nonexplosive ecthetic system which is shown to be sound and complete with respect to all valid syllogistic arguments with a consistent set of premises.

Recently, the second author, Briseid, and Safarik introduced nonstandard Dialectica, a functional interpretation capable of eliminating instances of familiar principles of nonstandard arithmetic—including overspill, underspill, and generalizations to higher types—from proofs. We show that the properties of this interpretation are mirrored by first-order logic in a constructive sheaf model of nonstandard arithmetic due to Moerdijk, later developed by Palmgren, and draw some new connections between nonstandard principles and principles that are rejected by strict constructivism. Furthermore, we introduce a variant of the Diller–Nahm interpretation with two different kinds of quantifiers, similar to Hernest’s light Dialectica interpretation, and show that one can obtain nonstandard Dialectica by weakening the computational content of the existential quantifiers—a process called herbrandization. We also define a constructive sheaf model mirroring this new functional interpretation, and show that the process of herbrandization has a clear meaning in terms of these sheaf models.

We show that, under $\mathsf{PFA}\left(S\right)$, a coherent Suslin tree forces that every two Aronszajn trees are club-isomorphic.

We present a diamond principle ${◊}_{\mathbb{R}}$ concerning all subsets of $\Theta$, the supremum of the ordinals that are the surjective image of $\mathbb{R}$. We prove that ${◊}_{\mathbb{R}}$ holds in Steel’s core model $\mathbf{K}\left(\mathbb{R}\right)$, a canonical inner model for determinacy.

We give modest upper bounds for consistency strengths for two well-studied combinatorial principles. These bounds range at the level of subcompact cardinals, which is significantly below a ${\kappa }^{+}$-supercompact cardinal. All previously known upper bounds on these principles ranged at the level of some degree of supercompactness. We show that by using any of the standard modified Prikry forcings it is possible to turn a measurable subcompact cardinal into ${\aleph }_{\omega }$ and make the principle ${\square }_{{\aleph }_{\omega },<\omega }$ fail in the generic extension. We also show that by using Lévy collapse followed by standard iterated club shooting it is possible to turn a subcompact cardinal into ${\aleph }_2$ and arrange in the generic extension that simultaneous reflection holds at ${\aleph }_2$, and at the same time, every stationary subset of ${\aleph }_3$ concentrating on points of cofinality $\omega$ has a reflection point of cofinality ${\omega }_1$.

We identify complete fragments of the simple theory of types with infinity ($\mathrm{TSTI}$) and Quine’s new foundations ($\mathrm{NF}$) set theory. We show that $\mathrm{TSTI}$ decides every sentence $\varphi$ in the language of type theory that is in one of the following forms:

(A) $\varphi =\forall x_1^{r_1}\cdots \forall x_k^{r_k}\exists y_1^{s_1}\cdots \exists y_l^{s_l}\theta$ where the superscripts denote the types of the variables, $s_1>\cdots >s_l$, and $\theta$ is quantifier-free,

(B) $\varphi =\forall x_1^{r_1}\cdots \forall x_k^{r_k}\exists y_1^s\cdots \exists y_l^s\theta$ where the superscripts denote the types of the variables and $\theta$ is quantifier-free.

This shows that $\mathrm{NF}$ decides every stratified sentence $\varphi$ in the language of set theory that is in one of the following forms:

(A${}^{\prime }$) $\varphi =\forall x_1\cdots \forall x_k\exists y_1\cdots \exists y_l\theta$ where $\theta$ is quantifier-free and $\varphi$ admits a stratification that assigns distinct values to all of the variables $y_1,\dots ,y_l$,

(B${}^{\prime }$) $\varphi =\forall x_1\cdots \forall x_k\exists y_1\cdots \exists y_l\theta$ where $\theta$ is quantifier-free and $\varphi$ admits a stratification that assigns the same value to all of the variables $y_1,\dots ,y_l$.

Let $G$ be a group, and let $X$ be an infinite transitive $G$-space. A free ultrafilter $\mathcal{U}$ on $X$ is called $G$-selective if, for any $G$-invariant partition $\mathcal{P}$ of $X$, either one cell of $\mathcal{P}$ is a member of $\mathcal{U}$, or there is a member of $\mathcal{U}$ which meets each cell of $\mathcal{P}$ in at most one point. We show that in ZFC with no additional set-theoretical assumptions there exists a $G$-selective ultrafilter on $X$. We describe all $G$-spaces $X$ such that each free ultrafilter on $X$ is $G$-selective, and we prove that a free ultrafilter $\mathcal{U}$ on $\omega$ is selective if and only if $\mathcal{U}$ is $G$-selective with respect to the action of any countable group $G$ of permutations of $\omega$.

A free ultrafilter $\mathcal{U}$ on $X$ is called $G$-Ramsey if, for any $G$-invariant coloring $\chi :[X{]}^2\to \{0,1\}$, there is $U\in \mathcal{U}$ such that $[U{]}^2$ is $\chi$-monochromatic. We show that each $G$-Ramsey ultrafilter on $X$ is $G$-selective. Additional theorems give a lot of examples of ultrafilters on $\mathbb{Z}$ that are $\mathbb{Z}$-selective but not $\mathbb{Z}$-Ramsey.