We show that from a supercompact cardinal $\kappa$, there is a forcing extension $V\left[G\right]$ that has a symmetric inner model $N$ in which $\mathrm{ZF}+\neg \mathrm{AC}$ holds, $\kappa$ and ${\kappa }^{+}$ are both singular, and the continuum function at $\kappa$ can be precisely controlled, in the sense that the final model contains a sequence of distinct subsets of $\kappa$ of length equal to any predetermined ordinal. We also show that the above situation can be collapsed to obtain a model of $\mathrm{ZF}+\neg {\mathrm{AC}}_{\omega }$ in which either (1) ${\aleph }_1$ and ${\aleph }_2$ are both singular and the continuum function at ${\aleph }_1$ can be precisely controlled, or (2) ${\aleph }_{\omega }$ and ${\aleph }_{\omega +1}$ are both singular and the continuum function at ${\aleph }_{\omega }$ can be precisely controlled. Additionally, we discuss a result in which we separate the lengths of sequences of distinct subsets of consecutive singular cardinals $\kappa$ and ${\kappa }^{+}$ in a model of $\mathrm{ZF}$. Some open questions concerning the continuum function in models of $\mathrm{ZF}$ with consecutive singular cardinals are posed.

For sentences $\varphi$ of $L_{{\omega }_1,\omega }$, we investigate the question of absoluteness of $\varphi$ having models in uncountable cardinalities. We first observe that having a model in ${\aleph }_1$ is an absolute property, but having a model in ${\aleph }_2$ is not as it may depend on the validity of the continuum hypothesis. We then consider the generalized continuum hypothesis (GCH) context and provide sentences for any $\alpha \in {\omega }_1\setminus \{0,1,\omega \}$ for which the existence of a model in ${\aleph }_{\alpha }$ is nonabsolute (relative to large cardinal hypotheses). Finally, we present a complete sentence for which model existence in ${\aleph }_3$ is nonabsolute.

We introduce a family of rank functions and related notions of total transcendence for Galois types in abstract elementary classes. We focus, in particular, on abstract elementary classes satisfying the condition known as tameness, where the connections between stability and total transcendence are most evident. As a byproduct, we obtain a partial upward stability transfer result for tame abstract elementary classes stable in a cardinal $\lambda$ satisfying ${\lambda }^{{\aleph }_0}>\lambda$, a substantial generalization of a result of Baldwin, Kueker, and VanDieren.

If $\mathbf{S}$ is a semilattice with operators, then there is an implicational theory $\mathcal{Q}$ such that the congruence lattice $Con\left(\mathbf{S}\right)$ is isomorphic to the lattice of all implicational theories containing $\mathcal{Q}$.

Through contact algebras we study theories of mereotopology in a uniform way that clearly separates mereological from topological concepts. We identify and axiomatize an important subclass of closure mereotopologies (CMT) called unique closure mereotopologies (UCMTs) whose models always have orthocomplemented contact algebras (OCAs), an algebraic counterpart. The notion of MT-representability, a weak form of spatial representability but stronger than topological representability, suffices to prove that spatially representable complete OCAs are pseudocomplemented and satisfy the Stone identity. Within the resulting class of contact algebras the strength of the algebraic complementation delineates two classes of mereotopology according to the key ontological choice between mereological and topological closure operations. All closure operations are defined mereologically if and only if the corresponding contact algebras are uniquely complemented while topological closure operations highly restrict the contact relation but allow not uniquely complemented and nondistributive contact algebras. Each class contains a single ontologically coherent theory that admits discrete models.

We study arithmetic and hyperarithmetic degrees of categoricity. We extend a result of E. Fokina, I. Kalimullin, and R. Miller to show that for every computable ordinal $\alpha$, ${\mathbf{0}}^{\left(\alpha \right)}$ is the degree of categoricity of some computable structure $\mathcal{A}$. We show additionally that for $\alpha$ a computable successor ordinal, every degree $2$-c.e. in and above ${\mathbf{0}}^{\left(\alpha \right)}$ is a degree of categoricity. We further prove that every degree of categoricity is hyperarithmetic and show that the index set of structures with degrees of categoricity is ${\Pi }_1^1$-complete.

We show that the fact that the first player (“white”) wins every instance of Galvin’s “racing pawns” game (for countable trees) is equivalent to arithmetic transfinite recursion. Along the way we analyze the satisfaction relation for infinitary formulas, of “internal” hyperarithmetic comprehension, and of the law of excluded middle for such formulas.

We develop a semantics for independence logic with respect to what we will call general models. We then introduce a simpler entailment semantics for the same logic, and we reduce the validity problem in the former to the validity problem in the latter. Then we build a proof system for independence logic and prove its soundness and completeness with respect to entailment semantics.