We show, assuming PD, that every complete finitely axiomatized second-order theory with a countable model is categorical, but that there is, assuming again PD, a complete recursively axiomatized second-order theory with a countable model which is non-categorical. We show that the existence of even very large (e.g., supercompact) cardinals does not imply the categoricity of all finitely axiomatizable complete second-order theories. More exactly, we show that a non-categorical complete finitely axiomatized second-order theory can always be obtained by (set) forcing. We also show that the categoricity of all finite complete second-order theories with a model of a certain singular cardinality $\kappa $ of uncountable cofinality can be forced over any model of set theory. Previously, Solovay had proved, assuming $V=L$, that every complete finitely axiomatized second-order theory (with or without a countable model) is categorical, and that in a generic extension of L there is a complete finitely axiomatized second-order theory with a countable model which is non-categorical.

Let $\mathsf {KP}$ denote Kripke–Platek Set Theory and let $\mathsf {M}$ be the weak set theory obtained from $\mathsf {ZF}$ by removing the collection scheme, restricting separation to $\Delta _0$-formulae and adding an axiom asserting that every set is contained in a transitive set ($\mathsf {TCo}$). A result due to Kaufmann [9] shows that every countable model, $\mathcal {M}$, of $\mathsf {KP}+\Pi _n\textsf {-Collection}$ has a proper $\Sigma _{n+1}$-elementary end extension. We show that for all $n \geq 1$, there exists an $L_\alpha $ (where $L_\alpha $ is the $\alpha ^{\textrm {th}}$ approximation of the constructible universe L) that satisfies $\textsf {Separation}$, $\textsf {Powerset}$ and $\Pi _n\textsf {-Collection}$, but that has no $\Sigma _{n+1}$-elementary end extension satisfying either $\Pi _n\textsf {-Collection}$ or $\Pi _{n+3}\textsf {-Foundation}$. Thus showing that there are limits to the amount of the theory of $\mathcal {M}$ that can be transferred to the end extensions that are guaranteed by Kaufmann’s theorem. Using admissible covers and the Barwise Compactness theorem, we show that if $\mathcal {M}$ is a countable model $\mathsf {KP}+\Pi _n\textsf {-Collection}+\Sigma _{n+1}\textsf {-Foundation}$ and T is a recursive theory that holds in $\mathcal {M}$, then there exists a proper $\Sigma _n$-elementary end extension of $\mathcal {M}$ that satisfies T. We use this result to show that the theory $\mathsf {M}+\Pi _n\textsf {-Collection}+\Pi _{n+1}\textsf {-Foundation}$ proves $\Sigma _{n+1}\textsf {-Separation}$.

We prove that every $\Sigma ^0_2$ Gale-Stewart game can be won via a winning strategy $\tau $ which is $\Delta _1$-definable over $L_{\delta }$, the $\delta $th stage of Gödel’s constructible universe, where $\delta = \delta _{\sigma ^1_1}$, strengthening a theorem of Solovay from the 1970s. Moreover, the bound is sharp in the sense that there is a $\Sigma ^0_2$ game with no strategy $\tau $ which is witnessed to be winning by an element of $L_{\delta }$.

Answering a question of Kaye, we show that the compositional truth theory with the full collection scheme is conservative over Peano Arithmetic. We demonstrate it by showing that countable models of compositional truth which satisfy the internal induction or collection axioms can be end-extended to models of the respective theory.

We investigate the primitive recursive content of linear orders. We prove that the punctual degrees of rigid linear orders, the order of the integers $\mathbb {Z}$, and the order of the rationals $\mathbb {Q}$ embed the diamond (preserving supremum and infimum). In the cases of rigid orders and the order $\mathbb {Z}$, we further extend the result to embed the atomless Boolean algebra; we leave the case of $\mathbb {Q}$ as an open problem. We also show that our results for the rigid orders, in fact, work for orders having a computable infinite invariant rigid sub-order.

We provide a complete characterization of theories of tracial von Neumann algebras that admit quantifier elimination. We also show that the theory of a separable tracial von Neumann algebra $\mathcal {M}$ is never model complete if its direct integral decomposition contains $\mathrm {II}_1$ factors $\mathcal {N}$ such that $M_2(\mathcal {N})$ embeds into an ultrapower of $\mathcal {N}$. The proof in the case of $\mathrm {II}_1$ factors uses an explicit construction based on random matrices and quantum expanders.