Theoria, Volume 87, Issue 4, Page 971-985, August 2021.

A theory T is tight if different deductively closed extensions of T (in the same language) cannot be bi-interpretable. Many well-studied foundational theories are tight, including $\mathsf {PA}$ [39], $\mathsf {ZF}$, $\mathsf {Z}_2$, and $\mathsf {KM}$ [6]. In this article we extend Enayat’s investigations to subsystems of these latter two theories. We prove that restricting the Comprehension schema of $\mathsf {Z}_2$ and $\mathsf {KM}$ gives non-tight theories. Specifically, we show that $\mathsf {GB}$ and $\mathsf {ACA}_0$ each admit different bi-interpretable extensions, (...) and the same holds for their extensions by adding $\Sigma ^1_k$ -Comprehension, for $k \ge 1$. These results provide evidence that tightness characterizes $\mathsf {Z}_2$ and $\mathsf {KM}$ in a minimal way. (shrink)

This paper focuses on the categoricity of arithmetic and determinacy of arithmetical truth. Several ‘internal’ categoricity results have been discussed in the recent literature. Against the background of the philosophical position called internalism, we propose and investigate truth-theoretic versions of internal categoricity based on a primitive truth predicate. We argue for the compatibility of a primitive truth predicate with internalism and provide a novel argument for (and proof of) a truth-theoretic version of internal categoricity and internal determinacy with some positive (...) properties. (shrink)

In this article we present a notion of “logical perfection”. We first describe through examples a notion of logical perfection extracted from the contemporary logical concept of categoricity. Categoricity (in power) has become in the past half century a main driver of ideas in model theory, both mathematically (stability theory may be regarded as a way of approximating categoricity) and philosophically. In the past two decades, categoricity notions have started to overlap with more classical notions of robustness and smoothness. These (...) have been crucial in various parts of mathematics since the nineteenth century. We postulate and present the category of logical perfection. We draw on various notions of perfection from mathematics of the 19th and 20th centuries and then trace the relation to the concept of categoricity in power as a logical notion of what a “mathematically perfect” structure is. (shrink)