A metric set theory with a universal set

TL;DR

This paper introduces a new metric set theory, , inspired by model theory of metric structures, featuring bounded quantification, a metric extensionality axiom, and an approximate comprehension scheme, enabling development of classical mathematics with various models.

Contribution

It presents , a novel metric set theory with bounded quantification, a Hausdorff distance-based extensionality axiom, and an approximate comprehension scheme, expanding the foundations of metric set theory.

Findings

 suffices to develop classical mathematics with an added infinity axiom.

Construction of canonical representatives of well-order types.

Existence of ultrametric models with externally ill-founded ordinals.

Abstract

Motivated by ideas from the model theory of metric structures, we introduce a metric set theory, $\mathsf{MSE}$, which takes bounded quantification as primitive and consists of a natural metric extensionality axiom (the distance between two sets is the Hausdorff distance between their extensions) and an approximate, non-deterministic form of full comprehension (for any real-valued formula $\varphi(x,y)$, tuple of parameters $a$, and $r < s$, there is a set containing the class $\{x: \varphi(x,a) \leq r\}$ and contained in the class $\{x:\varphi(x,a) < s\}$). We show that $\mathsf{MSE}$ is sufficient to develop classical mathematics after the addition of an appropriate axiom of infinity. We then construct canonical representatives of well-order types and prove that ultrametric models of $\mathsf{MSE}$ always contain externally ill-founded ordinals, conjecturing that this is true of all models. To establish several independence results and, in particular, consistency, we construct a variety of models, including pseudo-finite models and models containing arbitrarily large standard ordinals. Finally, we discuss how to formalize $\mathsf{MSE}$ in either continuous logic or {\L}ukasiewicz logic.