Submission date: 4 February 2023.

Abstract:

Motivated by ideas from the model theory of metric structures, we introduce a metric set theory, 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 φ(x,y), tuple of parameters a, and r < s, there is a set containing the class {x: φ(x,a) ≤ r} and contained in the class {x:φ(x,a) < s}). We show that 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 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 MSE in either continuous logic or Łukasiewicz logic.

Mathematics Subject Classification: 03E70, 03B50, 03C66

Keywords and phrases:

Full text arXiv 2302.02258: pdf, ps.