A homotopy approach to set theory

TL;DR

This paper introduces a homotopy-theoretic perspective on set theory, proposing a homotopy-invariant version of the Generalised Continuum Hypothesis and exploring analogies with homotopy theory, PCF theory, and potential links to model theory and geometry.

Contribution

It presents a novel homotopy-theoretic framework for set theory, connecting it with homotopy equivalences and proposing a new variant of GCH that aligns with PCF theory.

Findings

Sets equal up to finitely many elements form a homotopy equivalence relation

A homotopy-invariant version of GCH is proposed and analyzed

Analogies between set theory, homotopy theory, and PCF theory are established

Abstract

We observe that the notion of two sets being equal up to finitely many elements is a homotopy equivalence relation in a model category, and suggest a homotopy-invariant variant of Generalised Continuum Hypothesis about which more can be proven within ZFC and which first appeared in PCF theory. The formalism allows to draw analogies between notions of set theory and those of homotopy theory, and we indeed observe a similarity between homotopy theory ideology/yoga and that of PCF theory. We also briefly discuss conjectural connections with model theory and arithmetics and geometry.