We shall observe that the notion of two sets being equal up to finitely many elements is a homotopy equivalence relation in a model category, a common axiomatic formalism for homotopy theory introduced by Quillen "to cover in a uniform way a large number of arguments in homotopy theories that were formally similar to well-known ones in algebraic topology. We show the same formalism covers some arguments in (naive) set theory, and naturally leads to define and consider a well-known set-theoretic invariant, the covering number \(\mathsf{cf}([\aleph_\omega]^{≤\aleph_0})=\mathsf{cov}(\aleph_ω,\aleph_1,\aleph_1,2)\), of PCF theory.

Further we observe a similarity between homotopy theory ideology/yoga and "artificially/naturality thesis" of Shelah (Logical Dreams, \(S5\)) claiming that "the various cofinalities are better measures" of size.

We shall argue that the formalism is curious as it suggests to look at a homotopy-invariant variant of Generalised Continuum Hypothesis about which more can be proven within ZFC and first appeared in PCF theory independently but with a similar motivation.

This is joint work with Assaf Hasson.