# 2010 seminar talk: A homotopy approach to some questions in set theory

Talk held by Misha Gavrilovich (KGRC) at the KGRC seminar on 2010-11-18.

### Abstract

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 *cf([ℵ _{ω}]^{≤ℵ0})=*cov

*(ℵ*, of PCF theory.

_{ω},ℵ_{1},ℵ_{1},2)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.