A homotopy approach to set theory

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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.