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Regular version of the site

Article

Exercises de style: A homotopy theory for set theory

Israel Journal of Mathematics. 2015. Vol. 209.
Gavrilovich M., Hasson A.

We construct a model category (in the sense of Quillen) for set
theory, starting from two arbitrary, but natural, conventions. It is the simplest
category satisfying our conventions and modelling the notions of niteness,
countability and in nite equi-cardinality. We argue that from the homotopy
theoretic point of view our construction is essentially automatic following basic
existing methods, and so is (almost all) the veri cation that the construction
works.


We use the posetal model category to introduce homotopy-theoretic intu-
itions to set theory. Our main observation is that the homotopy invariant
version of cardinality is the covering number of Shelah's PCF theory, and
that other combinatorial objects, such as Shelah's revised power function -
the cardinal function featuring in Shelah's revised GCH theorem | can be
obtained using similar tools. We include a small \dictionary" for set theory in
QtNaamen, hoping it will help in nding more meaningful homotopy-theoretic
intuitions in set theory.