Abstract
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 finiteness, countability and infinite 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 verification that the construction works.
We use the posetal model category to introduce homotopy-theoretic intuitions 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 finding more meaningful homotopy-theoretic intuitions in set theory.
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This paper is dedicated to the memory of Nikolai Aleksandrovich Shanin (25 May 1919 - 17 September 2011). He founded the St. Petersburg School of Logic and was an a honorable man.
The first author was partially supported by a MODNET (European Commission Research Training Network) grant and by the Skirball foundation as a postdoctoral fellow at Ben Gurion University.
The second author was partially supported by GIF grant No. 2266/2010 and by ISF grant No. 1156/10.
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Gavrilovich, M., Hasson, A. Exercices de style: A homotopy theory for set theory. Isr. J. Math. 209, 15–83 (2015). https://doi.org/10.1007/s11856-015-1211-7
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DOI: https://doi.org/10.1007/s11856-015-1211-7