Joshua Brot, Mengyang Cao, David J. Fernández-Bretón: Finiteness classes arising from Ramsey-theoretic statements in set theory without choice

Thursday, June 14, 2018, from 4 to 5:30pm
East Hall, room 4096

Speaker: Joshua Brot, Mengyang Cao, David J. Fernández-Bretón (University of Michigan)

Title: Finiteness classes arising from Ramsey-theoretic statements in set theory without choice

Abstract:

We investigate infinite sets that witness the failure of certain Ramsey-theoretic statements, such as Ramsey’s or (appropriately phrased) Hindman’s theorem; such sets may exist if one does not assume the Axiom of Choice. We will show very precise information as to where such sets are located within the hierarchy of infinite Dedekind-finite sets. The proofs involve both very pleasant combinatorial arguments (to establish certain implications) and the Fränkel-Mostowski technique to obtain permutation models of ZFA (to prove that certain other implications are not provable).

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