This will be an online talk for the Leeds Set Theory Seminar, 21 May 2025 1pm BST. Contact the organizers (Hope Duncan) for Teams access.

Abstract: One can find in the philosophical research literature surrounding Skolem’s paradox a certain claim, referred to as the transitive submodel theorem, according to which every transitive model of set theory admits a countable transitive submodel of the same theory. Although the statement may initially appear plausible—perhaps one thinks it follows from an application of the downward Löwenheim-Skolem theorem—nevertheless it turns out that as a mathematical claim, it is overstated. There is no such theorem. In this talk I shall give a full account of the countable transitive submodel proposition, taken as a principle of set theory, showing from suitable hypotheses that counterexamples are possible and characterizing exactly the circumstances in which the principle does hold. Ultimately, the countable transitive submodel proposition should be seen as a certain anti-large cardinal principle that is equiconsistent with but independent of ZFC and refuted by all the moderately strong large cardinal notions. This is joint work in progress with Timothy Button, with thanks to W. Hugh Woodin.