This will be a talk for the Rust Belt Workshop in the Philosophy of Logic, Language, and Mathematics, held at Ohio State University in Columbus, Ohio, February 8-9, 2025, University Hall (230 N Oval Mall, Columbus, OH) Room 386B.

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 quite 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, and there is no such theorem. It is a mistake, although an interesting mistake worth discussing. In this talk I shall give a full account of the countable transitive submodel proposition, taken as a principle of set theory, by showing from suitable hypotheses that counterexamples are possible, characterizing exactly the circumstances in which the principle does hold, and investigating the consistency strength of the proposition and also the consistency strength of its negation. Ultimately, the countable transitive submodel proposition should be seen as a certain anti-large cardinal principle that is equiconsistent with but independent of ZFC, refuted by all the moderately strong large cardinal notions.

This is joint work in progress with Timothy Button, with thanks to W. Hugh Woodin.

I will post a link to the paper when it is available.