Elliot Glazer (Harvard University)

Foundationless geology and a Foundation conservativity result

It is well-understood that the Axiom of Foundation has no "mathematical consequences" over ZFC - Foundation, since every mathematical structure is isomorphic to one whose universe is an ordinal by the well-ordering theorem. Over ZF - Foundation, there are mathematical consequences to adding Foundation, e.g. the sentence "if all orderable sets are well-orderable, then every set is well-orderable." In joint work with Asaf Karagila, we identify a precise sense in which there is no simpler consequence of adding Foundation. In particular, for any $\varphi$ a sentence in second-order logic, adding Foundation does not refute the existence of a set model of $\varphi.$ This talk will focus on applying techniques of set-theoretic geology in a context without Choice or Foundation, which is a key ingredient in the proof of this theorem. Slides