Philosophy of Mathematics seminar

Douglas Blue, Pittsburgh

Hierarchies of infinity

Large cardinals beyond Choice are large cardinal axioms incompatible with the Axiom of Choice but not known to be inconsistent in ZF. We will survey the Reinhardt and Berkeley hierarchies of such axioms, the implications their consistency would have for inner model theory, and recent theorems bearing on whether they are in fact consistent. In light of these results, we will discuss how the hierarchies of ZFC large cardinals, ZF large cardinals, and determinacy theories seem to relate to each other. We will describe a scenario in which ZFC together with mild large cardinals and Woodin's axiom V=Ultimate L dwarfs these hierarchies. Should this scenario be realized, then, by virtue of its giving access to consistency strengths otherwise inaccessible, we contend that it would constitute a novel kind of argument for a new axiom.