Description:
Abstract We show that various tameness assertions about abstract elementary classes imply the existence of large cardinals under mild cardinal arithmetic assumptions. For instance, we show that if 𝜅 is an uncountable cardinal such that 𝜇𝜔 < 𝜅 for every 𝜇 < 𝜅 and every AEC with Löwenheim-Skolem number less than 𝜅 is < 𝜅-tame, then 𝜅 is almost strongly compact. This is done by isolating a class of AECs that exhibits tameness exactly when sufficiently complete ultrafilters exist.