The Ultrapower Axiom (UA) roughly states that any pair of ultrapowers can be compared by internal ultrapowers. The Axiom was extensively studied by Gabriel Goldberg, leading to a series of striking results.

Goldberg asked whether UA is consistent with a measurable cardinal that violates GCH. The main challenge is that UA is not easily preserved under forcing constructions, especially ones that achieve violation of GCH on a measurable from large cardinal assumptions. For example, such forcings might create normal measures which are incomparable in the Mitchell order – a property that negates UA.
In this talk, we sketch the proof that the failure of GCH on the least measurable cardinal can indeed be forced while preserving UA, starting from the minimal canonical inner model carrying a (\kappa, \kappa^{++})-extender. We will present the forcing construction and sketch the main proof ideas. This is a joint work with Omer Ben-Neria.