This will be a talk for the Logic Advanced Class at the Mathematical Institute, University of Oxford. The topic concerns the extension of the axiom of determinacy as well as the axiom . It will take place on the 13th of June 2024 at 11 AM in the lecture room C3.

**Abstract.** Solovay defined the inner model in the context of by using it to define the supercompactness measure on naturally given by . Solovay speculated that stronger versions of this inner model should exist, corresponding to stronger versions of the measure . Woodin, in his unpublished work, defined which is arguably the ultimate version of the supercompactness measure that Solovay had defined. I will talk about in the context of and the axiom .

The event will be followed by another talk on the subject of extensions of the axiom of determinacy. Douglas Blue, from the University of Pittsburgh, will talk about *The iterability problem and the transfinite generalization of AD* for the Logic Seminar at the Mathematical Institute at 5PM in the lecture room L3.