An inner model theoretic proof of Becker's theorem

TL;DR

This paper presents a new inner model theoretic proof of Becker's theorem, establishing supercompactness of omega_2 under AD+V=L(R), and discusses open questions about supercompactness of Suslin cardinals.

Contribution

It provides a novel proof of Becker's theorem using inner model theory, expanding understanding of large cardinals under determinacy assumptions.

Findings

Omega_2 is kappa-supercompact for all kappa ≤ supremum of Suslin cardinals under AD+V=L(R)

Inner model theory can be used to prove supercompactness results

Open problem: supercompactness of Suslin cardinals under AD_R

Abstract

We give a new proof of a theorem of Becker that under AD+V=L(R), omega_2 is a kappa-supercompact for every kappa less than or equal to the supremum of all Suslin cardinals. Our proof uses inner model theory. It is still open whether one can prove, say under AD_R, that Suslin cardinals and their successors are <Theta-supercompact. That this is true is a theorem of Steve Jackson, who proved it using generic codes.