Successor Large Cardinals in Symmetric Extensions
Tanmay Inamdar

Abstract:
We give an exposition in modern language (and using partial orders) of
Jech's method for obtaining models where successor cardinals have
large cardinal properties. In such models, the axiom of choice must
necessarily fail. In particular, we show how, given any regular
cardinal and a large cardinal of the requisite type above it, there is
a symmetric extension of the universe in which the axiom of choice
fails, the smaller cardinal is preserved, and its successor cardinal
is measurable, strongly compact or supercompact, depending on what we
started with. The main novelty of the exposition is a slightly more
general form of the Levy-Solovay Theorem, as well as a proof that fine
measures generate fine measures in generic extensions obtained by
small forcing.

Keywords: Set Theory;  Large Cardinals;  Forcing;  Axiom of Choice;  Symmetric Extensions