Successor Large Cardinals in Symmetric Extensions ∗

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 Lévy-Solovay Theorem, as well as a proof that fine measures generate fine measures in generic extensions obtained by small forcing.