Matthew Foreman : "Generic embeddings"

Techniques related to ideals and their associated generic elementary embeddings are becoming ubiquitous in set theory. These lectures seek to expose them from several perspectives: their use, how to construct them and their potential significance in the foundations of mathematics.

The lectures will start with elementary techniques relating generic ultrapowers, ideals and generic elementary embeddings. The "three parameters" will be introduced and the ideas of "natural" and "induced" ideals will be discussed. We will then move to applications of generic large cardinal embeddings to some classical problems in set theory and some applications in algebra and topology. After this we will consider some special cases, including natural ideals such as the nonstationary ideal on the first uncountable cardinal.

The second part of the lectures will deal with the existence of generic elementary embeddings. The comments will have two directions: outright proofs of the existence of generic elementary embeddings from large cardinals and relative consistency results.

Links:

Matt Foreman gave a related series of lectures at the Singular Cardinal Combinatorics meeting in Gainesville. He has agree that we post his slides although he warns that they contain errors, typos and problems of attribution and should not be considered authoritative.

Foreman has written a chapter on generic embeddings for the forthcoming Handbook of Set Theory, which is nearing completion.

Instructions on how to apply for funds to attend this workshop

Travel to Harrisonburg

Harrisonburg lodging options

Getting to the campus and the workshop room

Nearby recreational activities