July 12, 2024


July-December 2012 Thematic Program on Forcing and its Applications
Distinguished Lecture Series
November 7-9, 2012 at 3:30 p.m.
Fields Institute, Room 230

Matthew D. Foreman
University of California, Irvine

November 7, 2012 at 3:30 pm.
Large Cardinals: Who are they? What are they doing here? Why won't they go away? (slides of presentation 67Mb)

This lecture will discuss the roots of large cardinals, (starting from Euclid), trace their evolution and survey some present day results. Aimed at a general audience, the talk will avoid technical language as much as possible. While no one may change their mind about large cardinals, everyone will leave having a better insight into what they are.

November 8, 2012 at 3:30 pm.
Does set theory have anything to do with mathematics? (
slides of presentation 60 Mb )

We discuss the relationship of Set Theory with other branches of mathematics and the role it has historically played. We will give some recent examples and discuss one--the classification problem for ergodic measure preserving transformations--in some depth.

November 9, 2012 at 3:30 pm.
Generic Elementary Embeddings
(slides of presentation 35 Mb)

Conventional large cardinals have been codified to have a certain form--postulating class sized objects. Though these are well-understood to have "equivalent" statements in ZFC, they don't actually "live in V". One can stipulate some very similar objects that can be thought of as "generic" large cardinals. The equivalent ZFC versions of these objects can have small cardinalities. As a result they are directly relevant to questions such as the Continuum Hypothesis. Moreover, generic elementary embeddings have become an essential technique for extracting consequences of large cardinals involving sets of small cardinality.
This lecture will show that a broad class of generic elementary embeddings is equiconsistent with their analogous large cardinals. The results include equiconsistency results between combinatorial properties of the first few uncountable cardinals and huge cardinals.

Matthew Dean Foreman is a set theorist at University of California, Irvine. He has made contributions in widely varying areas of set theory, including descriptive set theory, forcing, and infinitary combinatorics.

Foreman earned his Ph.D. in 1980 at University of California, Berkeley under the direction of Robert M. Solovay, with a dissertation on Large Cardinals and Model Theoretic Transfer Properties.

Speakers in the Distinguished Lecture Series (DLS) have made outstanding contributions to their field of mathematics. The DLS consists of a series of three one-hour lectures.

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