November 7-9, 2012 at 3:30 p.m.
Fields Institute, Room 230
Matthew D. Foreman
University of California, Irvine
Large Cardinals: Who are they? What are they doing here? Why won’t they go away?
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?
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
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.