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 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.
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