Large cardinals, L-like universes and the Inner model
hypothesis
Sy Friedman (7. Logic Colloquium)
Kurt Godel Reseach Center, Vienna
Abstract:
There are many different ways to extend the axioms of
ZFC. One attractive way is to adjoin the axiom V=L,
asserting that every set is constructible. This axiom has
many desirable consequences, such as the generalised continuum
hypothesis, the existence of a definable wellordering
of the class of all sets, as well as strong combinatorial
principles, such as Diamond, Square and Morass.
However V=L adds no consistency strength to ZFC. As many
interesting set-theoretic statements have consistency
strength beyond ZFC, it is now common in set theory to
assume the existence of ``large'' inner models of the
set-theoretic universe, i.e., inner models containing large
cardinals.
Can we simultaneously have the advantages of both the axiom of
constructibility and the existence of large cardinals? One
way to achieve this is via the Inner Model Program,
whose goal is to show that in any universe with large
cardinals, there is an L-like inner model with large
cardinals. An alternative approach is given by the
Outer Model Program, whose goal is to show that any
universe with large cardinals is contained in a larger
universe which is L-like and contains large cardinals. We
shall describe some results that have been obtained with
these approaches.
We also present a natural statement of ``largeness'' for the
set-theoretic universe, the Inner Model Hypothesis, which
implies the existence of inner models with large cardinals.
Date and time: November 10, 2005 at 13.3o
Place: Mathematical Institute, Zitna 25, Praha 1
Visit sponsored by the
Eduard Cech Center.
