Large cardinals, L-like universes and the Inner model hypothesis

Sy Friedman (7. Logic Colloquium)
Kurt Godel Reseach Center, Vienna


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

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