Title:		Covering Properties of Core Models

Speaker:	Ernest Schimmerling
		Department of Mathematics
		University of California, Irvine

Abstract: 

Jensen showed that if 0-sharp does not exist, then L has the "covering
property": every uncountable set of ordinals is contained in a
constructible set of the same cardinality. The covering property implies
the "weak covering property": if \alpha \geq \omega_2 is a successor
cardinal of L, then the cofinality of \alpha is equal to the cardinality
of \alpha.  The weak covering property implies that L computes successors
of singular cardinals correctly. 

Kunen found that if 0-sharp does not exist, then L computes successors of
weakly compact cardinals correctly. 

Extensions to these results must respect the limitations implied by Prikry
forcing. One kind of extension is illustrated by:

Theorem 1 (Mitchell and Schimmerling)  
Assume that every set has a sharp and that there is no model with a Woodin
cardinal.  Then K has the weak covering property.

Theorem 2 (Schimmerling and Steel)
Assume that every set has a sharp and that there is no model with a Woodin
cardinal.  Then K computes successors of weakly compact cardinals
correctly. 

One sees a different approach in:

Theorem 3 (Schimmerling and Woodin)  
If W is an iterable weasel without measurable cardinals, and W-sharp does
not exist, then W has the covering property. 

The terminology, history, references, main ideas, and applications will be
discussed in the talk.