Analysis of supercompactness measures by mice
by
Richard Ketchersid
University of North Texas
Coauthors: Steve Jackson

I will present a basis theorem of the following form: Let P be a P¹₃ subset of countable subsets of w_n and let m_n be the unique supercompactness measure on countable subsets of w_n then the following are equivalent (1) For m_n-almost all s, P(s) holds and (2) M models some sentence f where M is a 1-Woodin iterable mouse which is minimal in some sense.

The point is that by replacing ïterable" by some other approximation of iterability we should get closure of P¹₃ under the supercompactness measure m_n. This would be a generalization of a classic theorem of Kechris and Martin whose proof should now generalize to pointclasses for which currently no proof is known.

The techniques involved are part of a larger project to improve descriptive set theoretic results using directed systems of mice. Related work in this direction has been carried out by Woodin, Steel, Hjorth, and Neeman.

Date received: March 8, 2005