Logic Seminar
Paul Larson
Miami University
Building models with iterated generic elementary embeddings.
Abstract: Iterations of generic elementary embeddings with critical point $\omega_1$
are the fundamental construction underlying Woodin's
P$_{max}$ forcing, and they can be used to prove a number of $\Sigma_1$
absoluteness results with respect to the uncountable.
We will present proofs of the following using this method : (1) The
existence of a model for a statement of L$_{\omega_1, \omega}$(Q)
is forcing-absolute (2) If a PC_delta over L$_{\omega_1, \omega}$(aa) class
(forceably) has an uncountable model satisying uncountably
many types over a countable fragment of the language, then it has
2$^{\aleph_1}$ many uncountable models, each satisfying uncountably many
types over this
fragment, but pairwise satisfying just countably many in common. Time
permitting, we will discuss an extension of the Magidor-Malitz Theorem
using
this method.
Thursday April 21, 2011 at 3:00 PM in SEO 612