Logic Seminar
John Baldwin
UIC
Constructing Atomic Models in the continuum
Abstract: I will discuss several applications of a method of Shelah to build a
model in the continuum from a countable model satisfying certain
geometric conditions. In particular, this provides a streamlined
argument for the first part of the Ackerman, Freer, and Patel paper
discussed earlier in the seminar.
Theorem. Let $B$ be a countable model with a
totally trivial closure operation. Then there is an uncountable Borel
model which `strongly witnesses' Th($M$).
Another application is to the notion recently introduced by Shelah,
which I will call pseudoclosure, pcl.
Theorem. If there is a quasiminimal $M \in \mathbf{K}_T$,
where pcl satisfies exchange, with cardinality $\aleph_1$, then there is an $N \in \mathbf{K}_T$ with cardinality $2^{\aleph_0}$.
A key point that I will mention in the seminar is the reduction from
a complete sentence of $L_{\omega_1,\omega}$ to the class of atomic
models of an associated first order theory. The details are in
section 6.1 of my monograph, summarized in Theorem 6.1.8.
http://homepages.math.uic.edu/~jbaldwin/pub/AEClec.pdf
Tuesday November 20, 2012 at 4:00 PM in SEO 427