John Baldwin : Constructing Atomic Models in the continuum

Posted by Lynn Scow , part of the Logic Seminar.

At

Nov. 20, 2012, 4 p.m.

In

SEO 427

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