# MSCS Seminars Today

## Calendar for Tuesday October 20, 2020

Tuesday October 20, 2020

**Midwest Model Theory Seminar**

Using Ultraproducts to Compare Continuous Structures

H. Jerome Keisler (University of Wisconsin)

3:00 PM in the internet

We revisit two research programs that were proposed in the
1960's, remained largely dormant for five decades, and then become hot
areas of research in the last decade.
The monograph ``Continuous Model Theory'' by Chang and Keisler, Annals
of Mathematics Studies (1966), studied structures with truth values in
[0,1], with formulas that had continuous functions as connectives, sup
and inf as quantifiers, and equality. In 2008, Ben Yaacov, Bernstein,
Henson, and Usvyatsev introduced the model theory of metric
structures, where equality is replaced by a metric, and all functions
and predicates are required to be uniformly continuous. This has led
to an explosion of research with results that closely parallel first
order model theory, with many applications to analysis. In my
forthcoming paper ``Model Theory for Real-valued Structures'', the
"Expansion Theorem" allows one to extend many model-theoretic results
about metric structures to general [0,1]-valued structures--the
structures in the 1966 monograph but without equality.
My paper ``Ultrapowers Which are Not Saturated'', J. Symbolic Logic
32 (1967), 23-46, introduced a pre-ordering $\mathcal
M\trianglelefteq\mathcal N$ on all first-order structures, that holds
if every regular ultrafilter that saturates $\mathcal N$ saturates
$\mathcal M$, and suggested using it to classify structures. In the
last decade, in a remarkable series of papers, Malliaris and Shelah
showed that that pre-ordering gives a rich classification of simple
first-order structures. Here, we lay the groundwork for using the
analogous pre-ordering to classify [0,1]-valued and metric structures.

This seminar is online using Cisco Webex - for login information, please email jfreitag@uic.edu.