Logic Seminar
John Baldwin
UIC
Interactions of Set Theory with $L_{\omega_1,\omega}$
Abstract: I will discuss some aspects of continuing joint work with Friedman,
Koerwein, Larson, Laskowski, and Shelah. This work uses forcing
techniques to prove model theoretic results in ZFC. Force a model
theoretic result to be consistent by a tool such as Martin's axiom,
collapsing cardinals or a specific forcing with high model theoretic
content. Then use iterated elementary embeddings of the model of set
theory to show the model theoretic result is absolute between V and a
well-chosen model. Deduce it holds in ZFC. Applications include
various extensions of results for $L_{\omega_1,\omega}$ to
analytically presented AEC, a new proof of Harrington's theorem on
Scott rank of counterexamples to Vaught's conjecture and the
development of a new notion of algebraic closure for
$L_{\omega_1,\omega}$ that better explains $\aleph_1$-categoricity.
I will briefly contrast this with results about the characterization
of cardinals. The circle is closed by concluding (95\% now) from
arguments of the first sort that if a sentence of
$L{\omega_1,\omega}$ characterizes a cardinal below the continuum
then it has $2^{\aleph_1}$ models in $\aleph_1$.
Tuesday November 25, 2014 at 4:00 PM in SEO 427