Logic Seminar
John Baldwin
UIC MSCS
Categoricity, $\omega$-stability, and large models in $L_{\omega_1,\omega}$
Abstract: My approach is historical, beginning with Morley's theorem, and continuing with the connections of $L_{\omega_1,\omega}$
with atomic models of a first order theory.
I discuss work Laskowsi, Shelah, and I have been working on since at
least 2013 on whether an $L_{\omega_1,\omega}$ class $K$ in a countable vocabularyhas
unboundedly large models and/or is stable in $\aleph_0$.
A first order
theory is categorical in $\aleph_1$ iff it is $\omega$-stable and has no
two cardinal models; this characterization is easily seen to be absolute.
Already in [Sh87] Shelah had given an example of an
$L_{\omega_1,\omega}(Q)$ sentence that is categorical under MA but not
under $2^{\aleph_0}<2^{\aleph_1}$. Whether there is similar example for
$L_{\omega_1,\omega}$ remains open.
%The recent work goes in the other direction, trying to find tractable absolute conditions equivalent to $\aleph_1$-categoricity.
In \cite{BLS16}, we reformulated the problem as the study of atomic models of first order
theories and introduced the notions of
pseudo-algebracity and of a pseudo-minimal set as an analog for
strong minimality. We proved: If a countable first order theory $T$ has
an atomic model and fewer than $2^{\aleph_1}$ models in $\aleph_1$ then
the pseudo-minimal types are dense.
In [BLS24] we showed that a sentence of $L_{\omega_1,\omega}$
that is categorical in $\aleph_0$ and $\aleph_1$ and has a model in
$\beth_1^+$ is $\omega$-stable. I will expound this line of work and the role of
the essential set-theoretic argument forcing argument (within ZFC). The most recent development
is an analog of $U$-rank in this setting.
The papers are on my website and in the arkiv. If any one wants background in
advance, drop me an e-mail.
[BLS16] Constructing many atomic models in
$\aleph_1$. Journal of Symbolic Logic, 81:1142–1162, 2016.
[BSL24] When does $\aleph_1$-categoricity imply $\omega$-stability Model Theory
Tuesday February 25, 2025 at 3:30 PM in 636 SEO