Logic Seminar
John Baldwin
UIC
What is Morley's method.
Abstract:
We provide some context for a number of variants on the Ehrenfeucht-Mostowski construction and expound the construction of tree indiscernibles for sentences of $L_{\omega_1,\omega}$. Two results of Baldwin-Shelah on the stability spectrum for $L_{\omega_1,\omega}$ use these methods.
Here ${S_i}^m(M)$ denotes an appropriate notion ($at$ or ${mod}$) of Stone space of $m$-types over $M$.
Theorem A. Suppose that for some positive integer $m$ and for every $\alpha< \delta(T)$,
there is an $M \in K$ with $|{S^m}_i(M)| > |M|^{beth_\alpha(|T|)}$.
Then for every $\lambda \geq |T|$, there is an $M$ with $|{S^m}_i(M)| > |M|$.
Theorem B. Suppose that for every $\alpha<\delta(T)$, there is $M_\alpha \in K$ such
that $\lambda_\alpha = |M_{\alpha}| \geq beth_\alpha$ and $|{S^m}_i(M_\alpha)| >
\lambda_\alpha$. Then for any $\mu$ with $\mu^{\aleph_0}>\mu$, $K$ is not $i$-stable in $\mu$.
These results provide a new kind of sufficient condition for the unstable case and shed some light on the spectrum of strictly stable theories in this context. The methods avoid the use of compactness in the theory under study.
seminar begins with tea.
Tuesday February 2, 2010 at 4:00 PM in SEO 612