Isaac Goldbring




Monday, March 6, 2017 - 4:00pm


RH 440R

Morley introduced the notion of a scattered sentence of $L_{\omega_1,\omega}$.  Roughly speaking, $\varphi$ is scattered if it does not have a perfect set of countable models.  He then showed that scattered sentences have at most $\aleph_1$ many countable models (up to isomorphism) whilst non-scattered sentences have continuum many nonisomorphic countable models.  The absolute Vaught conjecture states that scattered sentences have only countably many countable models up to isomorphism.  Unlike the original Vaught conjecture (which holds trivially under CH), the absolute Vaught conjecture does not depend on the model of set theory in question and is in fact equivalent to the original Vaught conjecture under the negation of CH.

Keisler connected the notion of scattered sentences with randomizations of structures.  Randomizations are models of a certain continuous theory, called the pure randomization theory, and in such a randomization, one can define the probability that $\varphi$ holds.  A randomization of $\varphi$ is a randomization in which $\varphi$ holds with probability one.  An example of a randomization of $\varphi$ is a basic randomization of $\varphi$, which consists of a collection of "measurable" random variables taking values in a countable family of models of $\varphi$.  $\varphi$ is is said to have few separable randomizations if every randomization of $\varphi$ is a basic randomization of $\varphi$.   

Keisler showed that if $\varphi$ has few separable randomizations, then $\varphi$ is scattered.  Moreover, he showed that, assuming Martin's axiom for $\aleph_1$, the converse holds.  Andrews, Goldbring, Hachtman, Keisler, and Marker were able to remove the use of Martin's axiom by an absoluteness argument.  Thus, it is a theorem of ZFC that $\varphi$ is scattered if and only if $\varphi$ has few separable randomizations.

In this talk, I will try to define most of the above results in more detail and sketch the ideas behind the proofs of the theorems alluded to above.