Roland Walker : Overview of Recursive Saturation, Scott Sets, and $S$-Saturation

Posted by Roland Walker , part of the Louise Hay Logic Seminar.

At

Nov. 13, 2019, 4 p.m.

In

427 SEO

Abstract

Recursive saturation and $S$-saturation are interesting concepts at the intersection of model theory and recursion theory. In this talk, we will define and discuss recursively saturated models, Scott sets, effectively perfect theories, standard systems, and $S$-saturated models. We will prove that any recursively saturated model of an effectively perfect theory is $S$-saturated for exactly one Scott set $S$. It is natural to ask if all Scott sets arise in this fashion. In 1982, Knight and Nadel showed that all Scott sets of size at most $\aleph_1$ arise as the standard system of some nonstandard model of Peano Arithmetic. We will prove a more general version of this result which applies to any computable theory. This closes the question under CH. We will discuss recent progress which closes the question for specific theories when CH is not assumed.