2012 seminar talk: An example illustrating a theorem of Gregory

Talk held by Julia Knight (University of Notre Dame, USA) at the KGRC seminar on 2012-07-11.

Abstract

John Gregory showed (in 1970) that for a countable admissible set $A$, if $T$ is set of $L_A$ sentences that is $\Sigma_1$ on $A$ and $T$ has models $\mathcal{M}$ and $\mathcal{N}$ such that $\mathcal{N}$ is a proper $L_A$-elementary extension of $\mathcal{M}$, then $T$ has an uncountable model. I will describe an example showing that the result fails if we drop the assumption that $T$ is $\Sigma_1$ on $A$. This is joint work with my three current students, Jesse Johnson, Victor Ocasio, and Steven VanDenDriessche. The construction involves iterated forcing.