# Models of arithmetic with two expansions to a model of ACA_0

This is a pair of talks for CUNY’s MoPA seminar on 5 Oct and 26 Oct 2016.

First talk: In this talk and its sequel I will construct models of arithmetic with exactly two expansions to a model of $\mathsf{ACA}_0$. To do so, I will use a modified version of Keisler’s construction of a rather-classless model. This talk will focus on this construction, while in Part 2 I will show how to use this construction to get the result.

Second talk: In this talk and its prequel I construct models of arithmetic with exactly two expansions to a model of $\mathsf{ACA}_0$. Last time, we saw how to build models of arithmetic which are A-rather classless for some class $A$ of the model. In this talk, I will use a kind of forcing argument to show how to pick this $A$ so that the resulting model has exactly two expansions to $\mathsf{ACA}_0$. Time permitting, I will explain the difficulties in moving from two to three.

Unfortunately, Roman Kossak spotted an error in the second talk which I have since been unable to fix. To my knowledge it is currently open whether there is a model of $\mathsf{PA}$ with precisely two $\mathsf{ACA}_0$-realizations.