### Models of arithmetic with two expansions to ACA_0

#### by kamerynwilliams

This was 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 . 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 . Last time, we saw how to build models of arithmetic which are -rather classless for some class of the model. In this talk, I will use a kind of forcing argument to show how to pick this so that the resulting model has exactly two expansions to . 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 with precisely two -realizations.