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 \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.

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