Independence in Arithmetic: The Method of $(\mathcal L, n)$-Models

Corey Bacal Switzer

Published 2019-06-10Version 1

I develop in depth the machinery of $(\mathcal L, n)$-models originally introduced by Shelah \cite{ShelahPA} and, independently in a slightly different form by Kripke (cf \cite{put2000}, \cite{quin80}). This machinery allows fairly routine constructions of true but unprovable sentences in $\mathsf{PA}$. I give three applications: 1. Shelah's alternative proof of the Paris-Harrington theorem, 2. The independence over $\mathsf{PA}$ of a $\Pi^0_1$ Ramsey theoretic statement about colorings of finite sequences of structures and 3. The independence over $\mathsf{PA}$ of both a $\Pi_2^0$-statement and a $\Pi^0_1$-statement about choice functions for sequences of numbers and finite structures respectively. The latter is reminiscent of Shelah's example.