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

TL;DR

This paper develops the machinery of $(\mathcal L, n)$-models to construct true but unprovable sentences in Peano Arithmetic, providing new proofs and independence results in mathematical logic.

Contribution

It extends the theory of $(\mathcal L, n)$-models and applies it to give alternative proofs and new independence results in arithmetic and Ramsey theory.

Findings

Shelah's alternative proof of the Paris-Harrington theorem

Independence of a new $\Pi^0_1$ Ramsey statement over PA

Development of $(\mathcal L, n)$-models machinery

Abstract

I develop in depth the machinery of $(\mathcal L, n)$-models originally introduced by Shelah and, independently in a slightly different form by Kripke. This machinery allows fairly routine constructions of true but unprovable sentences in $\mathsf{PA}$. I give two applications: 1. Shelah's alternative proof of the Paris-Harrington theorem, and 2. The independence over $\mathsf{PA}$ of a new $\Pi^0_1$ Ramsey theoretic statement about colorings of finite sequences of structures.