The Paris–Harrington principle and second-order arithmetic—bridging the finite and infinite Ramsey theorem

Abstract

The Paris–Harrington principle ($\mathrm{PH}$) is known as one of the earliest examples of “mathematical” statements independent from the standard axiomatization of natural numbers called Peano Arithmetic ($\mathrm{PA}$). In this article, we discuss various variations of $\mathrm{PH}$ and examine the relations between finite and infinite Ramsey’s theorem and systems of arithmetic.