Mathematical Incompleteness Results in First-Order Peano Arithmetic: A Revisionist View of the Early History

TL;DR

This paper reevaluates the history of mathematical incompleteness results in first-order Peano arithmetic, arguing that Gentzen's work predates and is more foundational than the Paris-Harrington theorem.

Contribution

It challenges the common historical narrative by asserting Gentzen's incompleteness result as the earliest, revising the understanding of the development of mathematical logic.

Findings

Gentzen's incompleteness result predates Paris-Harrington

Goodstein restated Gentzen's result in a number-theoretic form

The paper questions the significance of Paris-Harrington as the first incompleteness result

Abstract

In the Handbook of Mathematical Logic, the Paris-Harrington variant of Ramsey's theorem is celebrated as the first result of a long 'search' for a purely mathematical incompleteness result in first-order arithmetic. This paper questions the existence of any such search and the status of the Paris-Harrington result as the first mathematical incompleteness result. In fact, I argue that Gentzen gave the first such result, and that it was restated by Goodstein in a number-theoretic form.