A lower bound on Gowers' FIN_k theorem

TL;DR

This paper demonstrates that Gowers' FIN_k theorem, a Ramsey-type result related to Banach space theory, cannot be derived from the logical system ACA_0, highlighting its logical independence.

Contribution

It establishes the non-derivability of Gowers' FIN_k theorem from ACA_0, revealing new insights into its logical strength.

Findings

Gowers' FIN_k theorem is independent of ACA_0.

The theorem cannot be proved within the system ACA_0.

This result clarifies the logical complexity of Gowers' theorem.

Abstract

Gowers' FIN$_k$ theorem, also called Gowers' pigeonhole principle or Gowers' theorem, is a Ramsey-type theorem. It first occurred in the study of Banach space theory and is a natural generalization of Hindman's theorem. In this short note, we will show that Gowers' FIN$_k$ theorem does not follow from ACA$_0$.