Iterated hyper-extensions and an idempotent ultrafilter proof of Rado's theorem

TL;DR

This paper introduces a nonstandard analysis approach using iterated hyper-extensions to manipulate ultrafilters, providing new foundations and a short proof of Rado's theorem in Ramsey theory.

Contribution

It develops a novel formalism for ultrafilter manipulation via nonstandard analysis, enabling concise proofs of key theorems in Ramsey theory.

Findings

Short proof of Milliken-Taylor's Theorem

Ultrafilter version of Rado's theorem

New foundations for ultrafilter manipulation

Abstract

By using nonstandard analysis, and in particular iterated hyper-extensions, we give foundations to a peculiar way of manipulating ultrafilters on the natural numbers and their pseudo-sums. The resulting formalism is suitable for applications in Ramsey theory of numbers. To illustrate the use of our technique, we give a (rather) short proof of Milliken-Taylor's Theorem, and a ultrafilter version of Rado's theorem about partition regularity of diophantine equations.