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By using nonstandard analysis, and in particular iterated hyperextensions, 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

Iterated hyper-extensions and an idempotent ultrafilter proof of Rado’s Theorem