  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.
