Hypernatural Numbers as Ultrafilters

Abstract

Abstract. In this paper we present a use of nonstandard methods in the theory of ultrafilters and in related applications to combinatorics of numbers. Introduction.Ultrafilters are really peculiar and multifaced mathematical objects, whose study turned out a fascinating and often elusive subject. Researchers may have diverse intuitions about ultrafilters, but they seem to agree on the subtlety of this concept; e.g., read the following quotations: "The space βω is a monster having three heads" (J. van Mill [41]); ". . . the somewhat esoteric, but fascinating and very useful objectThe notion of ultrafilter can be formulated in diverse languages of mathematics: in set theory, ultrafilters are maximal families of sets that are closed under supersets and intersections; in measure theory, they are described as {0, 1}-valued finitely additive measures defined on the family of all subsets of a given space; in algebra, they exactly correspond to maximal ideals in rings of functions F I where I is a set and F is a field. Ultrafilters and the corresponding construction of ultraproduct are a common tool in mathematical logic, but they also have many applications in other fields of mathematics, most notably in topology (the notion of limit along an ultrafilter, the Stone-Čech compactification βX of a discrete space X, etc.), and in Banach spaces (the so-called ultraproduct technique).In 1975, F. Galvin and S. Glazer found a beautiful ultrafilter proof of Hindman's theorem, namely the property that for every finite partition of the natural numbers N = C 1 ∪ . . . ∪ C r , there exists an infinite set X and a piece C i such that all sums of distinct elements from X belong to C i . Since this time, ultrafilters on N have been successfully used also in combinatorial number theory and in Ramsey theory. The key fact is that the compact space βN of ultrafilters on N can be equipped 2000 Mathematics Subject Classification. 03H05, 03E05, 54D80.

Ramsey properties of nonlinear Diophantine equations

“…Therefore, for every h, k ≥ 2 we can apply the closure property of F to the system y j = z n is in F with full injectivity. 10 Let us notice that for n = h = k = 2, Example 2.14 reduces to…”

“…In the following, we will work in a c + -saturated extension of N. In addition to the fundamental principles of nonstandard analysis, our proofs will also use properties of the relation of u-equivalence on hypernatural numbers, as introduced by the first named author in [10]. (See also [11], where uequivalent pairs are named indiscernible, and [9], [32], where many algebraic properties of u-equivalence are proved by means of iterated hyperextensions.)…”

Fermat-Like Equations that are not Partition Regular

“…In addition to the usual basic nonstandard tools, we will consider the following equivalence relation on the set * N of hypernatural numbers. (See [7]; see also [6] where u-equivalent numbers are called indiscernible. )…”

“…We will use the following three basic properties of u-equivalence, whose proofs can be found in §2 of [7].…”

Ramsey’s Coheirs

“…Our approach is similar, except that, unlike Newelski, we do not pursue connections with topological dynamics, but rather offer an alternative realm of application. The investigation of alternative methods in the study of regularity phenomena has been called for by Di Nasso [5, Open problem #1]. This article contains a possible answer.…”