In this paper two new combinatorial principles in nonstandard analysis are isolated and applications are given. The second principle provides an equivalent formulation of Henson’s isomorphism… (More)

By using nonstandard analysis, we prove embeddability properties of di↵erences A B of sets of integers. (A set A is “embeddable” into B if every finite configuration of A has shifted copies in B.) As… (More)

We consider the possibility of a notion of size for point sets, i.e. subsets of the Euclidean spaces Ed( R) of all d-tuples of real numbers, that satisfies the fifth common notion of Euclid’s… (More)

The methods of nonstandard analysis are presented in elementary terms by postulating a few natural properties for an infinite “ideal” number α. The resulting axiomatic system, including a… (More)

The näıve idea of “size” for collections seems to obey both to Aristotle’s Principle: “the whole is greater than its parts” and to Cantor’s Principle: “1-to-1 correspondences preserve size”.… (More)

In this paper we survey various set-theoretic approaches that have been proposed over the last thirty years as foundational frameworks for the use of nonstandard methods in mathematics. Introduction.… (More)

We present a short proof of Jin’s theorem which is entirely elementary, in the sense that no use is made of nonstandard analysis, ergodic theory, measure theory, ultrafilters, or other advanced… (More)

By allowing values in non-Archimedean extensions of the unit interval, we consider finitely additive measures that generalize the asymptotic density. The existence of a natural class of such “fine… (More)

In this paper we introduce the notion of elementary numerosity as a special function defined on all subsets of a given set Ω which takes values in a suitable non-Archimedean field and satisfies the… (More)

An introduction of nonstandard analysis in purely algebraic terms is presented. As an application, we give a nonstandard proof of a characterization theorem for compact subsets of Sym(N).