A simple algebraic characterization of nonstandard extensions

Abstract

We introduce the notion of functional extension of a set X, by means of two natural algebraic properties of the operator " * " on unary functions. We study the connections with ultrapowers of structures with universe X, and we give a simple characterization of those functional extensions that correspond to limit ultrapower extensions. In particular we obtain a purely algebraic proof of Keisler's characterization of nonstandard (= complete elementary) extensions.

A topological interpretation of three Leibnizian principles within the functional extensions

“…It allows for obtaining correct results about, say, the real numbers by using ideal elements, like actual infinitesimal or infinite numbers. In this section we review the main features of the functional extensions introduced in the paper [11] (see also [4]), with the goal of characterizing all nonstandard (= complete elementary) extensions by means of a few simple properties of an operation * that assigns an appropriate extension * f : * X → * X to each function f : X → X.…”

“…The importance of the property (dir), called coherence in [8], is due to the fact that, by providing an "internal coding of pairs", it allows for extending multivariate functions "parametrically": this possibility is essential in order to get the full principle Tran, which involves properties, relations, and functions of any arities. More precisely, the following facts that hold in every functional extension * X of X allow for considering only unary functions (see Subsection 3.2 of [11]):…”

“…• In Section 1, we recall the precise definition of functional extension together with the main properties stated in [11]. In particular, we establish the fundamental result that the functional extensions are all and only the complete elementary (=nonstandard) extensions of any model M .…”

“…In this paper, we try and give a precise mathematical formulation of these principles in the context of the Functional Extensions of [11], structures which generalize at once compactifications and completions of topological spaces, and nonstandard extensions of models (see also [4]). Given a set M , a functional extension of M is a superset * M of M such that every function f : M → M has a distinguished extension * f : * M → * M that preserves compositions and equalizers.…”

“…We shall see below that the "real" properties of M , when extended to * M , may be taken as the clopen subsets of a topology on * M (the so called S-topology), and so the above principles Ind and Poss turn out to be respectively a property of separation and a property of compactness of the S-topology of * M . On the other hand, the principle Tran requires all the conditions postulated for the * -extension in [11], including the order-theoretic property of directeness, so as to take care of properties and relations of arbitrary arities.…”