Nonstandard Analysis by Means of Ideal Values of Sequences

A notion of ideal value of N -sequences is axiomatized through elementary properties. The resulting theory ZFC[α] provides nonstandard analysis with a general foundational framework. 1991 MSC. Primary: 26E35, 03E70, 03H05. Secondary: 03C20, 03E35. In this paper, the axiomatic system ZFC[α] is presented. It is a generalization in a set theoretic context of an approach to nonstandard analysis recently given by V. Benci in [Be]. In that paper, nonstandard analysis is presented by postulating, besides a nonstandard embedding ∗ : V (X) → V (X) from a superstructure into itself, the existence of a pre-extension for each function defined on the set N of natural numbers. Axioms are then given that correlate pre-extensions with the enlarging map ∗. Here, we formalize the idea of preextension by means of a notion of ideal value. Precisely, we enrich the language of set theory with a constant symbol α (to be intended as a new ideal natural number) and axiomatize the ideal value φ(α) of N -sequences φ. The idea of adding a new logical constant to behave as an “infinite” natural number is not new. Precisely, it goes back to the pioneering work by C. Schmieden and D. Laugwitz [SL], appeared before Robinson’s presentation of nonstandard analysis. A general way of formally giving a “nonstandard extension” of a given first order theory by means of a new constant symbol, can be found in Laugwitz’ work [La]. In this paper, an approach is given in the full generality of set theory. As a consequence of our axioms, a nonstandard embedding ∗ : V → V is determined for the universal class of all sets, and all basic principles of nonstandard analysis, including transfer and countable saturation, are proved. All principles of mathematics as formalized by Zermelo-Fraenkel set theory with choice are assumed, with the only exception of regularity. Thus, working in ZFC[α], one can manipulate (external) sets as in ordinary mathematics with no restrictions. Also, nonstandard arguments can be iterated, because the enlarging map ∗ is defined for all sets. For instance, one can consider the hyper-hyperreal numbers