An Axiomatic Presentation of The Nonstandard Methods in Mathematics

A nonstandard set theory ∗ZFC is proposed that axiomatizes the nonstandard embedding ∗. Besides the usual principles of nonstandard analysis, all axioms of ZFC except regularity are assumed. A strong form of saturation is also postulated. ∗ZFC is a conservative extension of ZFC. Introduction. For practitioners, the most popular foundational approach to nonstandard methods is the so-called superstructure approach, introduced by A. Robinson and E. Zakon in [RZ]. Roughly speaking, it consists of two superstructures V (X) and V (Y ), namely the standard model and the nonstandard model, and of a mapping ∗ : V (X) → V (Y ), namely the nonstandard embedding, which provides a transfer principle for elementary properties. The κ-saturation property is also usually assumed, for a suitable cardinal κ. 1 Though well-suited for applications, such an approach reveals limitations from a foundational point of view. Superstructures consist of sets of finite rank in the cumulative hierarchy, thus modeling only a fragment of set theory, and different superstructures are needed to treat different problems. Another criticism is that, in principle, nonstandard methods are not concerned with superstructures. Also, it seems esthetically desirable to include the nonstandard techniques within a unified axiomatic system. Since the seventies, the search for a general framework for nonstandard methods led to the formulation of several axiomatic approaches. 2 Those studies revealed serious foundational difficulties. Most notably, the core problem that every nonstandard theory has to deal with is Hrbàc̆ek paradox [Hr], namely the inconsistency of the usual principles of nonstandard analysis and set theory, in the presence of unlimited levels of saturation. In this paper we show that the above paradox can be overcome because universes of full nonstandard set theory can be constructed which have, in some precise sense, the “feel” of unlimited saturation. The axiomatic theory ∗ZFC presented in this paper is an extension of the traditional foundational framework of mathematics, namely ZFC, where 1For an introduction to nonstandard analysis and the foundations of the superstructure approach, we refer the reader to [HL] and [CK] §4.4, respectively. 2A survey of nonstandard set theories is given in [D3].