Hyperordinals and Nonstandard α-Models

Abstract

In this paper we extend the traditional set-theoretic notion of standard models and nonstandard models going up to α levels in the cumulative hierarchy, α any given limit ordinal. A proof of the representation theorem is given and the structure of nonstandard models is studied where the ”transfer principle” holds for every (not necessarily bounded) formula. These models preserve a stratified structure which is investigated by means of ”pseudo-rank” functions taking linearly ordered values (hyperordinals). In particular, such functions show a ”rigidity” property of the internal sets, in that each external set has a pseudo-rank which is greater than the pseudo-rank of any internal set. 1 Nonstandard α-Models We shall work in ZF−C, i.e. in Zermelo-Fraenkel axiomatic set theory ZF with choice but without the axiom Fnd of foundation (regularity). Thus the axioms of our set theory are precisely: Ext (Extensionality); Pair (Pairing); Sep (Separation Schema); Un (Union); Pow (Power Set); Rep (Replacement Schema); Inf (Infinity) and AC (Choice). For unexplained set-theoretic and model-theoretic notions and notations we refer to [J] and [CK] respectively.