Hyperordinals and nonstandard alpha-models

In this paper we extend the traditional set-theoretic notion of standard models and nonstandard models going up to alpha levels in the cumulative hierarchy, alpha 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