Pseudo-Superstructures as Nonstandard Universes

A definition of nonstandard universe which gets over the limitation to the finite levels of the cumulative hierarchy is proposed. Though nec-essarily nonwellfounded, nonstandard universes are arranged in strata in the likeness of superstructures and allow a rank function taking linearly ordered values. Nonstandard universes are also constructed which model the whole ZFC theory without regularity and satisfy the κ-saturation property. Introduction. The traditional superstructure approach to the nonstandard methods is the one introduced by A. Robinson and E. Zakon in [RZ] and further developed in [Z]. It is based on the well-known fact that virtually all ordinary mathematics can be thought as embedded in set theory. Roughly speaking, it consists of