Permanence, Comprehensiveness, Saturation

Abstract

Nonstandard analysis introduces a brave new world of mathematical entities. It also has a number of distinctive structural features and principles of reasoning that can be used to explore this world. Already in the context of subsets of *ℝ we have examined several of these principles: permanence, internal induction, overflow, underflow, saturation. Now we will see that in the context of a universe embedding U → U’ they occur in a much more powerful form, since they apply to properties that may refer to any internal entities in U’. We assume from now on that we are dealing with such an embedding for which *ℕ - ℕ ≠ Ø.