Internal set theory: A new approach to nonstandard analysis

Internal set theory. We present here a new approach to Abraham Robinson's nonstandard analysis [10] with the aim of making these powerful methods readily available to the working mathematician. This approach to nonstandard analysis is based on a theory which we call internal set theory (1ST). We start with axiomatic set theory, say ZFC (Zermelo-Fraenkel set theory with the axiom of choice [1]). In addition to the usual undefined binary predicate E of set theory we adjoin a new undefined unary predicate standard. The axioms of 1ST are the usual axioms of ZFC plus three others, which we will state below.