An Approach to Non-standard Analysis

This note has two principal aims: to portray an essence of Non-Standard Analysis as a particular structure (which we call lim-rim), noting its interplay with the notion of ultrapower, and to present a construction of Non-Standard Analysis, viewed as a matter of mathematics, where the set A of non-standard elements of a set A – “the adjunction of all possible limits” is a “good” kind of lim-rim which plays a role analogous to that of the algebraic closure of a field – “the adjunction of all roots of polynomials”. In the same spirit as with algebraic closures, one has uniqueness up to isomorphism, and also universality and homogeneity, provided one has enough General Continuum Hypothesis. The cardinality of A will be something like 2 |A| – the same as that of the set of ultrafilters in A, and one has a high degree of saturation.