Non-standard analysis provides a framework to carry out formally calculus proofs in a much more intuitive manner than in the 2 − δ formulation of Weierstraß. This paper introduces the notions of proof-planning and rippling, and gives an example showing how they can be applied to non-standard analysis to produce readable proofs. We present the related work in the area and give an outline of the ...

The aim of this paper is to show that the representation with the help of Nonstandard Analysis of a real phenomenon, presenting different observation scales, allows an important simplification of language. Indeed, it is convenient to have available the concept of infinitely small and infinitely large quantities in dealing with the macroscopic effects of microscopic phenomena. This is illustrate...

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 ...

3 Construction of Star Types Using Ultrafilters 14 3.1 A Free Ultrafilter over the Naturals . . . . . . . . . . . . . . . 15 3.2 Definition of star type constructor . . . . . . . . . . . . . . . 15 3.3 Transfer principle . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.4 Standard elements . . . . . . . . . . . . . . . . . . . . . . . . 17 3.5 Internal functions . . . . . . . . . . . . . ....

2 Filter: Filters and Ultrafilters 13 2.1 Definitions and basic properties . . . . . . . . . . . . . . . . . 13 2.1.1 Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.2 Ultrafilters . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.3 Free Ultrafilters . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Collect properties . . . . . . . . . . . . . . . . . . . . . . . . . ...