In this paper, we propose a new approach to nonstandard analysis without using the ultrafilters. This method is very simple in practice. Moreover, we construct explicitly the total order relation in the new field of the infinitesimal numbers. To illustrate the importance of this work, we suggest comparing  a few applications of this approach with the former methods.

We propose a new model of computation based on Nonstandard Analysis. Intuitively, the role of ‘algorithm’ is played by a new notion of finite procedure, called Ω-invariance and inspired by physics, from Nonstandard Analysis. Moreover, the role of ‘proof’ is taken up by the Transfer Principle from Nonstandard Analysis. We obtain a number of results in Constructive Reverse Mathematics to illustra...

In this paper we propose a new approach to realizability interpretations for nonstandard arithmetic. We deal with analysis in the context of (semi)intuitionistic realizability, focusing on Lightstone-Robinson construction model through an ultrapower. particular, consider extension $\lambda$-calculus memory cell, that contains integer (the state), order indicate which slice ultrapower $\cal{M}^{...

Let (X,T ) be a topological space, and X a non–standard extension of X . There is a natural “standard” topology T on X generated by G, where G ∈ T . The topological space ( X, T ) will be used to study, in a systematic way, compactifications of (X,T ) . Mathematics Subject Classification: 031105, 54J05, 54D35, 54D60.

A general method is developed by using nonstandard analysis for formulating and proving a theorem about upper Banach density parallel to each theorem about Shnirel'man density or lower asymptotic density. There are many interesting results about Shnirel'man density or lower asymptotic density (see [4, Chapter 1] for example) in additive number theory. There are also a few interesting results ab...

A nonstandard set theory ∗ZFC is proposed that axiomatizes the nonstandard embedding ∗. Besides the usual principles of nonstandard analysis, all axioms of ZFC except regularity are assumed. A strong form of saturation is also postulated. ∗ZFC is a conservative extension of ZFC. Introduction. For practitioners, the most popular foundational approach to nonstandard methods is the so-called super...