A notion of ideal value of N -sequences is axiomatized through elementary properties. The resulting theory ZFC[α] provides nonstandard analysis with a general foundational framework. 1991 MSC. Primary: 26E35, 03E70, 03H05. Secondary: 03C20, 03E35. In this paper, the axiomatic system ZFC[α] is presented. It is a generalization in a set theoretic context of an approach to nonstandard analysis rec...

Some important concepts in the nonstandard analysis theory of turbulence are presented in this article. The structure of point, on which differential equations are defined, is analyzed. The distinction between the uniform point and the non-uniform point, as well as between the standard point and the nonstandard point, is showed. A new kind of equations, which differ essentially from those in ex...

Using Nonstandard Analysis, we will provide a rigorous computation for the harmonic oscillator Feynman path integral. The computation will be done without having prior knowledge of the classical path. We will see that properties of classical physics falls out naturally from a purely quantum mechanical point of view. We will assume that the reader is familiar with Nonstandard Analysis.

The conceptual gulf between constructive analysis (in the sense of Bishop [1]) and nonstandard analysis (in the sense of Robinson [7]) is vast, despite the historical genesis of both methodologies from foundational considerations. Those who use nonstandard arguments often say of their proofs that they are “constructive modulo an ultrafilter”; implicit in this statement is the suggestion that su...