A weak theory nonstandard analysis, with types at all finite levels over both the integers and hyperintegers, is developed as a possible framework for reverse mathematics. In this weak theory, we investigate the strength of standard part principles and saturation principles which are often used in practice along with first order reasoning about the hyperintegers to obtain second order conclusio...

Nonstandard analysis is the branch of mathematics concerned with constructing a number system that includes infinitesimals. These are numbers smaller than any real number yet still greater than zero. Although nonstandard analysis was developed in 1966 by Abraham Robinson, it is based off of Gottfried Wilhelm Leibniz’s construction of the calculus, created some 300 years earlier. Nonstandard ana...

An introduction of nonstandard analysis in purely algebraic terms is presented. As an application, we give a nonstandard proof of a characterization theorem for compact subsets of Sym(N).

A number of authors have attempted to apply Nonstandard Analysis to Probability Theory. Unfortunately, the nonstandard reformulations heretofore proposed have retained most of the essential difficulties inherent in the standard formulations. As a result, the application of nonstandard techniques has met with limited success. Hersh [4] produced a nonstandard analogue of Wiener measure. His "meas...

Nonstandard models of electrical-electronics circuits with instantaneous on/off switching are proposed. Nonstandard analysis allows to ground new approaches of circuits models development when their analyses need only partial qualitative study as well bring out the calculation algorithm.

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...

In the present work we approach the study of surfaces using Nonstandard Analysis, by providing first a nonstandard characterization of a surface. Further, the tangent space to a surface is defined as well. M.S.C. 2000: 26E35, 53A05, 53A07.

Nonstandard analysis is often used to prove that certain objects exist, i.e., that certain sets are not empty. In the literature one can find many existence theorems whose only known proofs use nonstandard analysis; see, for example, [AFHL]. This article will survey a new method for existence proofs, based on the concept of a neometric space. We shall state definitions and results (usually with...