In the present paper, we propose a new axiomatic approach to nonstandard analysis and its application general theory of spatial structures in terms category theory. Our framework is based on idea internal set theory, while make use an endofunctor U topos sets S together with natural transformation υ, instead as “standard”, “internal”, or “external”. Moreover, notion space called U-space, USpace...

Recently, Dag Normann and the author have established a connection between higher-order computability theory and Nonstandard Analysis; new results in both fields are obtained by exploiting this connection ([28]). Now, on the side of computability theory, the technique known as Grilliot’s trick constitutes a template for explicitly defining the Turing jump functional (∃) in terms of a given effe...

In this paper, we highlight a new computational aspect of Nonstandard Analysis relating to higher-order computability theory. In particular, we prove that the Gandy-Hyland functional equals a primitive recursive functional involving nonstandard numbers inside Nelson’s internal set theory. From this classical and ineffective proof in Nonstandard Analysis, a term from Gödel’s system T can be extr...

In this paper, we introduce systems of nonstandard second-order arithmetic which are conservative extensions of systems of second-order arithmetic. Within these systems, we do reverse mathematics for nonstandard analysis, and we can import techniques of nonstandard analysis into analysis in weak systems of second-order arithmetic. Then, we apply nonstandard techniques to a version of Riemann’s ...

In this paper we give an introduction to nonstandard analysis, starting with an ultrapower construction of the hyperreals. We then demonstrate how theorems in standard analysis “transfer over” to nonstandard analysis, and how theorems in standard analysis can be proven using theorems in nonstandard analysis.

This paper concerns with the modeling and construction of a fifth order method for two dimensional acoustic wave equation in heterogenous media. The method is based on a standard discretization of the problem on smooth regions and a nonstandard method for nonsmooth regions. The construction of the nonstandard method is based on the special treatment of the interface using suitable jump conditio...

We present Nonstandard Analysis by three axioms: the Extension, Transfer and Saturation Principles in the framework of the superstructure of a given innnite set. We also present several applications of this axiomatic approach to point-set topology. Some of the topological topics such as the Hewitt realcompactiication and the nonstandard characterization of the sober spaces seem to be new in the...