A topological approach to non-Archimedean Mathematics

TL;DR

This paper introduces a topological framework called Lambda-theory for non-Archimedean mathematics, enabling the construction of generalized solutions to differential equations and variational problems lacking classical solutions.

Contribution

It presents a novel topological approach to nonstandard analysis, providing an alternative to ultrapower constructions and applying it to solve variational problems.

Findings

Constructed non-Archimedean extensions of the reals via topological completions.

Demonstrated the use of Lambda-theory to solve a calculus of variations problem.

Connected Lambda-theory with existing nonstandard analysis frameworks.

Abstract

Non-Archimedean mathematics (in particular, nonstandard analysis) allows to construct some useful models to study certain phenomena arising in PDE's; for example, it allows to construct generalized solutions of differential equations and variational problems that have no classical solution. In this paper we introduce certain notions of non-Archimedean mathematics (in particular, of nonstandard analysis) by means of an elementary topological approach; in particular, we construct non-Archimedean extensions of the reals as appropriate topological completions of $\mathbb{R}$. Our approach is based on the notion of $\Lambda $-limit for real functions, and it is called $\Lambda $-theory. It can be seen as a topological generalization of the $\alpha $-theory presented in \cite{BDN2003}, and as an alternative topological presentation of the ultrapower construction of nonstandard extensions (in the sense of \cite{keisler}). To motivate the use of $\Lambda $-theory for applications we show how to use it to solve a minimization problem of calculus of variations (that does not have classical solutions) by means of a particular family of generalized functions, called ultrafunctions.