A Semi-Constructive Approach to the Hyperreal Line

TL;DR

This paper introduces a semi-constructive nonstandard analysis framework using possibility semantics, defining the $F$-hyperreal line with key properties like transfer and saturation, avoiding the full Axiom of Choice.

Contribution

It presents a novel semi-constructive approach to nonstandard analysis via possibility semantics, defining the $F$-hyperreal line with fundamental properties.

Findings

Shares key properties with classical hyperreal line

Avoids reliance on full Axiom of Choice

Addresses philosophical concerns in nonstandard analysis

Abstract

Using a recent alternative to Tarskian semantics for first-order logic, known as $\textit{possibility semantics}$, I introduce an alternative approach to nonstandard analysis that remains within the bounds of \textit{semi-constructive} mathematics, i.e., does not assume any fragment of the Axiom of Choice beyond the Axiom of Dependent Choices. I define the $F\it{-hyperreal}$ $\it{line}$ ${^\dagger\!\mathcal{R}}$ as a possibility structure and show that it shares many fundamental properties of the classical hyperreal line, such as a Transfer Principle and a Saturation Principle. I discuss the technical advantages of ${^\dagger\!\mathcal{R}}$ over some other alternative approaches to nonstandard analysis and argue that it is well-suited to address some of the philosophical and methodological concerns that have been raised against the application of nonstandard methods to ordinary mathematics.