The unreasonable effectiveness of Nonstandard Analysis

TL;DR

This paper demonstrates that classical Nonstandard Analysis contains extensive computational content, converting theorems into effective forms and applying this to foundational classifications in Reverse Mathematics.

Contribution

It introduces a template $rak{CI}$ that systematically extracts effective theorems from nonstandard proofs, bridging Nonstandard Analysis and computable mathematics.

Findings

$rak{CI}$ applies to all Big Five categories in Reverse Mathematics

The template converts nonstandard theorems into effective, computable forms

Herbrandisations imply the original nonstandard theorems

Abstract

As suggested by the title, the aim of this paper is to uncover the vast computational content of classical Nonstandard Analysis. To this end, we formulate a template $\mathfrak{CI}$ which converts a theorem of 'pure' Nonstandard Analysis, i.e. formulated solely with the nonstandard definitions (of continuity, integration, differentiability, convergence, compactness, et cetera), into the associated effective theorem. The latter constitutes a theorem of computable mathematics no longer involving Nonstandard Analysis. To establish the vast scope of $\mathfrak{CI}$, we apply this template to representative theorems from the Big Five categories from Reverse Mathematics. The latter foundational program provides a classification of the majority of theorems from 'ordinary', that is non-set theoretical, mathematics into the aforementioned five categories. The Reverse Mathematics zoo gathers exceptions to this classification, and is studied in [70,71] using $\mathfrak{CI}$. Hence, the template $\mathfrak{CI}$ is seen to apply to essentially all of ordinary mathematics, thanks to the Big Five classification (and associated zoo) from Reverse Mathematics. Finally, we establish that certain 'highly constructive' theorems, called Herbrandisations, imply the original theorem of Nonstandard Analysis from which they were obtained via $\mathfrak{CI}$.