Affiliation:
1. Department of Mathematics, TU Darmstadt, 64289, Germany
Abstract
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 ${\mathfrak{C}\mathfrak{I}}$ which converts a theorem of ‘pure’ Nonstandard Analysis, i.e. formulated solely with the nonstandard definitions (of continuity, integration, differentiability, convergence, compactness, etc.), into the associated effective theorem. The latter constitutes a theorem of computable mathematics no longer involving Nonstandard Analysis. To establish the huge scope of ${\mathfrak{C}\mathfrak{I}}$, 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’, i.e. non-set theoretical, mathematics into the aforementioned five categories. The Reverse Mathematics zoo gathers exceptions to this classification, and is studied in [ 74, 77] using ${\mathfrak{C}\mathfrak{I}}$. Hence, the template ${\mathfrak{C}\mathfrak{I}}$ 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 Herbrandizations, also imply the original theorem of Nonstandard Analysis from which they were obtained via ${\mathfrak{C}\mathfrak{I}}$.
Publisher
Oxford University Press (OUP)
Subject
Logic,Hardware and Architecture,Arts and Humanities (miscellaneous),Software,Theoretical Computer Science
Reference106 articles.
1. Pure and Applied Mathematics;Albeverio,1986
2. Gödel’s functional (“Dialectica”) interpretation;Avigad;Handbook of Proof Theory,1998
3. A functional interpretation for nonstandard arithmetic;van den Berg;Annals of Pure and Applied Logic,2012
4. Reverse mathematics and parameter-free transfer;van den Berg;Annals of Pure and Applied Logic,2019
5. Uniform Heyting arithmetic;Berger;Annals of Pure and Applied Logic,2005
Cited by
4 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献