2015/08/31 by Sam Sanders · 2 citations
Computer Science · Mathematics · Neuroscience · #Algebra over a field #Calculus (dental) #Cognitive Science and Education Research #Compact space #Computability #Computability, Logic, AI Algorithms #Computable analysis #Computer science #Constructive #Convergence (economics) #Differentiable function #Discrete mathematics #Mathematical and Theoretical Analysis #Mathematics #Pure mathematics #Real analysis #Reverse mathematics #Scope (computer science) #Set (abstract data type) #math.LO
paper · pdf · doi:10.1093/logcom/exaa019
published as Journal of Logic and Computation, Volume: 30, Issue: 1, Jan. 2020, pages 459 - 524 · 61 pages
openalex publication_date 2020/01/01 · arxiv created 2020/12/15 · arxiv updated 2020/12/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 \mathfrakC\mathfrakI 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 \mathfrakC\mathfrakI, 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 \mathfrakC\mathfrakI. Hence, the template \mathfrakC\mathfrakI 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 \mathfrakC\mathfrakI.