Heyting valued considerations on some fundamental existence theorems in modern analysis
type: Item
instance of: scholarly article
title: Heyting valued considerations on some fundamental existence theorems in modern analysis
zbMATH Open document ID: 0664.03036
author: Hirokazu Nishimura
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1988-01-01T00:00:00Z
review text: \textit{C. Rousseau} [Lect. Notes Math. 753, 623-659 (1979; Zbl 0433.32003)] has shown that an intuitionistic division theorem in one variable, interpreted in the Heyting valued set theory \(V^{(\Omega)}\) with \(\Omega\) being the topology \({\mathfrak O}({\mathbb{C}}^{n-1})\) of \({\mathbb{C}}^{n-1}\), gives rise to the classical Weierstrass division theorem in n variables. Following these lines, we will show that the inverse function theorem, demonstrated intuitionistically and interpreted in the Heyting valued set theory \(V^{(\Omega)}\) with \(\Omega\) being the topology of some appropriately chosen space of parameters, is nearly the classical implicit function theorem. Similarly it will be demonstrated that Cauchy's celebrated local existence theorem for ordinary differential equations, established intuitionistically and interpreted in the Heyting valued set theory \(V^{(\Omega)}\) with \(\Omega\) being the topology of some appropriately chosen space of parameters, gives rise to the classical theorems concerning the smooth dependence of local solutions on parameters and on initial conditions.
reviewed by: Hirokazu Nishimura
Mathematics Subject Classification ID: 03F55
zbMATH DE Number: 4085638
zbMATH Keywords: Cauchy's local existence theorem for ordinary differential equations
zbMATH Keywords: Heyting valued set theory
zbMATH Keywords: inverse function theorem
zbMATH Keywords: implicit function theorem
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195174869
OpenAlex ID: W2042766146
cites work: Q3803111
cites work: Concepts of general topology in constructive mathematics and in sheaves
cites work: Concepts of general topology in constructive mathematics and in sheaves, II
cites work: Heyting valued set theory and fibre bundles
cites work: Q3871177
cites work: Q3037435
cites work: Q3939796
DOI: 10.2977/PRIMS/1195174869
Recommended article: Some connections between Heyting valued set theory and algebraic geometry. Prolegomena to intuitionistic algebraic geometry
Recommended article: Two questions from Dana Scott: Intuitionistic topologies and continuous functions
Recommended article: Q3337467
Recommended article: Topological forcing semantics with settling
Recommended article: Global intuitionistic analysis
Recommended article: On the affirmative content of Peano's theorem on differential equations.
Recommended article: Heyting valued set theory and Sato hyperfunctions
Recommended article: Q3722491
Recommended article: Heyting-valued interpretations for constructive set theory
Recommended article: Globalization of intuitionistic set theory
description: scientific article; zbMATH DE number 4085638