Equivalents of the (weak) fan theorem
type: Item
instance of: scholarly article
title: Equivalents of the (weak) fan theorem
zbMATH Open document ID: 1060.03081
author: Iris Loeb
published in: Annals of Pure and Applied Logic
publication date: 2005-01-11T00:00:00Z
review text: The author proves that several theorems are equivalent to the fan theorem over an intuitionistic base system. The base system, WKV, is a weakened version of an axiom system used by \textit{S. C. Kleene} and \textit{R. E. Vesley} [The foundations of intuitionistic mathematics. Especially in relation to recursive functions. Amsterdam: North-Holland Publishing Company (1965; Zbl 0133.24601)]. Theorems that are shown to be equivalent to the fan theorem include the Heine-Borel theorem for \([0,1]\), various properties of continuous functions on closed intervals (uniform continuity, Riemann integrability, and boundedness), the Lindenbaum Lemma, and completeness and compactness theorems of propositional calculus. Intuitionistically, the fan theorem is not equivalent to weak König's lemma, so the preceding results are distinct from the parallel results of reverse mathematics as presented by \textit{S. G. Simpson} [Subsystems of second order arithmetic. Berlin: Springer (1999; Zbl 0909.03048)].
reviewed by: Jeffry L. Hirst
Mathematics Subject Classification ID: 03F35
Mathematics Subject Classification ID: 03B30
Mathematics Subject Classification ID: 03F55
zbMATH DE Number: 2126477
zbMATH Keywords: reverse mathematics
zbMATH Keywords: intuitionistic mathematics
zbMATH Keywords: second-order arithmetic
zbMATH Keywords: WKV
zbMATH Keywords: Heine-Borel
zbMATH Keywords: compactness
MaRDI profile type: Publication
full work available at URL: j.apal.2004.07.002
OpenAlex ID: W1982061352
cites work: Q3679172
cites work: Q3754620
cites work: Q5343325
cites work: Heine-Borel does not imply the Fan Theorem
cites work: Q4220572
cites work: Metamathematical investigation of intuitionistic arithmetic and analysis. With contributions by C. A. Smorynski, J. I. Zucker and W. A. Howard
DOI: 10.1016/J.APAL.2004.07.002
description: scientific article; zbMATH DE number 2126477