An analysis of the continuum hypothesis
File(s)
Published version
Author(s)
Powell, Andrew
Type
Journal Article
Abstract
This paper analyzes the Continuum Hypothesis, that the cardinality of a set of real numbers is either finite, countably infinite, or the same as the cardinality of the set of all real numbers. It argues (i) that the real numbers are as similar to the natural numbers as possible in the sense that the relationship between any general method of deciding membership of a set of real numbers and the cardinality of the set should be a natural generalization of the case of the same relationship in the case of a set of natural numbers; and (ii) that CH is a very strong choice principle that is maximally efficient as a principle for deciding whether a real number is in a set of real numbers in the sense that it is uniform in deciding membership for every real number in a countable number of steps. The approach taken is to formulate principles equivalent to or weaker than the Continuum Hypothesis and to use techniques from computer science (infinite binary search), information theory, and set theory to prove theorems that support theses (i) and (ii).
Date Issued
2025-02-20
Date Acceptance
2025-02-17
Citation
Axioms, 2025, 14 (3)
ISSN
2075-1680
Publisher
MDPI AG
Start Page
154
End Page
154
Journal / Book Title
Axioms
Volume
14
Issue
3
Copyright Statement
© 2025 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
License URL
Publication Status
Published
Article Number
ARTN 154
Date Publish Online
2025-02-20