{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T00:56:30Z","timestamp":1760057790685,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,2,20]],"date-time":"2025-02-20T00:00:00Z","timestamp":1740009600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>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).<\/jats:p>","DOI":"10.3390\/axioms14030154","type":"journal-article","created":{"date-parts":[[2025,2,20]],"date-time":"2025-02-20T11:03:37Z","timestamp":1740049417000},"page":"154","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["An Analysis of the Continuum Hypothesis"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3470-1038","authenticated-orcid":false,"given":"Andrew","family":"Powell","sequence":"first","affiliation":[{"name":"Institute for Security Science and Technology, Imperial College, South Kensington Campus, London SW7 2AZ, UK"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,20]]},"reference":[{"key":"ref_1","first-page":"321","article-title":"Axioms of Set Theory","volume":"Volume 89","author":"Barwise","year":"1977","journal-title":"Handbook of Mathematical Logic"},{"key":"ref_2","unstructured":"G\u00f6del, K. (1940). The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory, Princeton University Press."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1143","DOI":"10.1073\/pnas.50.6.1143","article-title":"The Independence of the Continuum Hypothesis","volume":"50","author":"Cohen","year":"1963","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"150","DOI":"10.2307\/2267778","article-title":"On Notation for Ordinal Numbers","volume":"3","author":"Kleene","year":"1938","journal-title":"J. Symb. Log."},{"key":"ref_5","unstructured":"Kanamori, A. (1994). The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings, Springer."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1080\/00029890.2002.11919865","article-title":"The Continuum Problem","volume":"109","author":"Stillwell","year":"2002","journal-title":"Am. Math. Mon."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1007\/s00407-014-0142-8","article-title":"How Woodin changed his mind: New thoughts on the Continuum Hypothesis","volume":"69","author":"Rittberg","year":"2015","journal-title":"Arch. Hist. Exact Sci."},{"key":"ref_8","first-page":"793","article-title":"Martin\u2019s Maximum++ implies Woodin\u2019s axiom","volume":"192","author":"Schindler","year":"2021","journal-title":"Ann. Math."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Moore, J.T. (2010, January 19\u201327). The Proper Forcing Axiom. Proceedings of the International Congress of Mathematicians 2010, ICM 2010, Hyderabad, India.","DOI":"10.1142\/9789814324359_0038"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11511-013-0089-7","article-title":"Forcing Axioms and the Continuum Hypothesis","volume":"210","author":"Aspero","year":"2013","journal-title":"Acta Math."},{"key":"ref_11","unstructured":"Todorcevic, S. The Power-Set of \u03c91 and the Continuum Hypothesis. Proceedings of the EFI Final Report, 2013., Available online: https:\/\/www.math.toronto.edu\/~stevo\/Todorcevic_Structure5.pdf."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Caicedo, A.E., and Cummings, J. (2017). What makes the continuum \u21352. Foundations of Mathematics, American Mathematical Society.","DOI":"10.1090\/conm\/690"},{"key":"ref_13","unstructured":"Aumann, J.R. (1957). Set Theory, Chelsea. [3rd ed.]."},{"key":"ref_14","first-page":"783","article-title":"Descriptive set theory: Projective sets","volume":"90","author":"Barwise","year":"1977","journal-title":"Handbook of Mathematical Logic"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Kechris, A.S. (1995). Classical Descriptive Set Theory, Springer.","DOI":"10.1007\/978-1-4612-4190-4"},{"key":"ref_16","unstructured":"Zalta, E.N. (2019). The Continuum Hypothesis. The Stanford Encyclopedia of Philosophy, Spring 2019 ed., Metaphysics Research Lab, Stanford University."},{"key":"ref_17","unstructured":"Jech, T. (2002). Set Theory: The Third Millenium Edition, Revised and Expanded, Springer."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Halmos, P.R. (1974). Naive Set Theory, Undergraduate Texts in Mathematics; Springer.","DOI":"10.1007\/978-1-4757-1645-0"},{"key":"ref_19","first-page":"168","article-title":"Conceptions of the Continuum","volume":"51","author":"Feferman","year":"2009","journal-title":"Intellectica"},{"key":"ref_20","unstructured":"Spival, M. (2006). Calculus, Cambridge University Press."},{"key":"ref_21","unstructured":"Dauben, J.W. (1979). Georg Cantor: His Mathematics and His Philosophy of the Infinite, Princeton University Press."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Li, M., and Vitanyi, P. (1997). An Introduction to Kolmogorov Complexity and Its Applications, Springer.","DOI":"10.1007\/978-1-4757-2606-0"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A Mathematical Theory of Communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_24","unstructured":"Stone, J.V. (2022). Information Theory: A Tutorial Introduction, Sebtel Press."},{"key":"ref_25","unstructured":"Rathjen, M. (2006, January 22\u201330). The art of ordinal analysis. Proceedings of the International Congress of Mathematicians, European Mathematical Society, Madrid, Spain."},{"key":"ref_26","first-page":"345","article-title":"About the axiom of choice","volume":"90","author":"Barwise","year":"1977","journal-title":"Handbook of Mathematical Logic"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.4064\/fm-34-1-1-5","article-title":"L\u2019hypoth\u00e8se g\u00e9n\u00e9ralis\u00e9e du continu et l\u2019axiome du choix","volume":"34","year":"1947","journal-title":"Fundam. Math."},{"key":"ref_28","unstructured":"Horowitz, E., and Sahni, S. (1978). The Fundamentals of Algorithms, Pitman."},{"key":"ref_29","first-page":"403","article-title":"Forcing","volume":"90","author":"Barwise","year":"1977","journal-title":"Handbook of Mathematical Logic"},{"key":"ref_30","unstructured":"Kunen, K. (1980). Set Theory: An Introduction to the Independence Proofs, Elviser Science Publisher."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1016\/S0049-237X(08)71685-9","article-title":"Semantical Analysis of I","volume":"40","author":"Crossley","year":"1965","journal-title":"Formal Systems and Recursive Functions"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/3\/154\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:39:08Z","timestamp":1760027948000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/3\/154"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,20]]},"references-count":31,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,3]]}},"alternative-id":["axioms14030154"],"URL":"https:\/\/doi.org\/10.3390\/axioms14030154","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,2,20]]}}}