https://jdh.hamkins.org/how-ch-could-have-been-fundamental/ Skip to primary content Joel David Hamkins mathematics and philosophy of the infinite [DALL] Search [ ] [Search] Main menu * Home + About + My Curriculum Vita + Contact + Comment Board * Publications + Publication list + Recent publications + Publications by topic o Automorphism towers o Infinitary computability o Infinitary utilitarianism o Large cardinals + My Research Collaborators * Talks + Talks + Recent and Upcoming Talks + Videos * Appointments and Grants + About Me + My Academic Appointments + Grants and Awards * Teaching + About My Courses * Students + About My Graduate Students + List of My Graduate Students * Mathematical Shorts * Math for Kids Post navigation - Previous How the continuum hypothesis could have been a fundamental axiom Posted on July 3, 2024 by Joel David Hamkins Joel David Hamkins, "How the continuum hypothesis could have been a fundamental axiom," Mathematics arXiv (2024), arxiv:2407.02463. [DALL] Abstract. I describe a simple historical thought experiment showing how we might have come to view the continuum hypothesis as a fundamental axiom, one necessary for mathematics, indispensable even for calculus. The full pdf article is available at arxiv.org/pdf/2407.02463. Share: * Click to share on Twitter (Opens in new window) * Click to share on Facebook (Opens in new window) * Click to share on Reddit (Opens in new window) * Click to share on WhatsApp (Opens in new window) * Click to email a link to a friend (Opens in new window) * Click to print (Opens in new window) * More * * Click to share on LinkedIn (Opens in new window) * Click to share on Tumblr (Opens in new window) * Click to share on Pocket (Opens in new window) * Click to share on Pinterest (Opens in new window) * This entry was posted in Publications and tagged categoricity, CH, continuum hypothesis, hyperreal numbers, Leibniz, Newton, thought experiment by Joel David Hamkins. Bookmark the permalink. 6 thoughts on "How the continuum hypothesis could have been a fundamental axiom" 1. [6cff4613]Joseph Shipman on July 3, 2024 at 2:19 pm said: I agree with Solovay that ~CH could also have been seen as fundamental via an intuition for the existence of a countably additive real valued measure. But there's an important difference. None of the alternative foundational schemes you treat in this paper buy us anything new in the arithmetical realm (or more generally anything covered by Shoenfield absoluteness), unless you want to make it second-order and get a little bit further to Grothendieck universes. But taking RVM as fundamental has far more in the way of concrete consequences, up to consistency of measurables for arithmetical sentences and settling most of the wall-known descriptive set theory questions left open by ZFC. Reply | 2. [6cff4613]Joseph Shipman on July 3, 2024 at 2:37 pm said: It's also interesting to take this in the other direction and ask what alternative historical developments could have led us to adopt axioms inconsistent with ZFC. This is well-covered ground in the case of alternatives to AC, but those don't get us anything concrete (Shoenfield absoluteness again). But what developments might have made ZF implausible? Reply | 3. [6cff4613]Joseph Shipman on July 3, 2024 at 2:40 pm said: Thesis: no conceivably plausible alternative historical development of mathematics would contradict ZF about any *arithmetical* statements. Reply | + [88f0]Joel David Hamkins on July 3, 2024 at 3:09 pm said: This seems reasonable, although it is an implicit commitment to Con(ZF), which some have doubted. For example, Silver conjectured the negation, and someone with that view would support, say, PA + not Con(ZF). We know that this is consistent relative to ZF, so one can't really object to the basic coherence of it. Certainly it is plausible. And if one had a strong form of it, denying consistency at the level of $\Sigma_{1000}$-replacement, for example, then this would be an arithmetic statement directly contradicting ZF. Reply | 4. Pingback: The continuum hypothesis could have been a fundamental axiom, CFORS Grad Conference, Oslo, June 2024 | Joel David Hamkins 5. Pingback: How the continuum hypothesis could have been a fundamental axiom, UC Irvine Logic & Philosoph of Science Colloquium, March 2024 | Joel David Hamkins Leave a Reply Cancel reply Your email address will not be published. Required fields are marked * [ ] [ ] [ ] [ ] [ ] [ ] [ ] Comment * [ ] Name * [ ] Email * [ ] Website [ ] [ ] Save my name, email, and website in this browser for the next time I comment. [ ] Notify me of follow-up comments by email. [ ] Notify me of new posts by email. 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