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Joel David Hamkins

mathematics and philosophy of the infinite

Joel David Hamkins

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Tag Archives: Irvine

How the continuum hypothesis could have been a fundamental axiom, UC Irvine Logic & Philosoph of Science Colloquium, March 2024

Posted on February 12, 2024 by Joel David Hamkins
8

This will be a talk for the Logic and Philosophy of Science Colloquium at the University of California at Irvine, 15 March 2024.

Abstract. With a simple historical thought experiment, I should like to describe how we might easily have come to view the continuum hypothesis as a fundamental axiom, one necessary for mathematics, indispensable even for calculus.

Slides-CH-could-have-been-fundamental-Hamkins-Irvine-March-2024Download

The paper is now available at How the continuum hypothesis could have been a fundamental axiom.

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Posted in Talks | Tagged continuum hypothesis, Irvine, multiverse, pluralism, thought experiment | 8 Replies

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Proof and the Art of Mathematics, MIT Press, 2020

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  • Comment by Joel David Hamkins on Closed connected subset of a connected set
    I had meant that $A$ included the limit point, so it is closed.
  • Comment by Joel David Hamkins on Can a Hamkins infinite time Turing Machine with infinite Super Turing jumps (from higher type oracles) get the power to decide $\Sigma_1^2$ sets?
    @aznek Every ITTM computation either halts at some countable ordinal stage or enters a repeating loop of computation at a countable ordinal, so one never needs uncountable ordinals for these computations. Some people, not me, find it easier to conceive of infinite computations when they take place in a finite amount of real time, and […]
  • Comment by Joel David Hamkins on Are there large cardinal axioms compatible with choice yet not with class well ordering principle?
    I edited to clarify.
  • Comment by Joel David Hamkins on Are there large cardinal axioms compatible with choice yet not with class well ordering principle?
    I am answering the question in the title. The answer is negative, since any large cardinal that is consistent with KM (which includes global choice) is also consistent with KM + the class-well-order principle and vice versa. This version of the question seems to have nothing to do with Reinhardt or choiceless large cardinals.
  • Answer by Joel David Hamkins for Are there large cardinal axioms compatible with choice yet not with class well ordering principle?
    The answer to the title question is negative. Every first-order expressible large cardinal axiom that is consistent with KM is also consistent with KM plus the class-well-order principle. What I claim more specifically is that KM is equiconsistent with KM plus your hyper-class-well-order principle, and furthermore, every model of KM has a submodel to a […]
  • Answer by Joel David Hamkins for What are the known large cardinal axioms for which weaker and stronger set theories "catch up"?
    Let me post this additional answer with what may be relevant interesting context for the general question that your theory is about. Namely, the situation is much better if you use $\newcommand\CC{\text{CC}}\newcommand\KM{\text{KM}}\KM$ instead of $\newcommand\ZFC{\text{ZFC}}\ZFC$. Theorem. The following theories are equiconsistent: Kelley-Morse set theory $\KM$. $\ZFC^-$ plus there is an inaccessible cardinal. The theorem admits […]
  • Comment by Joel David Hamkins on What good naming alternatives are there for the schemas?
    For example, this page is a little better: en.wikipedia.org/wiki/Redundancy_theory_of_truth, linked from your page. But in my view, Tarski should be much more prominent on that page, and also they don't even mention his definition of satisfaction $M\models\varphi$, which is the principle mathematizing instance of this theory.
  • Comment by Joel David Hamkins on What good naming alternatives are there for the schemas?
    That wikipedia page does not seem to be about the disquotational theory of truth, which is 100% about Tarski and his recursive definition of truth in a model. The sentence "snow is white" is true iff snow is white is the canonical example of this theory. It is the same idea behind the T-scheme, which […]

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