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Joel David Hamkins

mathematics and philosophy of the infinite

Joel David Hamkins

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Tag Archives: Digital Gnosis

Frege’s philosophy of mathematics—Interview with Nathan Ormond, December 2021

Posted on October 10, 2021 by Joel David Hamkins
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I was interviewed by Nathan Ormond for a discussion on Frege’s philosophy of mathematics for his YouTube channel, Digital Gnosis, on 10 December 2021 at 4pm.

The interview concludes with a public comment and question & answer session.

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Posted in Events, Talks, Videos | Tagged Digital Gnosis, Frege, philosophy of mathematics | Leave a reply

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  • Answer by Joel David Hamkins for Decimal expansion definition of real numbers, constructively
    The topic is related to a certain famous mistake that Alan Turing had made in his famous 1936 article, On computable numbers, with an application to the Entscheidungsproblem. (See my blog post, Alan Turing, On computable numbers, where I discuss the issue.) The paper is an incredible achievement. He accomplishes so much: he defines and […]
  • Comment by Joel David Hamkins on Almost disjoint families of true cardinality ${\frak c}$
    If $\mathcal{A}$ has only one member (or is empty), then it will trivially have true cardinality $\frak{c}$, so the question reduces to whether there is MAD family not of true cardinality $\frak{c}$.
  • Comment by Joel David Hamkins on Why doesn't BukovskĂ˝'s classification of forcing extensions solve Shelah's dream?
    I don't think you've stated the result correctly, since we could just take $g$ to be the constant 0 function on that domain and fulfill those properties trivially, even when $N$ is not a $\kappa$-c.c. extension of $M$. Probably you want to insist also that $f(i)\in g(i)$.
  • Comment by Joel David Hamkins on Does type theory have an equivalent to large cardinal axioms?
    Meanwhile, of course, there are different kinds of models, and model-existence assertions become more valuable when the models have other nice features. For example, having a transitive model of a large cardinal set theory is much stronger than mere consistency, and having the large cardinal in some $V_\theta$ is stronger still, in many cases equivalent […]
  • Comment by Joel David Hamkins on Does type theory have an equivalent to large cardinal axioms?
    @AndrejBauer Well, the completeness theorem says that a first-order theory is consistent if and only if it has a model, so in this sense, yes---whenever a theory is consistent, then we can cook up a model. The objection to the instrumentalist dodge that Steel and others mount is that it is the actual existence and […]
  • Comment by Joel David Hamkins on Does type theory have an equivalent to large cardinal axioms?
    This move, namely, replacing a large cardinal existence assertion with a consistency statement, is known as the "instrumentalist dodge," and it is strongly criticized in some quarters of the philosophy of set theory, notably by John Steel.
  • Comment by Joel David Hamkins on Does type theory have an equivalent to large cardinal axioms?
    Meanwhile, every large cardinal axiom is dominated in consistency strength by an arithmetic assertion, namely, the assertion that that large cardinal axiom is consistent with ZFC. And can't we easily import any given arithmetic assertion into type theory? This would amount at bottom to adopting not the actual existence of the large cardinal, but the […]
  • Comment by Joel David Hamkins on Can hereditarily non-well-orderable sets exist?
    @WillBrian Ah, it seems we had essentially the same idea. Your answer folds in the ZF interpretation of the anti-foundational model. I was trying AFA first, using a tree with no leaves and no infinite branch, but bisimilarities allow things to collapse.

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