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

The Book of Infinity

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  • collin237 on A model of set theory with a definable copy of the complex field in which the two roots of -1 are set-theoretically indiscernible
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  • Comment by Joel David Hamkins on Why do we need a transitive model in forcing arguments?
    The Łoś theorem for Boolean ultrapower is the statement that $V^{\mathbb{B}}/U\models\varphi\iff[\![\varphi]\!]\in U$, and this does not use genericity. The existential case uses the fullness property of names, namely that $[\![\exists x\,\varphi(x)]\!]=[\![\varphi(\tau)]\!]$ for some name $\tau$. The supremum is realized. In particular, fullness shows that your situation does not happen.
  • Comment by Joel David Hamkins on Clarification on the relationship of dream mathematics to ZFC and its potential as a synthetic measure theory
    Solovay's construction uses an inaccessible to get a model of ZF+DC + every set is Lebesgue measurable, and Shelah proved that the inaccessible is necessary. If there are enough large cardinals (proper class of Woodin cardinals is enough), then L(ℝ) has all the properties you claim, and that is the model that set theorists usually […]
  • Comment by Joel David Hamkins on Clarification on the relationship of dream mathematics to ZFC and its potential as a synthetic measure theory
    I see that you have edited, but your claims are still not exactly correct. The situation is more subtle than you describe. To get AD in L(ℝ) as an outright consequence of large cardinals, you need more than just infinitely many Woodin cardinals, although with infinitely many Woodin cardinals then one can construct a model […]
  • Comment by Joel David Hamkins on Clarification on the relationship of dream mathematics to ZFC and its potential as a synthetic measure theory
    One cannot produce a model of determinacy just from a model of ZFC, since the consistency strength of ZF+AD is far higher than ZFC. One needs infinitely many Woodin cardinals to build models of AD.
  • Comment by Joel David Hamkins on Cofinal trees in $({}^\omega \omega , \leq^\ast )$
    I'm less sure of that comment above now, since perhaps that replacing process gets in trouble at limit stages.
  • Comment by Joel David Hamkins on Cofinal trees in $({}^\omega \omega , \leq^\ast )$
    Nice question! One trivial observation: cofinal trees of width $
  • Comment by Joel David Hamkins on Impact of the axiom of replacement on finite sets
    Yes, sorry, I had meant the theory without replacement. I have edited.
  • Comment by Joel David Hamkins on Which cardinal $\kappa\geq \omega_1$ is critical for the following property...?
    If it is true for $\kappa$, then isn't it also true for any smaller $\kappa$, including $\kappa=\omega_1$? Given any set of size $\omega_1$, first extend it to a set of size $\kappa$, get the $X_\alpha$'s, and then cut back down to the original set. Or have I misunderstood? Oh, maybe when you cut down, you […]

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