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Joel David Hamkins

mathematics and philosophy of the infinite

Joel David Hamkins

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Tag Archives: Sean Carroll

Puzzles of reality and infinity, Mindscape Podcast

Posted on July 15, 2024 by Joel David Hamkins
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I was interviewed by Sean Carroll for his Mindscape Podcast, broadcast 15 July 2024.

282 | Joel David Hamkins on Puzzles of Reality and Infinity

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Posted in Talks, Videos | Tagged Gödel, incompleteness, Mindscape, multiverse, philosophy of mathematics, Sean Carroll | Leave a reply

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Proof and the Art of Mathematics, MIT Press, 2020

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  • Comment by Joel David Hamkins on Forcing with strong binary trees
    Can one implement the fusion arguments? It seems delicate to enforce the strong splitting requirement...
  • Comment by Joel David Hamkins on Interpretability and relative consistency with Kolmogorov randomness axioms
    Can you tell us what is $R$?
  • Comment by Joel David Hamkins on What tools are there, apart from using the countable chain condition, to show that forcing preserves cardinals?
    Perhaps someone should collect the various ideas in the comments and post an answer? I think the site works better when answers are posted as answers.
  • Comment by Joel David Hamkins on What tools are there, apart from using the countable chain condition, to show that forcing preserves cardinals?
    In case you are not aware, the generalized Delta-system lemma (theorem 9.19 in Jech) is extremely useful for proving instances of $\delta$-c.c. for higher cardinals, including your case. Also, an often useful variation of closure would be strategic closure, which shows that no new sequences of a certain length over the ground model are added, […]
  • Comment by Joel David Hamkins on Full name (in the sense of forcing) for a partial order
    This is also possible, since we can mix the names $\tau$ that I had used, with condition $p$, with the name $1_\pi$ having value $\neg p$. That was my first idea, actually, but I realized this complication was not needed for the version of fullness you had stated.
  • Answer by Joel David Hamkins for Full name (in the sense of forcing) for a partial order
    Suppose that we have a $\newcommand\P{\mathbb{P}}\P$-name of a partial order $\langle\pi,\leq_\pi,1_\pi\rangle$. So it is forced that this is a partial order and also that $1_\pi\in\pi$. Let $\pi'$ consist of all nice names $\tau$ for elements of $\pi$. These are the names $\tau$ for which there is a condition $p$ and maximal antichain of conditions $r\leq […]
  • Comment by Joel David Hamkins on Does Con(ZF + Reinhardt) really imply Con(ZFC + I0)?
    @tlonuqbar Sorry, I don't recall where I heard this first exactly, but my source might have been Laver's complaints about the terminology, since I do recall him making such a statement in a lecture talk.
  • Comment by Joel David Hamkins on How "natural" is Paris-Harrington?
    To my way of thinking, we should rather ground our notions of "natural" in mathematical features than in sociological ones. For example, perhaps we should say that properties are natural when they exhibit certain kinds of absoluteness or uniformity or preservation and so on. The difficulty is to find the right notions that align in […]

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