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Joel David Hamkins

mathematics and philosophy of the infinite

Joel David Hamkins

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Tag Archives: Mindscape

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 Can we have an external automorphism that sends the cardinality of a set to the cardinality of its powerset?
    By considering saturated models, the question is equivalent to asking whether there can be a Scott-cardinality $\kappa$ with the same 1-type as the Scott-cardinality $2^\kappa$. In other words, can set theory with $\kappa$ named be elementarily equivalent to the theory with $2^\kappa$ named? It isn't true in ZFC, since the $\beth$-index with have a different […]
  • Comment by Joel David Hamkins on Determine if there are infinite prime pairs $(p, q)$, such that $p^2+q^2$ is prime
    Sorry, the question as written still makes no sense. The title and question do not express what you intend.
  • Comment by Joel David Hamkins on In 3-valued logic, is there a finite set of connectives whose associated consequence relation is not finitely based?
    Probably best to provide the definition of what it means to be finitely based, and also to explain more fully the consequence relation.
  • Comment by Joel David Hamkins on Determine if there are infinite prime pairs $(p, q)$, such that $p^2+q^2$ is prime
    The title and question don't make sense, since it suggests p and q are infinite sets but also prime numbers. Can you fix the wording?
  • Comment by Joel David Hamkins on Is the classification of finite simple groups provable without the axiom of choice?
    Warning: the usage of "independent of AC" in the question is not the usual meaning of the word "independent" in logic, since the OP is asking whether one can prove the classification result without AC. And the answer to this is yes as Emil mentions, since any arithmetic statement provable in ZFC is also provable […]
  • Comment by Joel David Hamkins on Modal logic of a bounded Cohen-type frame over a countable model of ZFC + CH
    The models you can reach by your iterations can be grouped into exactly 3 theories (or 2 if $M\equiv M[c]$), making a linear chain. So the modal logic here consists exactly of the validities of this frame. I am not sure how most modal logicians would want to name it, but to my way of […]
  • Answer by Joel David Hamkins for Modal logic of a bounded Cohen-type frame over a countable model of ZFC + CH
    The modal logic of forcing with iterations of $\newcommand\Add{\text{Add}}\Add(\omega,1)$ and $\Add(\omega,\omega_2)$ is equal to the trivial logic of validities on a three-element linearly ordered frame $\{0,1,2\}$ with $0
  • Comment by Joel David Hamkins on Extracting a first order proof from a non first order proof
    We always have an effective procedure to find a first-order proof by searching through all proofs.

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