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Joel David Hamkins

mathematics and philosophy of the infinite

Joel David Hamkins

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Tag Archives: Rahul Sam

The Gödel incompleteness phenomenon, interview with Rahul Sam

Posted on January 14, 2024 by Joel David Hamkins
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Please enjoy my conversation with Rahul Sam for his podcast, a sweeping discussion of topics in the philosophy of mathematics—potentialism, pluralism, Gödel incompleteness, philosophy of set theory, large cardinals, and much more.

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Posted in Talks, Videos | Tagged large cardinals, multiverse, philosophy of mathematics, pluralism, potentialism, Rahul Sam | Leave a reply

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  • Comment by Joel David Hamkins on A generically absolute variant of HOD?
    @GabeGoldberg Can't countable models (even of full ZFC) be wrong about $0^\sharp$? But in your case, I guess, they also have the real $0^\sharp$, which might be enough to rule out the mistaken cases. I know that with mere $\omega$-models, the true $0^\sharp$ can sit unrecognized while an imposter is given honors. I guess transitivity […]
  • Comment by Joel David Hamkins on A generically absolute variant of HOD?
    A very nice answer.
  • Comment by Joel David Hamkins on A generically absolute variant of HOD?
    That is exactly the argument that I had realized on my walk. I'm home now and about to post it.
  • Comment by Joel David Hamkins on A generically absolute variant of HOD?
    I figured out how to use the coding erasing forcing to provide a model where HgOD is properly contained in gHOD. I'll post soon.
  • Answer by Joel David Hamkins for A generically absolute variant of HOD?
    As I mentioned in the comment, there is another concept known as the generic $\newcommand\HOD{\text{HOD}}\HOD$, denoted $\newcommand\gHOD{\text{gHOD}}\gHOD$ and defined to be the intersection of the $\HOD$s of all the forcing extensions $$\gHOD=\bigcap_G\HOD^{V[G]}.$$ Since every forcing extension is absorbed into a collapse extension via the collapse forcing $\text{Coll}(\omega,\delta)$, which is homogeneous and thus makes the $\HOD$ […]
  • Comment by Joel David Hamkins on A generically absolute variant of HOD?
    After $X$ is collapsed to be countable, then one doesn't ever need to do additional forcing, since when $X$ is countable then handling all $Y$ is the same as handling all countable $Y$, and this is absolute to further extensions.
  • Comment by Joel David Hamkins on A generically absolute variant of HOD?
    Seems natural to compare with what is called the generic HOD, denoted gHOD, defined as the intersection of all HOD of all forcing extensions V[G]. This is discussed in the set-theoretic geology paper.
  • Comment by Joel David Hamkins on Is cardinal assignment impossible in ZC?
    Sometimes people include foundation in ZC, but could you clarify whether you do?

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