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Joel David Hamkins

mathematics and philosophy of the infinite

Joel David Hamkins

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Tag Archives: NWO

Modal logics in set theory, NWO grants, 2006 – 2008

Posted on April 30, 2006 by Joel David Hamkins
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Modal logics in set theory, (with Benedikt Löwe), Nederlandse Organisatie voor Wetenschappelijk (B 62-619), 2006-2008.

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Posted in Grants and Awards | Tagged Amsterdam, NWO | Leave a reply

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Proof and the Art of Mathematics, MIT Press, 2020

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  • Answer by Joel David Hamkins for About forcing method
    Yes, part of your perspective is correct—we can make sense of forcing over any model of set theory. We can in effect internalize the concepts of forcing and express everything we need inside ZFC rather than in the metatheory. The assertion of "φ is forceable", meaning that it is true in some forcing extension, is […]
  • Comment by Joel David Hamkins on Is it possible to transform a statement of unsolvabilty to an equivalent one by using a bounded universal quantifier
    OK, I have posted the argument I had in mind for the multi-variable case.
  • Answer by Joel David Hamkins for Is it possible to transform a statement of unsolvabilty to an equivalent one by using a bounded universal quantifier
    Let me answer negatively for the case where the polynomial p is a polynomial in several variables p(x1,…,xn) over the integers. To begin, for any given program q, consider the c.e. set Eq that undertakes the algorithm of checking whether q(0) halts, then whether q(1) halts, then whether q(2) halts, and so forth, and each […]
  • Comment by Joel David Hamkins on How might mathematics have been different?
    Thanks for mentioning my CH thought experiment, Mike. The published version is open access online at doi.org/10.36253/jpm-2936.
  • Comment by Joel David Hamkins on Is it possible to transform a statement of unsolvabilty to an equivalent one by using a bounded universal quantifier
    I can post a (negative) answer for the multi-variable case, if that's what you mean. Unfortunately, it doesn't seem to settle the single-variable case.
  • Answer by Joel David Hamkins for Do independent collections of infinitely many buttons and infinitely many switches exist in models other than V=L?
    Yes, there are many models of ZFC with infinite independent families of buttons and switches. For example, if you start with any model of V=L, and then switch the switches quite a lot, but not push any of the buttons, then you still have a family of independent buttons and switches, but now you don't […]
  • Comment by Joel David Hamkins on Is every external downshifting elementary embedding j with j(x)=j[x], an automorphism?
    I am a little unclear about whether the embeddings arising in Friedman's theorem need be elementary from M to M. I believe the proof uses back-and-forth realizing types in such a way that they would be, but I wonder whether this just follows easily from the fact that j is an isomorphism of M with […]
  • Answer by Joel David Hamkins for Is every external downshifting elementary embedding j with j(x)=j[x], an automorphism?
    Let me omit the need for any extra assumption beyond consistency of ZFC. In fact, every countable computably saturated model of ZFC is a counterexample. If M is a countable computably saturated model of ZFC, then there will be an ordinal θ in M for which VθM≺M. One can simply realize the type describing […]

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