Joel David Hamkins

mathematics and philosophy of the infinite

Joel David Hamkins

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Tag Archives: Science News

Quoted in Science News

Posted on August 31, 2003 by Joel David Hamkins
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I was quoted briefly inĀ Infinite Wisdom: A new approach to one of mathematics’ most notorious problems, Science News, by Erica Klarrreich, August 30, 2003, in an article about Woodin’s attempted solution of the continuum hypothesis.

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Posted in Asides | Tagged CH, in the news, Science News, W. Hugh Woodin | Leave a reply

The Book of Infinity

Recent Comments

  • 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
  • Joel David Hamkins on Lectures on the Philosophy of Mathematics
  • Arthur on Lectures on the Philosophy of Mathematics
  • Joel David Hamkins on Lectures on the Philosophy of Mathematics
  • Saul Schleimer on Lectures on the Philosophy of Mathematics

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  • 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 […]
  • Comment by Joel David Hamkins on Statements in differential geometry independent from ZFC
    @BenjaminSteinberg Every computably undecidable decision problem is saturated with logical undecidability, over any base theory, since otherwise we could solve the problem by searching for proofs. In this sense, any computably undecidable problem of differential geometry will provide an answer to the question, even if one uses much stronger theories than ZFC.
  • Comment by Joel David Hamkins on Dedekind-Peano axioms, but numbers have at most one successor
    Ah, that is helpful.
  • Comment by Joel David Hamkins on Dedekind-Peano axioms, but numbers have at most one successor
    One should think of it as a very weak theory, in which even exponentiation is problematic and induction is possible only for very local phenomena.
  • Comment by Joel David Hamkins on Dedekind-Peano axioms, but numbers have at most one successor
    Ah, I thought you were asking about the second order theory. Huge difference. Since the models of the first-order theory are exactly the cut-offs of models of $I\Delta_0$, as I explain in my answer, the question is whether those theorems are provable in $I\Delta_0$, and there are many open questions about that. For example, pigeon-hole […]
  • Comment by Joel David Hamkins on Dedekind-Peano axioms, but numbers have at most one successor
    It is problematic to speak of "provable" in a second-order theory, since we don't have a sound & complete proof system for second-order logic.
  • Comment by Joel David Hamkins on Dedekind-Peano axioms, but numbers have at most one successor
    Both of those will be provable in the second-order theory. For Bertrand, take the statement for every n, there is a prime between n and 2n. You interpret the numbers up to 2n using digit representation as in my answer. This is a valid consequence of PA2top because it is true in standard finite segments. […]

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