Joel David Hamkins

mathematics and philosophy of the infinite

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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

The Book of Infinity

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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
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Mathoverflow activity

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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