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

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Comment by Joel David Hamkins on Partitioning $\mathbb{R}^n$ into closed sets
I think several of these comments should be posted as answers.
Comment by Joel David Hamkins on Upward reflection of rank-into-rank cardinals
Yes, that's right.
Comment by Joel David Hamkins on Upward reflection of rank-into-rank cardinals
The embedding $j$ fixes every set up to the critical point, so the image of $\omega$ under $j$ is just $\omega$ itself. But the critical sequence $\kappa_0=cp(j)$, $\kappa_{n+1}=j(\kappa_n)$, which is definable from $j$, is cofinal in $\lambda$, and this is what causes the problem.
Comment by Joel David Hamkins on Absoluteness of Countability
Yes, I should have removed it when I added the forcing tag.
Comment by Joel David Hamkins on Upward reflection of rank-into-rank cardinals
No, that isn't right. The structure $V_\lambda$ cannot satisfy ZF in the language with $j$, because the critical sequence will be an $\omega$-sequence unbounded in $\lambda$, which would violate the replacement axiom in $\langle V_\lambda,\in,j\rangle$. In particular, $j$ is not definable in $V_\lambda$, and you cannot add it without destroying ZF.
Comment by Joel David Hamkins on What examples of existence forcing proofs are there?
@AndrejBauer I don't know of any formal accounts of what I have called the common usage of "constructive"; it seems to be used mainly informally. Of course, the informal usage is connected with the usage in constructive logic, which I view as a more extreme form of it. But I also think that the ordinary […]
Comment by Joel David Hamkins on What examples of existence forcing proofs are there?
I frequently use the word "constructive" to mean just its ordinary mathematical meaning, even when talking about proving something in ZFC using classical logic. For example, one proof of existence is more constructive than another, when the object claimed to exist is exhibited in a more direct way. This concept is an important mathematical idea, […]
Comment by Joel David Hamkins on What examples of existence forcing proofs are there?
@AndrejBauer Many mathematicians don't agree that the word "constructive" is owned by the intuitionistic logicians. This word is very commonly used in mathematics with a much looser meaning to mean something like "following a construction", even when that construction does not obey the rules of whatever constructive logic one might have in mind. In this […]

New York Logic

Title TBA
Resource Sharing Linear Logic, I
Sub-Kripkean epistemic models I
Generic logical semantics of justifications III
Generic logical semantics of justifications II
Generic logical semantics of justifications.
Cut Elimination
Functors and infinitary interpretations of structures
Recursive Reducts of PA III
Kripke completeness of strictly positive modal logics and related problems

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