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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Comment by Joel David Hamkins on Can relative constructibility of subsets of some set via set unions have a limit below the cardinality of that set?
You have new subsets arising, yes, but the union issue seems moot to me, since the union of set always appears with the set or before.
Comment by Joel David Hamkins on Can relative constructibility of subsets of some set via set unions have a limit below the cardinality of that set?
My point is that the union of any set in the hierarchy always appears at or before that same stage, as a general feature of these hierarchies. You never need to go higher to construct the union of a set you already have.
Comment by Joel David Hamkins on Can relative constructibility of subsets of some set via set unions have a limit below the cardinality of that set?
Most accounts of $L_\gamma(A)$ and $L_\gamma[A]$ are closed under unions of their sets. So this would be automatically true for any $A$ and any $\gamma$, and any set $u$, not just subsets of $A$. So I'm not sure what you intend to ask?
Comment by Joel David Hamkins on How many models of Peano arithmetic are isomorphic to the standard model and how many models of Peano arithmetic are non-standard?
Yes. There are $2^{\omega_1}$ many distinct $\omega_1$-like models of PA (and of ZFC), as explained in this article: jdh.hamkins.org/….
Comment by Joel David Hamkins on Does every countable set of Turing degrees have an upper bound, without AC?
I think it is good to concentrate the issue on mod-finite.
Comment by Joel David Hamkins on Does every countable set of Turing degrees have an upper bound, without AC?
In the proof of the small observation, I feel more should be said. The point is that if you had an upper bound for the mod-finite classes, then you could fix a representative, and use that to choose least-programs producing a suitable set in the nth Turing degree from that oracle. So the point isn't […]
Comment by Joel David Hamkins on Does every countable set of Turing degrees have an upper bound, without AC?
Is there some standard way to preserve $\langle[r_n]_{\text{fin}}\mid n\in\mathbb{N}\rangle$ in a symmetric extension, while lacking $\langle r_n\mid n\in\mathbb{N}\rangle$? (I feel you are about to explain the superiority of symmetric systems to me.)
Comment by Joel David Hamkins on Does every countable set of Turing degrees have an upper bound, without AC?
Yes, of course.

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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