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Tag Archives: Simons Foundation

Research in set theory, Simons Foundation, Collaborative Grant Award, 2011 – 2016

Posted on June 30, 2011 by Joel David Hamkins

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J. D. Hamkins, Research in set theory, Simons Foundation, Collaboration Grant Award, 2011-2016.

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Posted in Grants and Awards | Tagged Simons Foundation | Leave a reply

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Mathoverflow activity

Comment by Joel David Hamkins on Automorphisms over models of L?
*for any standard finite $k \geq 1$ ...
Comment by Joel David Hamkins on Automorphisms over models of L?
And similarly, you cannot move $κ$ to $κ^{+ +}$ or to $κ^{+ + +}$ or indeed to $κ^{+^{k}}$ for any standard finite $k$ , since the ordinal index would have a different value modulo $k + 1$ .
Comment by Joel David Hamkins on Which sets of sentences can be "continuously" decided in an ultraproduct?
Do you mean to take the set of ultrafilters for which the ultrapower satisfies $φ$ ?
Comment by Joel David Hamkins on Ordinal recursion by and before von Neumann 1928
Dedekind 1888 proved the recursion theorem from second-order induction for arithmetic. This was crucial for his famous categoricity result, showing that any two models of his theory of the successor are isomorphic. The isomorphism is defined by recursion, which he proved is successful.
Comment by Joel David Hamkins on Can a Borel set in the plane intersect every arc but contain none?
Related question: mathoverflow.net/q/93601/1946
Comment by Joel David Hamkins on Does every dense down-set in $P (ω) / (fin)$ contain a partition?
It is not a cBa, since no countably infinite antichain has a supremum, since by Hausdorff's method you can squeeze another set below any candidate. Every $(ω, ω)$ gap is filled.
Comment by Joel David Hamkins on Does every dense down-set in $P (ω) / (fin)$ contain a partition?
This won't imply DC, since one can have AC for all sets at this level, and a violation of DC only very high up in the set-theoretic universe.
Comment by Joel David Hamkins on Extracting partitions from dense open subsets of complete Boolean algebras without choice
Conceivably it is easier to find antichains and partitions in complete Boolean algebras than in partial orders. For example, every antichain in a cBa extends to a partition by taking the complement of the supremum. Do we know the question for partial orders is equivalent to the question for complete Boolean algebras?

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