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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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Comment by Joel David Hamkins on Is there a class choice principle over MK that is equivalent to class well ordering over MK?
I've been speaking with Victoria Gitman, who has worked at length on many of these issues, particularly with the interpreted versions in $\text{ZFC}^-$ models, and there is a big spectrum of principles in this vicinity, with everything open about how they relate to one another or whether they are equivalent or inequivalent. There are some […]
Comment by Joel David Hamkins on On the existence of a real which is not set-generic over $L$
@Asaf, the OP wants a model of ZFC that has a real that is not set generic over $L$. This is expressible in the language of set theory. Your argument about $\text{Col}(\omega,V)$ does not provide such a model, since that extension will not have ZFC. The coding the universe argument does provide such a model, […]
Comment by Joel David Hamkins on Extracting a common convergent indexing from an uncountable family of sequences?
Another cardinal characteristic! The size of the smallest family of functions such that there is no subfamily of the same size with the common convergent subsequence property.
Comment by Joel David Hamkins on Extracting a common convergent indexing from an uncountable family of sequences?
@FarmerS That will be true for regular cardinals above the continuum by pigeon-hole. Not sure about the continuum itself, though.
Comment by Joel David Hamkins on Extracting a common convergent indexing from an uncountable family of sequences?
Very nice. As I mentioned in the comments above, the argument seems to work in any separable space, by going value 0/1 depending on whether the value is within a rational distance of the $n$th point of the dense set. So this shows $𝕔=𝕔_{\mathbb{R}}=𝕔_{\{0,1\}}=\frak{s}$.
Comment by Joel David Hamkins on Extracting a common convergent indexing from an uncountable family of sequences?
@FarmerS I think that is right, and indeed, it seems $𝕔=𝕔_{\mathbb{R}}=𝕔_{\{0,1\}}=\frak{s}$.
Comment by Joel David Hamkins on Extracting a common convergent indexing from an uncountable family of sequences?
And that idea will seem to work for separable Banach spaces also, because we can say 0/1 depending on whether you are within a rational distance of the $n$th point or not. If this is right, we have $𝕔=𝕔_{\mathbb{R}}=𝕔_{\{0,1\}}=\frak{s}$.
Comment by Joel David Hamkins on Extracting a common convergent indexing from an uncountable family of sequences?
@FarmerS Could you clarify? I had some ideas in that direction, but didn't quite see it. I guess you want to replace each sequence with an omega sequence of 0/1 sequences, which give information about above/below a given rational target. If we can make those all converge, then the original sequence will also.

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