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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 How might mathematics have been different?
Thanks for mentioning my CH thought experiment, Mike. The published version is open access online at doi.org/10.36253/jpm-2936.
Comment by Joel David Hamkins on Is it possible to transform a statement of unsolvabilty to an equivalent one by using a bounded universal quantifier
I can post a (negative) answer for the multi-variable case, if that's what you mean. Unfortunately, it doesn't seem to settle the single-variable case.
Answer by Joel David Hamkins for Do independent collections of infinitely many buttons and infinitely many switches exist in models other than V=L?
Yes, there are many models of ZFC with infinite independent families of buttons and switches. For example, if you start with any model of V=L, and then switch the switches quite a lot, but not push any of the buttons, then you still have a family of independent buttons and switches, but now you don't […]
Comment by Joel David Hamkins on Is every external downshifting elementary embedding $j$ with $j (x) = j [x]$ , an automorphism?
I am a little unclear about whether the embeddings arising in Friedman's theorem need be elementary from $M$ to $M$ . I believe the proof uses back-and-forth realizing types in such a way that they would be, but I wonder whether this just follows easily from the fact that $j$ is an isomorphism of $M$ with […]
Answer by Joel David Hamkins for Is every external downshifting elementary embedding $j$ with $j (x) = j [x]$ , an automorphism?
Let me omit the need for any extra assumption beyond consistency of ZFC. In fact, every countable computably saturated model of ZFC is a counterexample. If $M$ is a countable computably saturated model of ZFC, then there will be an ordinal $θ$ in $M$ for which $V_{θ}^{M} ≺ M$ . One can simply realize the type describing […]
Comment by Joel David Hamkins on Is the set of theorems of a PA + “PA is inconsistent” equivalent to the halting set?
Yes, I agree, the method is very general.
Comment by Joel David Hamkins on Is it possible to transform a statement of unsolvabilty to an equivalent one by using a bounded universal quantifier
If your polynomial $P$ would allow several variables instead of just one, then the answer is negative. But you are interested in the case where there is only one $x$ ?
Answer by Joel David Hamkins for Is the set of theorems of a PA + “PA is inconsistent” equivalent to the halting set?
It is a very nice question. I interpret the theme of the question as: how much can we rely on the theorems of $PA + \neg Con (PA)$ ? In fact, I shall show that a simple transformation for $Σ_{1}$ assertions allows us to fully rely on it. In particular, the answers to both questions are affirmative. The proof will […]

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