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bio website jdh.hamkins.org
location New York City
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I am a professor at the City University of New York, at the College of Staten Island and the CUNY Graduate Center, living in midtown Manhattan. My main research interest lies in mathematical logic, particularly set theory, focusing on the mathematics and philosophy of the infinite. A principal concern has been the interaction of forcing and large cardinals, two central concepts in set theory. I have worked in group theory and its interaction with set theory in the automorphism tower problem, and in computability theory, particularly the infinitary theory of infinite time Turing machines. Recently, I am preoccupied with the set-theoretic multiverse, engaging with the emerging field known as the philosophy of set theory.


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awarded  Nice Answer
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comment Questions about ordering of reals and irrationals
Yes; so just use your favorite representation (say, the one without tailing $1$s), and there is no problem.
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comment Isomorphic subcategories of directed graphs and presets
Could you tell me what it means to be full?
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comment Are hyperreal numbers isomorphic to formal power series?
Todd, one may refer to $\sin(\omega)$ or $e^\omega$ for any hyperreal $\omega$, including infinite elements, simply by appealing to the transfer principle---every function on the reals has a nonstandard analogue, which satisfies all the same first-order expressible properties in the hyperreals that the original function did in the standard reals.
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revised Are hyperreal numbers isomorphic to formal power series?
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answered Are hyperreal numbers isomorphic to formal power series?
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awarded  Nice Answer
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revised Image of poset with Hausdorff interval topology
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revised Image of poset with Hausdorff interval topology
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answered Image of poset with Hausdorff interval topology
Jan
22
revised Is there a compendium of the consistency strength between the most important formal theories?
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Jan
22
comment Is there a compendium of the consistency strength between the most important formal theories?
Can anyone explain to me how to make the images a little smaller?
Jan
22
revised Is there a compendium of the consistency strength between the most important formal theories?
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Jan
22
comment Is there a compendium of the consistency strength between the most important formal theories?
@EmilJeřábek I suppose the phrase, "strong theories like ZFC" is a matter of perspective! In the large cardinal hierarchy, where of course consistency strength issues loom large, one views ZFC as a weak theory. :-)
Jan
22
answered Is there a compendium of the consistency strength between the most important formal theories?
Jan
21
awarded  Notable Question
Jan
19
comment A categorical method to, say, determine the cardinality of a group
For example, can you identify all and only the groups $G$ with cardinality exactly $\aleph_{\omega^2+6}$ by some purely category-theoretic feature? I expect that at least one such group is identifiable this way, and so if we had the equinumerosity relation, we'll have identified them all.
Jan
19
comment A categorical method to, say, determine the cardinality of a group
That is, we want to show $|G|\leq |H|$ if and only if $\varphi(G,H)$ holds in the category of groups, where $\varphi$ is some purely category-theoretic statement to be interpreted in that category in a suitable formal language (or prove that there is no such statement).
Jan
19
comment A categorical method to, say, determine the cardinality of a group
I'm not really satisfied with the $\text{Hom}(\mathbb{Z},G)$ manner of answering this question, since evaluating the cardinality of a Hom set seems to me to be a set-theoretic activity rather than a categorical one. A more satisfying answer would be to show that the relation $|G|\leq |H|$ for groups $G$ and $H$ is definable (or not) purely in the category of groups, in some formal language appropriate for that category, one able to mention certain kinds of diagrams and whatnot. The "straight-edge and compass" that needs to be defined is what precisely this formal language is.
Jan
16
comment A “mother of all groups”? What kind of structures have “mother of all”s?
How could I argue with that?