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bio website jdh.hamkins.org
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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.


2d
comment The (global) theory of Borel equivalence relations
A related question would be: what is known about the rigidity of $\cal B$?
Jul
23
comment How subset is a set is proved in ZF system?
The notion of "property" you are using is the second-order notion, used also by Zermelo in his original presentation of the ZF axioms. In time, though, set-theorists settled on the first-order theory. Unless you are WM, I would encourage you to take a look at the first-order/second-order distinction in set theory, which I believe is the key to resolving the issue you are asking about.
Jul
21
comment Inverse of a Borel surjection
I'm not quite sure, so if you have a specific question, I would encourage you to ask it.
Jul
20
comment Borel Sets in Sacks Generic Extension
Actually, the assertion is $\Sigma^1_1$, if you already know that $c$ is a Borel code.
Jul
20
comment Borel Sets in Sacks Generic Extension
The reason being, that the assertion that "the Borel set with code $c$ is nonempty" is a $\Sigma^1_2$ assertion, which is absolute between the extension and the ground model, but no ground model real can compute $G$. So the argument has nothing to do with Sacks forcing, and it works for any extension with a new real.
Jul
20
awarded  Nice Answer
Jul
20
comment Inverse of a Borel surjection
I think you mean $\text{id}_Y$ rather than $\text{id}_Z$.
Jul
20
revised Inverse of a Borel surjection
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Jul
19
revised Inverse of a Borel surjection
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Jul
19
revised Inverse of a Borel surjection
Uniformization
Jul
19
revised Inverse of a Borel surjection
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Jul
19
answered Inverse of a Borel surjection
Jul
18
comment Should all equations which appear in a thesis be numbered?
If one is inclined to label every mathematical expression in the dissertation, then, applying this rule to the labels themselves -- they are after all a part of the dissertation, as well as "mathematical" -- one is facing the task of writing an infinitely long dissertation!
Jul
18
comment Quotients of standard Borel spaces
@Burak Actually, I had had the first meaning in mind originally, but my question was "when does it happen?", since the case of $=$ and others show easily that it doesn't always happen. But I agree, the second understanding of surjective also makes an interesting question.
Jul
18
comment Quotients of standard Borel spaces
Obviously we cannot reduce $=$ to any other relation by a surjective reduction, if "surjective" means onto the base set. Are you speaking of "surjective" in the sense of hitting every $F$-class?
Jul
18
comment Can we define an “empirically generic” real number?
I like the topic, but your question is sweeping. In addition, it uses vague undefined terms such as "natural" and "empirical", and I expect that not everyone agrees with the claims you make about these notions. (For example, it seems quite reasonable to me for some to reject the idea that $e$ and $\pi$ are random in any truly robust sense, in light of the fact that they are computable numbers.) Perhaps you could focus the question on a more specific mathematical inquiry?
Jul
18
comment Quotients of standard Borel spaces
I see. By "apparently", you mean that you think this equivalence is true, but so far don't have a proof? OK.
Jul
18
comment Quotients of standard Borel spaces
Could you clarify your statement in the second-to-last paragraph? Does Kechris prove that the existence of a surjective reduction to $=$ is equivalent to having a Borel selector? Your statement about sufficiency suggests that he proves only one direction.
Jul
18
comment Quotients of standard Borel spaces
Yes, that's it, and I've approved your edit.
Jul
18
revised Quotients of standard Borel spaces
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