bio  website  jdh.hamkins.org 

location  New York City  
age  
visits  member for  4 years, 8 months 
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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 settheoretic multiverse, engaging with the emerging field known as the philosophy of set theory.
2d

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The (global) theory of Borel equivalence relations
A related question would be: what is known about the rigidity of $\cal B$? 
Jul 23 
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How subset is a set is proved in ZF system?
The notion of "property" you are using is the secondorder notion, used also by Zermelo in his original presentation of the ZF axioms. In time, though, settheorists settled on the firstorder theory. Unless you are WM, I would encourage you to take a look at the firstorder/secondorder distinction in set theory, which I believe is the key to resolving the issue you are asking about. 
Jul 21 
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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 
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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 
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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 
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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 
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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 
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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 
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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 
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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 
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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 
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Quotients of standard Borel spaces
Could you clarify your statement in the secondtolast 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 
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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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