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bio website jdh.hamkins.org
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I am professor of mathematics, philosophy and computer science at the City University of New York (CUNY Graduate Center & College of Staten Island). Currently I am visiting at New York University.

My research is mostly in logic, broadly construed, especially mathematical logic and set theory, with connections to other areas.

Find out more on my blog: mathematics and philosophy of the infinite

My new book is available on Amazon
A Mathematician's Year in Japan


1d
awarded  Good Answer
2d
awarded  Publicist
Apr
15
reviewed Approve Which recursively-defined predicates can be expressed in Presburger Arithmetic?
Apr
14
awarded  Nice Answer
Apr
14
comment A specific Model of ZFC
I edited to explain the contribution of Paul Larson at Bedlewo in 2007.
Apr
14
revised A specific Model of ZFC
Edited to explain contribution of Paul Larson at Bedlewo
Apr
10
comment Constructing unnatural transformations
My point was that you will not be able to prove that definable maps are natural, because there are models of set theory in which ALL maps (of any kind, from any set or class to another), including those coming from AC, are definable.
Apr
10
comment Constructing unnatural transformations
You seem in your question to presume that uses of AC are inherently non-definable. But this is not a correct intuition. In fact, the way that we first came to know that AC is consistent is through Godel's constructible universe $L$, where there is a definable well-ordering of the universe. Thus, one can use AC in this universe and still have a definable function. Worse, there are other models of ZFC and even GBC in which every set and class is definable without parameters. So using AC in such a model cannot produce a non-definable class. My view is that the question has a problem at its core.
Apr
9
comment What's the minimum amount of knowledge to start doing research?
Here is a possible instance where knowing too much prevents one from being open to certain truths: math.stackexchange.com/a/40266/413
Apr
9
comment Characterizing Inf and Sup sets
I have seen $R$ used as a partial order, but also I have used $\leq$ as a variable.
Apr
8
comment The role of the rigid relation principle ($RR$) in the Kunen inconsistency
@MathieuBaillif, Unfortunately, this question is still open. Why not give it a shot? There may still be some easy pickings about it...
Apr
7
comment Dedekind-finite arithmetic vs natural numbers arithmetic
Asaf, your answer shows that one cannot construct weird examples just from having an infinite Dedekind finite set, and this seems highly relevant for the question.
Apr
7
comment Dedekind-finite arithmetic vs natural numbers arithmetic
@AsafKaragila, why not post your comments as an answer?
Apr
6
comment The role of the rigid relation principle ($RR$) in the Kunen inconsistency
Your question 3 is about $\neg\text{RR}$, rather than $\text{RR}$, and this is the point of my comments about it. I don't know if we can get Erdos-Hajnal from $\text{RR}$. It would also suffice to get Ulam's theorem on stationary partitions, since this can be used to prove the Kunen inconsistency as in Woodin's argument (see jdh.hamkins.org/generalizationsofkuneninconsistency).
Apr
6
answered The role of the rigid relation principle ($RR$) in the Kunen inconsistency
Apr
6
comment The role of the rigid relation principle ($RR$) in the Kunen inconsistency
For question 3, I would have expected that you inquire whether the Kunen inconsistency can be proved in $\text{NBG}+\text{RR}$, that is, by weakening the use of $\text{AC}$ in the usual proofs to use merely $\text{RR}$ instead. This version of the question would amount to: can you prove the Kunen inconsistency using only the rigid relation principle instead of the axiom of choice? Is that what you had had in mind, or did you really intend to ask question 3 as you have stated it?
Apr
4
comment In set theory, is there a name for a function which maps the empty set to zero and all the others to one?
@PaulTaylor, thank you for withdrawing that comment. I have voted to reopen the question; I'd like to hear your perspective and mathematical ideas. (I'm considerably less interested in overstated opinions about a mathematical perspective thought to be unhelpful.)
Apr
2
comment divisible by all standard prime numbers
Andreas had posted an answer to the effect that there must be $\mathbb{Z}$ chains consisting of composites, but this doesn't by itself answer Jaap's question, and so he deleted it.
Apr
2
awarded  Enlightened
Apr
2
awarded  Nice Answer