bio  website  jdh.hamkins.org 

location  New York City  
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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
1d

awarded  Good Answer 
2d

awarded  Publicist 
Apr 15 
reviewed  Approve Which recursivelydefined 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 nondefinable. 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 wellordering 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 nondefinable 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 
Dedekindfinite 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 
Dedekindfinite 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 ErdosHajnal 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 