bio  website  jdh.hamkins.org 

location  New York City  
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visits  member for  5 years, 6 months 
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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

comment 
Can the epsilon induction condition be presented with successor case and limit case, similar to transfinite induction?
I think it would be clearer to ask the question also in the body of the question, and not just in the title. 
2d

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On sentences true in all finite groups
@BenjaminSteinberg Can that example be used to answer the current question (with the particular $\Pi_2$ or $\Sigma_2$ forms using only two variables)? 
May 23 
awarded  foundations 
May 22 
comment 
Borel coloring of a graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$
Yes, I agree, although actually that is part of what I meant when I said that it is a coloring of that graph, so that adjacent vertices get different colors. 
May 22 
comment 
Does van der Waerden's Theorem hold for $\omega_1$?
Will, regarding your recent edits, it seems to me that the whole discussion would be easier to follow if in the question you stuck to the original original question only, and made all your other updates as a separate answer below, giving a clear summary as an answer all the further information that you've learned about it. 
May 22 
comment 
Induction and nonstandard halting times of standard machines
François, could you explain a bit more? Why doesn't $s=0$ fulfill that existential trivially? 
May 22 
comment 
Does $Add(\kappa,1)^L$ ever collapse cardinals?
@NateAckerman Adding one Cohen subset to $\kappa$ is isomorphic to adding two such sets, so $\text{Add}(\kappa,1)\cong\text{Add}(\kappa,2)$. In general, $\text{Add}(\kappa,I)$ should consist of partial functions on $\kappa\times I\to 2$, with support of size $\kappa$. In the case of $1$, we simply suppress the extra coordinate as unnecessary. The superscript $L$ on the notation $\text{Add}(\kappa,1)^L$ means that Jonas is considering the version of the forcing notion as defined in $L$ (by taking only such conditions that are in $L$), but he wants to force with that poset overe $V$. 
May 21 
comment 
Borel coloring of a graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$
But meanwhile, of course, the forcingfree proof is obviously easier to grasp for those who don't know forcing! I think of the situation as analogous to the difference in perspective between using nonstandard analysis and arguing with actual nonstandard infinitesimals versus arguing with epsilons and deltas and sets in an ultrafilter. Sometimes it is clarifying to have the actual infinitesimal or the actual generic object, even in cases where the need for it is eliminable. 
May 21 
comment 
Borel coloring of a graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$
+1 Very nice. Despite first appearances, this answer and mine are actually rather close, using the same underlying idea, and one can view this as an instance of the connection of Cohen genericity with comeager sets: a function is $V$generic just in case it is in every groundmodel comeager set. So the clopen set here corresponds to the condition $p$ in my answer, with $B$ the set of functions having the color that $g$ gets, and $B+e_k$ corresponds to my argument with $g'$, which made a change to $g$ outside $p$. 
May 21 
answered  Borel coloring of a graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$ 
May 21 
comment 
Is it possible to partition $\mathbb R^3$ into unit circles?
Regarding whether AC is required, or whether there is a Borel partition of the various types, I asked the following question: mathoverflow.net/q/93601/1946 
May 21 
comment 
Graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$
Please go ahead! 
May 21 
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Graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$
This graph is Borel, and it would be interesting to ask whether there can be a Borel $\mathbb{N}$coloring. If your edge relation connected points that differed finitely, then I believe there can be no Borel coloring, since it would give rise to a Borel selector on $E_0$, which is impossible. (For example, there can be no Borel coloring of the type that Chris describes.) But your relation is finer than $E_0$ and so a coloring needn't color the entire $E_0$ equivalence class differently. 
May 20 
revised 
Is this version of van der Waerden's Theorem consistent with ZFC?
added 1 character in body 
May 20 
revised 
Is this version of van der Waerden's Theorem consistent with ZFC?
added 107 characters in body 
May 20 
comment 
Is this version of van der Waerden's Theorem consistent with ZFC?
Does anyone know if the function $T$ that I am using here has an established name? 
May 20 
revised 
Is this version of van der Waerden's Theorem consistent with ZFC?
Provide counterexample with natural arithmetic on ordinals instead of usual arithmetic 
May 19 
revised 
Does van der Waerden's Theorem hold for $\omega_1$?
added 1094 characters in body 
May 19 
comment 
Looking for reference or proof to some facts stated on Anand Pillay's book
Welcome to MathOverflow! 
May 19 
comment 
DedekindMacNeille completion of the strictly increasing members of $\omega^\omega$
My pleasure! I've been enjoying your questions. 