111,706 reputation
14302527
bio website jdh.hamkins.org
location New York City
age
visits member for 5 years, 6 months
seen 5 hours ago

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
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
comment 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 forcing-free 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 non-standard 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 ground-model 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
comment 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 Dedekind-MacNeille completion of the strictly increasing members of $\omega^\omega$
My pleasure! I've been enjoying your questions.