109,522 reputation
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bio website jdh.hamkins.org
location New York City
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visits member for 5 years, 4 months
seen 3 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


3h
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 Although by now I have become aware of your views, let me say that I don't really find those kinds of resentful comments about set theory to be helpful for the kind of civilized mathematical exchange that is usually valued here at MO. Instead, I find that I learn much more from you in the cases where you take pains to explain your perspective in positive assertions that elucidate the mathematical power that it brings to bear on a topic.
8h
comment Proof that no differentiable space-filling curve exists
The answer to your last question must be no, since we could trivially extend a space filling curve by defining it on an extra segment, on which it was very nice, or by inserting a nice smooth function on a segment in the middle, before picking up the space-filling behavior again right at the place we left off.
1d
comment Proof theory and the generalized Riemann hypothesis
Yes, I agree. I took him to be considering the conjecture for all $T>10$, and in the case of large $T$ with very low Kolmogorov complexity, like $T=2^{2^{2^n}}$, then it is consistent with ZF that the conjecture is false because of the $\neg$Con(ZF) argument.
2d
comment Proof theory and the generalized Riemann hypothesis
@EmilJeřábek We can describe huge values of $T$ with fewer than $\log T$ symbols. I can prove assertions about $2^{100!}$ with many fewer than that number of symbols.
2d
comment Proof theory and the generalized Riemann hypothesis
If I have understood, then your conjecture implies Con(ZF), since a ZF-inconsistency would allow us to make short proofs for large $T$. Thus, you won't be able to prove the conjecture in ZF or ZFC alone, unless ZF is inconsistent. In particular, it is relatively consistent with ZF that the conjecture is false, simply because Con(ZF) implies Con(ZF+$\neg$Con(ZF)). Probably one wants to make a consistency assumption in the background when framing the question.
Mar
26
comment $\text{ZFGC}^{\text{−f}}+\text{BAFA}+\exists\kappa(κ \text{ is Reinhardt})$ and its implication
I basically agree, except that we don't know that choice is required in the BAFA case (or that it is required for the Kunen inconsistency).
Mar
26
comment $\text{ZFGC}^{\text{−f}}+\text{BAFA}+\exists\kappa(κ \text{ is Reinhardt})$ and its implication
I don't really understand what you mean.
Mar
26
comment $\text{ZFGC}^{\text{−f}}+\text{BAFA}+\exists\kappa(κ \text{ is Reinhardt})$ and its implication
The only argument that I know uses the global choice principle, and this is stronger than mere AC.
Mar
25
comment Freiling's Axiom of Symmetry Concretized
I view this theorem as the essential content of Freiling's observation that the axiom of symmetry is equivalent to $\neg\text{CH}$; the argument is essentially the same.
Mar
25
revised Freiling's Axiom of Symmetry Concretized
edited body
Mar
25
comment Freiling's Axiom of Symmetry Concretized
Aha, we had the same idea!
Mar
25
answered Freiling's Axiom of Symmetry Concretized
Mar
25
comment Consequences of technically proving anything in Coq (on at least Linux) exploiting a bug?
I had just meant that it was circulating on G+ with Richard's post. I didn't mean any big deal about "news".
Mar
25
comment Consequences of technically proving anything in Coq (on at least Linux) exploiting a bug?
Richard had just posted a link to the first link. (And you don't generally need a Google account to see G+ posts, but I notice that Richard had only shared that privately.)
Mar
25
comment Consequences of technically proving anything in Coq (on at least Linux) exploiting a bug?
I heard about github.com/clarus/falso from Richard Zach plus.google.com/u/0/+RichardZach/posts/CFQJQSQtmRd.
Mar
25
comment Consequences of technically proving anything in Coq (on at least Linux) exploiting a bug?
I have voted to reopen. Coq bugs are in current news.
Mar
25
comment $\text{ZFGC}^{\text{−f}}+\text{BAFA}+\exists\kappa(κ \text{ is Reinhardt})$ and its implication
Our treatment of BAFA makes use of a Global Choice principle, which seems required to make the back-and-forth argument succeed. I am not aware of a treatment without any AC that gets elementary embeddings.
Mar
25
comment $\text{ZFGC}^{\text{−f}}+\text{BAFA}+\exists\kappa(κ \text{ is Reinhardt})$ and its implication
If you drop AC, then it is not known whether there can be such an embedding nontrivial on the well-founded sets. This is the situation of Reinhardt cardinals in ZF without AC.
Mar
24
comment Is countably complete lattice bounded?
Another counterexample: the collection of all countable sets of reals, under the subset relation.
Mar
24
comment Is $\mathcal{P}(\omega)/fin$ with the interval topology a connected space?
I see. Meanwhile, I had some fun figuring it out, and I think it is a good problem. Regarding your remark, it doesn't seem to suffice in general for hyperconnectivity to show merely that nonempty sub-basic open sets intersect. For example, consider any Hausdorff space with three points fixed, and consider the sub-basis consisting of all open sets containing two out of three of those points. This generates the same topology, but notice that any two of these subbasic open sets intersect, even when there can be disjoint open sets.