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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


13h
comment What is the fewest number of points you must delete from $\mathbb{R}^3$ to make it not simply connected?
@PietroMajer For path connectedness, you can use the same foliation idea in $\mathbb{R}^3$ that I mentioned above: just pick continuum many disjoint paths from $x$ to $y$ (disjoint except for $x$ and $y$), and most of them survive the deletion.
21h
awarded  Necromancer
21h
awarded  Nice Question
21h
comment What is the fewest number of points you must delete from $\mathbb{R}^3$ to make it not simply connected?
Great! Can we view this argument as implementing the (vague) suggestion made in my second comment on the question?
22h
comment What is the fewest number of points you must delete from $\mathbb{R}^3$ to make it not simply connected?
Certainly we can find lines like that, since for any line through $x$ there is a parallel line through $y$, and fewer than continuum many of the line pairs are bad. We can even find such lines in any desired parallel planes containing $x$ and $y$.
22h
comment What is the fewest number of points you must delete from $\mathbb{R}^3$ to make it not simply connected?
Perhaps one might hope to undertake an analogue of that argument in three dimensions? I suppose if I had better geometric insight into the space of homotopies, this might be obvious. Can we put in a continuum of surfaces (disjoint except for the boundary) along which we hope to contract, and then argue that if we have deleted fewer than continuum many points, one of these surfaces must be fine?
22h
comment What is the fewest number of points you must delete from $\mathbb{R}^3$ to make it not simply connected?
@YaakovBaruch The answer to that question is known, and it is the continuum. If you delete fewer than continuum many points in the plane, the resulting space remains path connected, because for any two points, there is a foliation of continuum many disjoint paths between them, and most of these paths avoid the deleted points. Does this help with the simply connected case in three dimensions?
22h
revised What is the fewest number of points you must delete from $\mathbb{R}^3$ to make it not simply connected?
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22h
revised What is the fewest number of points you must delete from $\mathbb{R}^3$ to make it not simply connected?
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22h
asked What is the fewest number of points you must delete from $\mathbb{R}^3$ to make it not simply connected?
1d
comment Is every pair of writable reals one-tape-ITTM-computable?
@Wojowu I had overlooked that, so I deleted my comment.
1d
revised Is every pair of writable reals one-tape-ITTM-computable?
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1d
answered Is every pair of writable reals one-tape-ITTM-computable?
1d
awarded  Great Answer
1d
awarded  Good Answer
Aug
29
comment Is $\mathbb{R}^3 \setminus \mathbb{Q}^3$ simply connected?
Does this argument generalize beyond countable to show also that $\mathbb{R}^3\setminus X$ is simply connected whenever $X$ has size less than $\text{cov}(\cal M)$? (The cardinal characteristic $\text{cov}(\cal M)$ is the covering number of the meager ideal, or in other words, the smallest number of meager sets that union to the whole space; it is more interesting when the continuum hypothesis fails.)
Aug
28
comment Why should we care about “higher infinities” outside of set theory?
Thanks, @JimConant, you are very kind.
Aug
28
comment Why should we care about “higher infinities” outside of set theory?
Kanamori's book The Higher Infinite is a great read for a graduate introduction to large cardinals.
Aug
28
comment Preservation of $\diamondsuit$ by ccc forcings of size $\leq \omega_1$
The question uses completely standard terminology and notation---I think it is perfectly understandable to any set theorist without change. The title could be improved. I'd suggest, "Is $\Diamond$ preserved by c.c.c. forcing of size $\omega_1$?"; and the forcing tag could be added.
Aug
27
awarded  Enlightened