bio  website  jdh.hamkins.org 

location  New York City  
age  
visits  member for  5 years, 9 months 
seen  27 mins ago  
stats  profile views  55,571 
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
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?
added 124 characters in body 
22h

revised 
What is the fewest number of points you must delete from $\mathbb{R}^3$ to make it not simply connected?
added 33 characters in body 
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 onetapeITTMcomputable?
@Wojowu I had overlooked that, so I deleted my comment. 
1d

revised 
Is every pair of writable reals onetapeITTMcomputable?
added 65 characters in body 
1d

answered  Is every pair of writable reals onetapeITTMcomputable? 
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 notationI 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 