Linked Questions
10 questions linked to/from Situations where “naturally occurring” mathematical objects behave very differently from “typical” ones
42
votes
7
answers
3k
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How would one even begin to try to prove that a simple number-theoretic statement is undecidable?
This question is closely related to this one: Knuth's intuition that Goldbach might be unprovable. It stems from my ignorance about non-standard models of arithmetic. In a comment on the other ...
- 29k
32
votes
5
answers
4k
views
How many of the true sentences are provable?
Is there a natural measure on the set of statements which are true in the usual model (i.e. $\mathbb{N}$) of Peano arithmetic which enables one to enquire if 'most' true sentences are provable or ...
- 5,339
43
votes
6
answers
2k
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Are there "natural" sequences with "exotic" growth rates? What metatheorems are there guaranteeing "elementary" growth rates?
A thing that consistently surprises me is that many "natural" sequences $f(n)$, even apparently very complicated ones, have growth rates which can be described by elementary functions $g(n)$ ...
16
votes
5
answers
6k
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Regular spaces that are not completely regular
In the undergraduate toplogy course we were given examples of spaces that are $T_i$ but not $T_{i+1}$ for $i=0,\ldots,4$. However, no example of a space which is $T_3$ but not $T_{3.5}$ was given. ...
- 2,104
17
votes
8
answers
2k
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Examples of ubiquitous objects that are hard to find?
I've been wrestling with a certain research problem for a few years now, and I wonder if it's an instance of a more general problem with other important instances. I'll first describe a general ...
17
votes
1
answer
2k
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What is known about the relationship between Fermat's last theorem and Peano Arithmetic?
As far as I know, whether Fermat's Last Theorem is provable in Peano Arithmetic is an open problem. What is known about this problem?
In particular, what is known about the arithmetic systems $PA + \...
- 6,353
5
votes
1
answer
531
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Which finite groups can be characterized by their subgroup orders?
Given a finite group $G$, we denote by $\pi_s(G)$ the set of orders of its subgroups. Which finite groups $G$ can be characterized by the set $\pi_s(G)$, i.e. $\pi_s(H)=\pi_s(G)$ implies $H\cong G$? ...
4
votes
1
answer
310
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Is the typical category/groupoid zigzag-connected?
Homotopy categories can be thought of (e.g. on p205-6 of Context) as having arrows built from equivalence classes of special zigzags of arrows in their underlying category. Following this answer and ...
- 436
23
votes
0
answers
699
views
Do most manifolds have symmetries? or not?
Let us say that a (closed, connected) manifold has a symmetry if it admits a non-trivial action by a finite group. Note that I am not asking the action to be free. So for example rotating the 2-sphere ...
- 27.5k
3
votes
1
answer
129
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Do sets of big returns contain sets of returns?
We say a subset $E$ of $\mathbb{N}$ is a set of returns if there is some measure preserving system $(X,\mathcal{B},\mu,T)$ and some $A\in\mathcal{B}$ with $\mu(A)>0$ such that $E=\{n\in\mathbb{N};\...
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