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Results tagged with gm.general-mathematics
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Questions about mathematics which don't fall into the other arXiv categories. If you have a general question about mathematics but it is not research level, it's off-topic but it might be welcomed on Mathematics Stack Exchange.
18
votes
When is 2 qualitatively different from 3?
Every group in which every non-identity element has order 2 is abelian.
3
votes
Terminology: product on strict preorders corresponding to direct product of preorders?
If one takes the reflexive order relation as fundamental, then this is just the strict product order.
It is good practice to take the reflexive order relation as the primary relation, since in the c …
- 236k
34
votes
A search for theorems which appear to have very few, if any hypotheses
Theorem. Every group has a terminating transfinite automorphism tower.
Start with any group $G$, compute $\text{Aut}(G)$ and $\text{Aut}(\text{Aut}(G))$ and so on, iterating transfinitely, mapping e …
105
votes
Accepted
Have you solved problems in your sleep?
On several occasions it has happened that I have made a key insight while sleeping or drifting in and out of sleep.
For example, one of the critical ideas in my paper
Joel David Hamkins, Gap forcing, …
12
votes
What would you want to see at the Museum of Mathematics?
See the water demonstration of the Pythagorean theorem:
[]1
https://www.youtube.com/watch?v=CAkMUdeB06o
36
votes
What would remain of current mathematics without axiom of power set?
Several standard theories intensely studied by set theorists do not have the power set axiom.
One of these is the theory ZFC without the power set axiom, usually denoted $\text{ZFC}^-$. One should t …
- 236k
37
votes
Examples of interesting false proofs
True Theorem The symmetric groups (consisting of all permutations) on infinite sets of different cardinalities are not isomorphic.
False proof: The two groups have different cardinalities, since the …
2
votes
Basic results with three or more hypotheses
The principle of transfinite induction is often stated as the following theorem.
Theorem. Suppose that $A$ is a class of ordinals. If
(zero) $0$ is in $A$,
(successor) whenever an ordi …
39
votes
Basic results with three or more hypotheses
The real line $\langle\mathbb{R},\lt\rangle$ is (up to isomorphism) the unique nonempty, separable, complete, dense, endless total order.
(All conditions are needed: Without separable we have for ex …
16
votes
Accepted
Is there a name for sets for which it is easier to test membership than to find members---an...
This phenomenon occurs both positively and negatively in
many parts of logic, but to my knowledge, there is no
particular adjective that is always used in such
situations.
In classical computability …
- 236k
46
votes
Theorems true but wrong.
I remember years ago sitting in Leo Harrington's office in Berkeley explaining my dissertation to him (he was on my committee), and he spent some time just scanning through the dissertation seeking ou …
30
votes
What is your favorite "strange" function?
The Ackermann function $A(n,m)$ is defined on the natural numbers by a very simple recursion, but the values grow enormously, almost beyond conception. This function completely transcends any simple-m …