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Results tagged with big-list
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user 1946
Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.
18
votes
When is 2 qualitatively different from 3?
Every group in which every non-identity element has order 2 is abelian.
- 236k
6
votes
Papers that debunk common myths in the history of mathematics
Theodor Nenu and I have a paper addressing the question of whether Alan Turing proved the undecidability of the halting problem in his seminal 1936 paper on computable numbers, in which he introduces …
- 236k
12
votes
Uniqueness results that follow from CH
Under CH, we have saturated models of size continuum of any consistent first-order theory in a countable language, and for a complete theory these are unique by the back-and-forth method.
(In my paper …
- 236k
33
votes
What are some reasonable-sounding statements that are independent of ZFC?
"The real line is the only endless dense complete linear order in which every family of disjoint intervals is countable."
This statement generalizes the familar characterization of the real line (due …
- 328
126
votes
The most outrageous (or ridiculous) conjectures in mathematics
W. Hugh Woodin, at a 1992 seminar in Berkeley at which I was present, proposed a new and ridiculously strong large cardinal concept, now called the Berkeley cardinals, and challenged the seminar audie …
- 103
17
votes
9
answers
4k
views
How should the Math Subject Classification (MSC) be revised or improved?
Most of us are familiar with the Math Subject
Classification
(MSC),
a coded index attempting to classify all mathematical
research areas by topic. The MSC, devloped jointly by the Math Reviews and Zen …
CommunityBot
- 1
173
votes
Most 'unintuitive' application of the Axiom of Choice?
I have enjoyed the other answers very much. But perhaps it
would be desirable to balance the discussion somewhat with
a counterpoint, by mentioning a few of the
counter-intuitive situations that can o …
- 4,713
31
votes
What could be some potentially useful mathematical databases?
There are several natural examples from set theory.
Here is a database on consequences of and equivalent formulations of the axiom of choice, which is searchable by keyword and axiom form, and which …
- 21.6k
27
votes
Accepted
Are there any good nonconstructive "existential metatheorems"?
Set theory provides a good example. It is often convenient in set theory to work with the concept of "classes" and treat them as mathematical objects of their own kind. The standard axiomatization of …
- 2,031
21
votes
What are some nice uses of ultraproducts/ultrapowers?
Here are a few common uses that come to mind:
Large cardinals. Ultrapowers are used pervasively in large cardinal set theory. Most of the familiar large cardinal concepts can be characterized by thei …
- 236k
135
votes
43
answers
38k
views
What are the most attractive Turing undecidable problems in mathematics?
What are the most attractive Turing undecidable problems in mathematics?
There are thousands of examples, so please post here only the most attractive, best examples. Some examples already appear on …
- 349
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, …
- 4,713
56
votes
How helpful is non-standard analysis?
The other answers are excellent, but let me add a few
points.
First, with a historical perspective, all the early
fundamental theorems of calculus were first proved via
methods using infinitesimals, r …
- 4,713
7
votes
What are some interesting applications/corollaries of Kleene's Recursion theorem?
Here is a further example, which I find to be one of the rather more philosophically profound results in computability theory. Namely, Rice's theorem.
The theorem states, at bottom, that no nontrivial …
- 236k
20
votes
What are some interesting applications/corollaries of Kleene's Recursion theorem?
My favorite use of the Kleene recursion theorem is the universal algorithm.
In the baby form, consider the program $e$ that (on any input) undertakes the following process: it looks for a proof from P …
- 236k