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Results tagged with big-list
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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.
252
votes
What are some examples of colorful language in serious mathematics papers?
Does merely transposing two words count? "It is also hard not to show that ..." [Arnold W. Miller, "Some Properties of measure and category," Trans. A.M.S. 266, 1981, p. 106]
61
votes
Accepted
What is Realistic Mathematics?
When Solovay showed that ZF + DC + "all sets of reals are Lebesgue measurable" is consistent (assuming ZFC + "there is an inaccessible cardinal" is consistent), there was an expectation among set-theo …
- 73.2k
31
votes
Mathematicians who made important contributions outside their own field?
Paul Cohen was an analyst but got a Fields Medal for his work in set theory, proving the independence of the continuum hypothesis from ZFC and the independence of the axiom of choice from ZF.
30
votes
Theorems first published in textbooks?
Long ago, I proved that every derivation from $C^{k+1}$ functions to $C^k$ functions is given by a $C^k$ vector field. (The same fact with $\infty$ in place of $k$ and $k+1$ is, of course, classical. …
22
votes
Examples of theorems with proofs that have dramatically improved over time
I described an example, Hindman's theorem, at https://mathoverflow.net/questions/94546 . The short version is that Hindman's original proof was unpleasantly complicated, whereas a later proof by Galv …
21
votes
Structures that turn out to exhibit a symmetry even though their definition doesn't
Consider the Desargues configuration. It consists of (1) two triangles, say $ABC$ and $A'B'C'$ such that the lines $AA'$, $BB'$, and $CC'$ all meet at a point $P$, and (2) the three points of intersec …
21
votes
Nonequivalent definitions in Mathematics
An extension of a group $A$ by a group $B$ can be either a group $G$ with a normal subgroup isomorphic to $B$ with $G/B$ isomorphic to $A$ or a group $G$ with a normal subgroup isomorphic to $A$ with …
21
votes
Never appeared forthcoming papers
Dana Scott and Robert Solovay, "Boolean-valued models of set theory"
20
votes
Noteworthy, but not so famous conjectures resolved recent years
Maryanthe Malliaris and Saharon Shelah proved that the cardinal characteristics $\mathfrak p$ and $\mathfrak t$ are equal, answering a question that goes back at least to the 1970's and probably (with …
18
votes
What are some reasonable-sounding statements that are independent of ZFC?
In the ring of bounded operators on (complex, separable) Hilbert space, the ideal of compact operators is the sum of two properly smaller ideals. (I mean 2-sided ideals, in the algebraic sense, not t …
13
votes
Pseudonyms of famous mathematicians
I'm not sure whether to count as pseudonyms the altered names that people took (often to avoid antisemitic prejudice) as replacements for their real names. For example, Alfred Tarski's last name was …
11
votes
Applications of finite continued fractions
Here's a lower-level but still useful application. A student came to me with some computer-produced 10-digit (maybe more than 10, I don't remember exactly) floating-point numbers, which I suspected w …
10
votes
Important results with one or more than one proof
The first example that occurs to me is Hindman's theorem: If the set of positive integers is partitioned into finitely many pieces, then there is an infinite set $H$ such that all sums of finitely man …
10
votes
Examples of $G_\delta$ sets
In the space of all subsets of $\mathbb{N}$ (identified via characteristic functions with $2^{\mathbb{N}}$ and topologized as the product of copies of the discrete 2-point space), the set of infinite …
- 73.2k
9
votes
German mathematical terms like "Nullstellensatz"
"Urelement" is used in set theory as a fancy name for an atom, i.e., something that can be a member of a set but is not itself a set.