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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.
17
votes
Well known theorems that have not been proved
I am not sure whether this qualifies as "well-known".
Anyway, in set theory, in the study of the partition calculus (transfinite generalizations of Ramsey's theorem), effort centered for a while in st …
4
votes
Atlas-like websites on specific areas of mathematics
A database of number fields, by Jürgen Klüners and Gunter Malle. (Note this is not the same as the one mentioned in this answer.)
The site also provides links to similar databases.
24
votes
What definitions were crucial to further understanding?
In set theory, definitely the notion of a Woodin cardinal.
First, it is not an entirely straightforward notion to guess. Significant large cardinals were up to that point defined as critical points …
21
votes
Which journals publish expository work?
There is a new journal of the European Mathematical Society that seems perfect for these articles: EMS Surveys in Mathematical Sciences. The description at the link reads:
The EMS Surveys in Mathemat …
11
votes
Important open problems that have already been reduced to a finite but infeasible amount of ...
This is an elaboration of a comment on Suvrit's answer.
Ramsey numbers can be defined for (infinite) ordinals, just as in the finite case: $r(\alpha,\beta)$ is the least $\gamma$ such that for any $ …
31
votes
What notions are used but not clearly defined in modern mathematics?
There are several examples in set theory; the three I mention are related so I will include them in a single answer rather than three.
1) Large cardinal notion.
I have seen in print many times t …
8
votes
Books you would like to read (if somebody would just write them…)
AD${}^+$ by Hugh Woodin.
7
votes
Constructions unique up to non-unique isomorphism
In recent work in set theory the concept of "canonical structure" has emerged, in connection with combinatorial work on pcf theory. The idea is that there are many constructions that depend on the axi …
13
votes
Open problems in Euclidean geometry?
In recent years there have been a good amount of surveys and publications on "computational" or "combinatorial" geometry, and looking at them may give you a good idea of current questions. Specificall …
6
votes
Easier induction proofs by changing the parameter
Gauss' "second" (1815) proof of the fundamental theorem of algebra (Werke, Volume 3, 33-56, or see Paul Taylor's translation, currently available here) follows an interesting pattern, similar to the o …
91
votes
Proofs that require fundamentally new ways of thinking
The method of forcing certainly fits here. Before, set theorists expected that independence results would be obtained by building non-standard, ill-founded models, and model theoretic methods would be …
23
votes
Examples of common false beliefs in mathematics
In descriptive set theory, we study properties of Polish spaces, typically not considered as topological spaces but rather we equip them with their "Borel structure", i.e., the collection of their Bor …
35
votes
Proofs of the uncountability of the reals
I thought about this question a while ago, while teaching a topics course. Since one can easily check that $${}|{\mathbb R}|=|{\mathcal P}({\mathbb N})|$$ by a direct construction that does not involv …
- 32.5k
6
votes
Most memorable titles
Marginalia to a theorem of Silver (see also this link) by Keith I. Devlin and R. B. Jensen, 1975. A humble title and yet, undoubtedly, one of the most important papers of all time in set theory.
9
votes
Examples of ZFC theorems proved via forcing
In descriptive set theory, there is a significant number of results that have been established using forcing; typically, dichotomy theorems such as Silver's (a $\Pi^1_1$ equivalence relation has only …