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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.
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 …
46
votes
Solutions to the Continuum Hypothesis
(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices pa …
- 32.5k
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
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 …
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 …
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 …
23
votes
Awfully sophisticated proof for simple facts
A Turing machine is a mathematical formalization of a computer (program). If $y\in(0,1)$, a Turing machine with oracle $y$ has access to the digits of $y$, and can use them during its computations. We …
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 …
17
votes
Quick proofs of hard theorems
I've lately have found myself admiring the proof of the fundamental theorem of algebra using linear algebra, due to H. Derksen, American Mathematical Monthly, 110 (7) (2003), 620–623.
He proves dire …
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 …
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 …
13
votes
Examples of ZFC theorems proved via forcing
The Baumgartner-Hajnal theorem, from "A proof (involving Martin’s Axiom) of a partition relation". Fund. Math., 78(3):193–203, 1973.
Actually, there is a very interesting mathematical story here, an …
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 $ …
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 …
8
votes
Books you would like to read (if somebody would just write them…)
AD${}^+$ by Hugh Woodin.