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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.
393
votes
What are some reasonable-sounding statements that are independent of ZFC?
"If a set X is smaller in cardinality than another set Y, then X has fewer subsets than Y."
Althought the statement sounds obvious, it is actually independent of ZFC. The statement follows from the …
263
votes
What are the most misleading alternate definitions in taught mathematics?
Many topics in linear algebra suffer from the issue in the
question. For example:
In linear algebra, one often sees the determinant of a
matrix defined by some ungodly formula, often even with
specia …
174
votes
Accepted
Solutions to the Continuum Hypothesis
Since you have already linked to some of the contemporary
primary sources, where of course the full accounts of those
views can be found, let me interpret your question as a
request for summary accoun …
- 236k
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 …
148
votes
Accepted
Nontrivial theorems with trivial proofs
Bertrand Russell proved that the general set-formation principle known as the Comprehension Principle, which asserts that for any property $\varphi$ one may form the set $\lbrace\ x \mid \varphi(x)\ \ …
136
votes
Has philosophy ever clarified mathematics?
I find the case of Alan Turing's development of the concept of computatibility to be an example. Before Turing, the logicians had no clear concept of what it means to say that a function is computable …
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 …
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 …
115
votes
Examples of common false beliefs in mathematics
"Either you can prove the statement, or you can find a counterexample."
This statement is usually applied to universal statements, those having the form $\forall x\ \varphi(x)$, where the concept o …
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, …
104
votes
Theorems with unexpected conclusions
My favorite example of this phenomenon is Goodstein's Theorem.
Take any positive number $a_2$, such as the number $73$, and write it in complete base $2$, which means write it as a sum of powers of $2 …
95
votes
Examples of eventual counterexamples
The essence of the phenomenon of eventual counterexamples is that a certain pattern that holds among small numbers, turns out not to be universal. In the very best examples, such as the examples provi …
94
votes
Mistakes in mathematics, false illusions about conjectures
Computer designers and programmers dreamed, from the earliest days of the computer, of a computer that could play chess and win. Even Alan Turing had that dream, and designed turochamp, the first ches …
88
votes
Examples of common false beliefs in mathematics
"It is impossible in principle to well-order the reals in a definable manner."
To be more precise, the belief I am talking about is the belief that well-orderings of the reals are provably chaotic in …
79
votes
What are some reasonable-sounding statements that are independent of ZFC?
"There is no definable well-ordering of the real numbers."
Although many mathematicians simply believe this statement to be true, actually, it is independent of ZFC. In Goedel's constructible univer …