Linked Questions

1
vote
0answers
70 views

Things you've believed for a long time were true, but are false in reality [duplicate]

Do you have any things (mathematical statements, statements about mathematics) you've believed for a long time were true, but now with enough mathematical knowledge you realize were wrong? For ...
518
votes
151answers
32k views

What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)

I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of mathematics. I recently read Paul Lockhart's essay "The Mathematician's ...
15
votes
3answers
825 views

How does intuition fail for higher dimensions?

From this answer: Now, Algebraic Geometry is one of the oldest, deepest, broadest and most active subjects in Mathematics with connections to almost all other branches in either a very direct ...
8
votes
5answers
5k views

Help: rules of a game whose details I don't remember!

In a probability course, a game was introduced which a logical approach won't yield a strategy for winning, but a probabilistic one will. My problem is that I don't remember the details (the rules of ...
18
votes
1answer
7k views

Who discovered this number-guessing paradox?

In this math.se post I described in some detail a certain paradox, which I will summarize: $A$ writes two distinct numbers on slips of paper. $B$ selects one of the slips at random (equiprobably), ...
9
votes
8answers
266 views

Examples of “transfer via bijection”

On some occasions I have seen the following situation: We want find out whether a set of a given cardinality $\varkappa$ has some property P. If this property is invariant under bijective maps, then ...
7
votes
4answers
1k views

How many smooth functions are non-analytic?

We know from example that not all smooth (infinitely differentiable) functions are analytic (equal to their Taylor expansion at all points). However, the examples on the linked page seem rather ...
3
votes
3answers
3k views

an example of a continuous function whose Fourier series diverges at a dense set of points

Please give me a link to a reference for an example of a continuous function whose Fourier series diverges at a dense set of points. (given by Du Bois-Reymond). I couldn't find this in Wikipedia.
6
votes
3answers
306 views

The limit of the derivative of an increasing and bounded function is always $0$?

Let $\,f : \mathbb{R} \rightarrow \mathbb{R}$ be a infinitely differentiable function that is increasing and bounded. Then is it true that $\lim_{x\to \infty}f'(x)=0$?
6
votes
3answers
265 views

Mathematical results that were generally accepted but later proven wrong?

I am giving a presentation on mathematical results that were widely accepted for a period of time and then later proven wrong, or vice versa. This talk is geared towards undergraduates who are likely ...
2
votes
2answers
1k views

Graph of discontinuous linear function is dense

$f:\mathbb{R}\rightarrow\mathbb{R}$ is a function such that for all $x,y$ in $\mathbb{R}$, $f(x+y)=f(x)+f(y)$. If $f$ is cont, then of course it has to be linear. But here $f$ is NOT cont. Then show ...
5
votes
1answer
119 views

Chain of length $2^{\aleph_0}$in $ (P(\mathbb{N}),\subseteq)$

How can I find a chain of length $2^{\aleph_0}$ in $ (P(\mathbb{N}), \subseteq )$. The only chain I have in mind is $$\{\{0 \},\{0,1 \},\{0,1,2 \},\{ 0,1,2,3\},...,\{\mathbb{N} \} \}$$ But the ...
1
vote
2answers
357 views

Paradox of General Comprehension in Set Theory, other than Russell's Paradox

As is well known, the General Set Comprehension Principle (any class is a set) leads to the Russell Paradox (the class $x \notin x$ cannot be a set). As a result, set theories must restrict the ...
2
votes
2answers
82 views

Is the Russell paradox the only possible contradiction to the axiom schema of comprehension due to Frege (1893)? $\{x:P(x)\}$ [duplicate]

Is the Russell paradox the only possible contradiction to the axiom schema of comprehension due to Frege (1893)? The axiom that says that if $\varphi$ is a property, then there exists a set $Y = \{X: ...
3
votes
2answers
176 views

Generating function: Probability regarding coin toss

If a coin is flipped 25 times with eight tails occurring, what is the probability that no run of six (or more) consecutive heads occur? Wasn't sure how to approach this and am quite positive my ...

15 30 50 per page