Linked Questions

187
votes
30answers
13k views

A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language

The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to ...
136
votes
91answers
38k views

Which one result in mathematics has surprised you the most? [closed]

A large part of my fascination in mathematics is because of some very surprising results that I have seen there. I remember one I found very hard to swallow when I first encountered it, was what is ...
41
votes
12answers
2k views

Examples of results failing in higher dimensions

A number of economists do not appreciate rigor in their usage of mathematics and I find it very discouraging. One of the examples of rigor-lacking approach are proofs done via graphs or pictures ...
37
votes
9answers
4k views

The Monty Hall problem

I was watching the movie 21 yesterday, and in the first 15 minutes or so the main character is in a classroom, being asked a "trick" question (in the sense that the teacher believes that he'll get the ...
34
votes
7answers
7k views

Proving that 1- and 2-d simple symmetric random walks return to the origin with probability 1

How does one prove that a simple (steps of length $1$ in directions parallel to the axes) symmetric (each possible direction is equally likely) random walk in $1$ or $2$ dimensions returns to the ...
33
votes
5answers
4k views

How can a structure have infinite length and infinite surface area, but have finite volume?

Consider the curve $\frac{1}{x}$ where $x \geq 1$. Rotate this curve around the x-axis. One Dimension - Clearly this structure is infinitely long. Two Dimensions - Surface Area = ...
11
votes
2answers
328 views

Constructing a number not in $\bigcup\limits_{k=1}^{\infty} (q_k-\frac{\epsilon}{2^k},q_k+\frac{\epsilon}{2^k})$

I have couple of questions related to the properties of real numbers. My first question is as follows. Let $S_{\epsilon} = \displaystyle \bigcup_{k=1}^{\infty} \left( ...
8
votes
2answers
229 views

How to find an irrational number in this case?

From Baire category theorem, we see that $\mathbb{Q}$ can not be a $G_{\delta}$. But consider the following construction: Let us consider $\mathbb{Q}\cap [0,1]$, putting all the elements in the set ...
3
votes
1answer
208 views

Finding an irrational not covered in standard proof that $\mu(\mathbb{Q} \cap [0,1]) = 0$ [duplicate]

Possible Duplicate: How would one go about proving that the rationals are not the countable intersection of open sets? Constructing a number not in $\bigcup\limits_{k=1}^{\infty} ...
2
votes
3answers
295 views

Confusion regarding Russell's paradox

Russell's paradox is about a set not in a set itself - but don't all sets are not in sets themselves? $x \in x$ is not true, as {$1,2,3$} $\in$ {$1,2,3$} is not true.. Can anyone explain this?
0
votes
1answer
155 views

Expected number of tosses to get T,T

Assume a coin has a probability p to get a head H. Suppose a coin is tossed until the partern T,T appear in the last 2 tosses. Once he got T,T then the game is finished. What is the expected number of ...
0
votes
1answer
65 views

A setting in which Rice's theorem is not true

In my class we call a set of computable functions $A$ recursive if its indexing set $I_A=\{e\in\mathbb N:\phi_e\in A \}$ is recursive, where $\phi$ is some known Gödel numbering of the computable ...