# Questions tagged [paradoxes]

Paradoxes are arguments which contradict logic or common sense, often by using false and implicit premises.

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### Understanding how different groupings of terms in an Infinite series can lead to different answers.

One of the most, conceptually speaking, for me to understand is the topic of infinite series. I have always had a hard time proving that an infinite series diverges or even finding a solution for the ...
0answers
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### Formally writing the Bertrand's paradox in logic?

I was trying to formalise the different methods of calculating probability in bertrand's paradox in logic, this is my attempt: \begin{align} \Phi \equiv x^2 + y^2 = 1\\ \Psi(m,c) \equiv y = mx + c\\ \...
2answers
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### Solving $\int_0^{\pi/2}x\cot x\,\mathrm{d}x$ while running into a $0\times\infty=0$ problem

While i have been trying to solve the integral $\int_0^{\pi/2} x\cot x \, \mathrm{d}x$ i have noticed that by trying integrating by parts using $u = x$ and $\mathrm{d}v = \cot x \, \mathrm{d}x$, i ...
3answers
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### Notation used in Russell's paradox

I am slightly confused by the notation used in Russell's paradox. I am following this text. I understand that $\phi (x)$ is this boolean function, which outputs either ...
1answer
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### Couples are equally likely to have 1 or 2 children. How likely does a randomly chosen person have a sibling?

Assume that every couple can only have exactly 1 child or two children, with those outcomes being equally likely. Ignore any silly extra factors (e.g. 1 child dying, another being alive). If I choose ...
4answers
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### Is $a$ bigger than $0$ or not?

Consider Goldbach's original conjecture (no need worry, because we don't talk about "Goldbach" itself): every integer $n> 2$ could be detached as a sum of three primes. (In Goldbach's ...
1answer
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### I get $\int fg' \, dx = 2fg + \int fg'\,dx + C$ while performing integration by parts. What did I do wrong?

I am reaching a paradox by using integration by parts. I must be going wrong but unable to figure where I went wrong. Can you help me understand what I did wrong here? We have two differentiable ...
0answers
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### Why wasn't the Russell paradox solved by creating a third kind of pertinence, say $\rlap {~\small+} \in$, beyond $\in$ and $\notin$?

[I am trying to understand the historical process of math, and also the principles in a larger scope than just learning the axioms. So I think this question is a good exercise in that direction.] I ...
1answer
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### Why do we assume the set of all sets that do not contain themselves exists?

It seems to me that the set of all sets that do not contain themselves is very similar to the set \begin{equation} X = \{x | \ x \not\in X\} \end{equation} which is again very similar to the number $x$...
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2answers
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### Probability of picking a real number randomly

If we randomly pick a real number from the number line, the probability of picking a number (say x) is 0. This is true for all real numbers x and it makes sense to me why this must be true. But ...
2answers
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### Can someone explain the relation between “Achilles chasing turtle” paradox and monotonic, bounded sequence to me?

Recently I have read a book called "Caculus - Basic Concepts for High School" The possibility of infinite but bounded sets was not known, for example, to ancient Greeks. Suffice it to ...
1answer
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### Paradox in definite integrals

I don't know whether this question is suitable or not for this forum because it is not exactly rigorous mathematics. We generally descibe geometrically absolute value of a definite integral as area ...
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### Hosting guests coming in infinite lines, each line infinite in length and compossed of infinitely large buses, each bus with infinite guests?

I'm refering to Hilbert's hotel problem. I know it's possible to host an infinite line of infinitely large buses where each one brings infinite passengers this way. As primer numbers are infinite we ...