Wikipedia says There exists a decomposition of the ball into a finite number of non-overlapping pieces which can then be put back together in a different way to yield two identical copies of the original ball. So my question is, how many pieces is the ball broken into, and what are their shapes like?

This is Perron’s paradox: Let $N$ be the largest integer. If $N > 1$, then $N^2 > N$, contradicting the definition of $N$. Hence $N = 1$. What does it mean? I get from it that a very large number does not exist or $\infty=1$. Am I right? Or maybe the paradox is wrong?

I suspect that there is a mistake in the Wikipedia article on the St Petersburg paradox, and I would like to see if I am right before modifying the article. In the section “Solving the paradox”, the formula for computing of the expected utility of the lottery for a log utility function is given to […]

As I understand it, undecidability means that there exists no proofs or contradictions of a statement. So if you’ve proved $X$ is undecidable then there are no contradictions to $X$, so $X$ always holds, so $X$ is true. Similarly though, if $X$ is undecidable then $\lnot X$ is undecidable. But again, this would mean $\lnot […]

“I have three boxes, each with two compartments. One has two gold bars One has two silver bars One has one gold bar and one silver bar” You choose a box at random, then open a compartment at random. If that bar is gold, what is the probability that the other bar in the box […]

I have been thinking about this: One can arrive at Russell’s paradox from Cantor’s argument, but can we go the other way round, i.e., can we prove Cantor’s diagonal argument(often referred to as Cantor’s paradox) from the conclusion of Russell’s paradox using the Axiom Schema of Specification/Sepration– there is no universal set. What do other […]

I notice that Russell’s paradox, Burali-Forti’s paradox, and even Cantor’s paradox, all depend on our tolerance of sets that contain themselves (at one level of depth or another). Thus, I was thinking if it wouldn’t be a good way to stop the paradoxes, to just prohibit sets containing themselves, via a modification in the axiom […]

I really like to use paradoxes in my math classes, in order to awaken the interest of my students. Concretely, these last weeks I am proposing paradoxes that achieve the conclusion that 2=1. After one week, I explain the solution in the blackboard and I propose a new one. For example, I posted the following […]

This question already has an answer here: “This statement is false” – Propositional Logic 1 answer

Please check where is the mistake in this following process. I could not make out. $$e^x=\sum_\limits{n=0}^\infty \frac{x^n}{n!}$$ $$\implies e^{x^2}=\sum_\limits{n=0}^\infty \frac{{(x^2)}^n}{n!}=\sum_\limits{n=0}^\infty \frac{x^{2n}}{n!}$$ $$\implies \int e^{x^2} dx=\int \sum_\limits{n=0}^\infty \frac{x^{2n}}{n!} dx$$ $$\implies \int e^{x^2} dx=\sum_\limits{n=0}^\infty \frac{x^{2n+1}}{n!(2n+1)} + c$$ But $e^{x^2}$ has no antiderivative as such. How is then thus possible? Is this correct or just a fallacy?

Intereting Posts

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$\bigcup \emptyset$ is defined but $\bigcap \emptyset$ is not. Why?
Can a group have more subgroups than it has elements?
On “familiarity” (or How to avoid “going down the Math Rabbit Hole”?)
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Show $a_{n+1}=\sqrt{2+a_n},a_1=\sqrt2$ is monotone increasing and bounded by $2$; what is the limit?
Proving that $\mathbb{Z}$ and $\mathbb{Q}$ are not isomorphic.
In a metric space, is every open set the countable union of closed sets?
Two Circles and Tangents from Their Centers Problem
Why is it that if “A is sufficient for B” then “B is necessary for A”?
Numerical estimates for the convergence order of trapezoidal-like Runge-Kutta methods
Showing $\left \lvert \sum_{k=1}^n x_k y_k \right \rvert \le \frac{1}{\alpha} \sum_{k=1}^n x_k^2 + \frac{\alpha}{4} \sum_{k=1}^n y_k^2 $
If $n\in\mathbb N$ and $k\in\mathbb Z$, solve $n^3-32n^2+n=k^2$.