Articles of paradoxes

What (and how many) pieces does the Banach-Tarski Paradox break a sphere into?

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?

Could you explain Perron's paradox to me, please?

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?

Mistake in Wikipedia article on St Petersburg paradox?

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 […]

How can any statements be proven undecidable?

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 […]

Simple question with a paradox

“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 […]

Can one come to prove Cantor's theorem (existence of higher degree of infinities) FROM Russell's paradox?

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 […]

Is there a branch of mathematics that requires the existence of sets that contain themselves?

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 […]

$2=1$ Paradoxes repository

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 statement is false.”

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

Antiderivative of $e^{x^2}$: Correct or fallacy?

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?