# Mathematical mysteries?

What are some intriguing mathematical mysteries?

What I desire are points like the Banach-Tarski mystery, Paradox of Zeno of Elea, Russel's mystery, and so on.

Modify : As an added constraint, allow us concentrate on mysteries that are not currently in the checklist at :

https://secure.wikimedia.org/wikipedia/en/wiki/Category:Mathematics_paradoxes

A wonderful mystery (in the feeling of violating the usual point of view) which is not because checklist is Arrow's theorem. Essentially, it mentions the adhering to. Allow be offered a set of individuals that elect on some concern, and also have a limited variety of choices (at the very least 3). Everyone orders the choices according to her choices ; the end result of the ballot is an order on the set of choices which is intended to mirror the usual agreement.

Extra officially, a choice is a complete order on the set of choices, and also a ballot system is a function which links per $n$ - uple of choices an additional choice. It ends up that the only ballot system which pleases some innocent - looking theory is the estimate on some variable, that is, the tyranny of among individuals.

One of the effects of Goedel's incompleteness theorem is that if $T$ is a finitely axiomatizable concept of math, after that

- $T$ confirms that $T$ corresponds

if and also just if

- $T$ is irregular!

The factor is that an irregular concept confirms anything, and also a regular concept never ever confirms its very own uniformity.

One instance not in the above checklist is Goodstein's theorem, a very nonintuitive concrete number logical theory which is unprovable in Peano math (or, in a similar way, the Hercules vs. Hydra video game). They basically inscribe induction approximately the ordinal $\epsilon_0 = \omega^{\omega^{\omega^{\cdot^{\cdot^{\cdot}}}}}$ - something that is never instinctive to those that are not accustomed to such ordinals - specifically their Cantor regular kind.

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