Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

This question is in the spirit of the question "Nice examples of groups which are not obviously groups".

There are many impressive finiteness results in mathematics. For example:

  1. The finiteness of $\text{Gal}(\overline{\mathbf R}/\mathbf R)$;
  2. The finite-generatedness of homotopy groups of spheres;
  3. The finiteness of the set of smooth structures on the $n$-sphere, for $n\neq 4$;
  4. The conjectured finiteness of Shavarevich-Tate groups;
  5. The finite-generatedness of Mordell-Weil groups;
  6. The finiteness of class numbers;
  7. The finiteness of the set of rational points on a curve $X/\mathbf Q$ when the genus $X>1$...

So, what are the nicest examples of finite sets which are not obviously finite?

share|cite|improve this question
Sorry for this very late comment, but I would like to know what $\overline{ \Bbb R}$ is. I don't think this denotes an algebraic closure of $\Bbb R$, because then it would be isomorphic to $\Bbb C$, and the finiteness of $\text{Gal}(\Bbb C / \Bbb R)$ is not hard to establish, in my opinion. – Watson Jun 13 at 11:31
Moreover (and this is completely unrelated), I found in that question a link to one of your blog post. I was wondering : are some articles of this blog still available anywhere ? – Watson Jun 13 at 11:32

The finiteness of the number of different $n$ for which Fermat's equation has a solution?

share|cite|improve this answer
Hehe...of the finiteness of the set of all real functions which are differentiable at some point yet not-continuous there, or the finiteness of all the basis of a vector space which are a linearly dependent set, or the finiteness of the set of all compact discrete infinite topological spaces, or...Anyway, +1 – DonAntonio Dec 8 '13 at 5:03

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.