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?

  • $\begingroup$ 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. $\endgroup$
    – Watson
    Jun 13 '16 at 11:31
  • $\begingroup$ 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 ? $\endgroup$
    – Watson
    Jun 13 '16 at 11:32

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

  • $\begingroup$ 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 $\endgroup$
    – DonAntonio
    Dec 8 '13 at 5:03

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