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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?

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1 Answer 1

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

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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

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