What are the most interesting results in mathematics that say there are only finitely many of something?
- If it's ever shown that there are only finitely many twin primes, that would fit here.
- The compactness theorem for first-order logic is an example. Suppose $S$ is a set of first-order sentences. Suppose $\varphi$ is a first-order sentence that is true in every structure in which every sentence in $S$ is true. Then for some finite $S_0\subseteq S$, $\varphi$ is true in every structure in which every sentence in $S_0$ is true.
- The Robertson–Seymour theorem is an example. If a class of finite graphs is closed under taking minors, then it is equal to the class of graphs that have no minor isomorphic to any of the members of a set of "forbidden minors". The theorem says that that set is always finite. (Among such classes of graphs are planar graphs, outer-planar graphs (embeddable in a plane with all vertices on a circle), graphs knotlessly embedable in $\mathbb R^3$, and graphs linklessly embedable in $\mathbb R^3$. In some cases, finding the finite set of forbidden minors is challenging.
- The Borel–Cantelli lemma considers an infinite sequence $E_1, E_2, E_3, \ldots$ of events in a probability space. It says that if $\Pr(E_1)+\Pr(E_2)+\Pr(E_3)+\cdots<\infty$ then only finitely many of $E_1, E_2, E_3, \ldots$ occur. (Or maybe more precisely,: the probability that only finitely many occur is $1$. The set of outcomes for which infnitely many of the events occur may be non-empty, but its measure is $0$.)
PS: Maybe two different sorts of results can fit here:
- Results that say, for example, exactly eleven of something exist, and maybe lists them. For example, there are only five regular polyhedra, and we all know which five. If only finitely many twin primes exist, then perhaps a theorem would say exactly which ones they are.
- Results that say that something must in every instance be finite, but do not and cannot give a finite upper bound. The The Borel–Cantelli lemma is of that sort. Even specifying which events are in the sequence and what their probabilities are does not make it possible to give a finite upper bound. The compactness theorems of logic are also of that sort. And maybe I should have mentioned compactness theorems of topology? Tychonoff's theorem, maybe? The Robertson–Seymour theorem in general is of that sort, but some of its concrete instances are substantial theorems in their own right. The one that says planar graphs are those that exclude two specified forbidden minors. I seem to recall a simple example in which their were something like 30 forbidden minors – maybe it was triangulations of a torus or something like that.
- Maybe in some cases the question of whether an example is of the first kind or the second might itself be a hard problem.
Despite my inclusion of the first bullet point above, somehow I hesitate to include uniqueness theorems in general. There are zillions of those, and I don't think anyone would want to compile a comprehensive list of uniqueness theorems in mathematics unless maybe they're creating a reference book.