Can you give examples of deep, important results that have only one known proof, and not just because the first proof is fairly recent, or because not many people really cared to think about it? How hard is the proof from the perspective of the non-expert in the field? In the opposite direction, can you give examples of important results, for which several genuinely different proofs were found? Are these proofs all considered equally hard, or some are much easier or, perhaps, even surprisingly easy?


For example, [Carleson's theorem](https://en.wikipedia.org/wiki/Carleson%27s_theorem#History) has more than one proof and, although the latest proof is much simplified, it is probably still quite technical. Poincare conjecture has one known proof, not easily accessible to non-experts. Kadison-Singer problem has one known proof, accessible to non-experts with a little effort. [Capset problem](http://arxiv.org/abs/1605.09223) has one known proof, fully accessible to non-experts. The last three examples are relatively recent results, and some of them are likely to remain the only known approaches for a long time, while maybe not others. Instead of trying to guess the future, what are interesting examples that are less recent?



Important results with one or more than one proof
I wonder whether 'deepness' is subjective or not. The Compactness Theorem of first-order logic has several proofs.



> **Theorem. (Gödel-Maltsev)** Given a language $L$ and a set $S$ of first-order sentences in that language, if every finite subset of $S$ has a model, then $S$ has model.


(A *language* is just a set of relational, function and constant symbols, for example the language of ordered fields is $+,\times,=,\leqslant,0,1$; a *first-order sentence* in this language is a finite word using this language and $\exists,\forall,\land(and),\lor(or)$ and paranthesis for ease of reading only, for instance the sentence $\forall x\exists y(y^2=x)$; a *model* of a sentence is just a set where the sentence can be interpreted in a 'true' way. For instance $\bf R$ is not a natural model of the above sentence as $-1$ does not have a squareroot there, but $\bf C$ is a model of it.)


(For one immediate application, one can deduce in a straightforewared way that there is an ordered field containing $\bf R$ as well as infinitesimal/infinite elements while having the same first order properties as $\bf R$) 


**Some Proofs.** (I believe there are many more)


(1) Gödel's original proof (for Gödel, in 1929, this Theorem was stated as a 'Remark') is from his Completeness Theorem, stated in the particular case where the language $L$ is countable, in which case the **axiom of choice is not needed**. Hard for me to say the nature of the proof. Grammatical maybe.


(2) I don't know precisely the nature of Maltsev's proof, published in German in 1936, and extending G.'s result to the case of an arbitrary signature $L$, using the axiom of choice.


(3) Los' proof via the 'explicit' construction of the model as ultraproduct of finite structures, using the 'axiom of the ultrafilter', which is weaker that the axiom of choice. I would say this proof is of a topological nature.


In the same (?) vein, Gromov's 'bounded version' of his Theorem stating that a finitely generated group of polynomial growth as a nilpotent subgroup of finite index has several (at least 2) proofs, apparently.



> **Theorem. (Gromov)** For any positive integers $k$, $d$, $n$, there exists a positive integer $m$ such that any $n$-generated group, in which for all $r= 1,\dots,m$ the size of the ball of radius $r$ centered at the identity is at most $kr^d$, has a subgroup of index and nilpotency class at most
> $m$.


I read that Gromov's original proof is a Compactness argument. Van den Dries and Wilkie gave an alternative proof using Gödel's Compactness Theorem. Belegradek recently provided a third proof using yet another kind of compactness argument, using very little model theory.


**Edit:** To match the OP's Edit, I'm just adding that the field concerning the first Theorem should be 'model theory' which is currently classified (according to Bairwise, if I am not mistaken) as the latest of the 4 branches of 'mathematical logic'. It is central/important in the sense that it more or less gave birth to the 'field', and is used more or less tacitly in pretty much every Theorem of model theory. The second Theorem belongs to the field of 'geometric group theory', and I do not know about it's being central. It seems to be important in the sense that many mathematicians seem to be interested in it. It also has a wikipedia page in 3 major mathematical languages: <https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Gromov_sur_les_groupes_%C3%A0_croissance_polynomiale>


