what are your favorite theorems? one of mine is "this is a theorem (of peano arithmetic)", it's expressible by the diagonal lemma and is a consequence of Löb's theorem. another is Kőnig's lemma, I think it's neat and it's especially interesting in reverse mathematics, where weaker forms of it lead to interesting systems.
the Sprague–Grundy theorem and the Euclid–Euler theorem are both quite good I think. also this may not be a theorem, but the sophomore's dream identity is quite nice
Hm, the compactness theorem and the completeness theorem are both pretty good. I'd also agree that Sprague-Grundy is pretty nice.
I don't know if this one has a name but probably my personal favorite is "there is no uncountable, well ordered (by $\le$) subset of the reals"
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