Which are the deepest theorems with the most elementary proofs?
I give two examples:
i) Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function
ii) Proof that the halting problem is undecidable using diagonalization
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Zev Chonoles
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GM2001
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5This should be community wiki, I think. – Cheerful Parsnip Oct 22 '11 at 02:33
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1Proving the incompleteness theorems requires a little more than just diagonalization. – anon Oct 22 '11 at 02:34
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Something like this got asked in MO IIRC. Now if only I could find it... – J. M. ain't a mathematician Oct 22 '11 at 02:42
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2Isn't depth, by definition, inversely proportional to the elementariness of proofs? – hmakholm left over Monica Oct 22 '11 at 02:43
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How do you define the "deapness" of a proof or of a theorem? – Doug Spoonwood Oct 22 '11 at 04:10
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2If you read Spivak's "Calculus on Manifolds", he specifically structures the whole book around making Stokes' Theorem trivial to prove... does that count? – detly Oct 22 '11 at 09:17
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1@detly I don't know the particular example to which you are referring, but I don't think it counts as "simple" if you tuck away all the hard work in lemma after lemma and use them produce a two-line proof of a big result. – Austin Mohr Oct 23 '11 at 01:58
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I am sure this is not what you had in mind, but I am tempted to mention the "elementary" proofs of the Prime Number Theorem due to Erdős and Selberg. :) – Srivatsan Oct 23 '11 at 14:51
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possible duplicate of Surprising Generalizations – JT_NL Oct 23 '11 at 18:37
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1Why is this closed? Reopen, thank you. Leave reason if you vote to close? – GM2001 Oct 26 '11 at 14:02
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4How about "reopen, please"?!? Also "PLZ reopen" in the title is obnoxious. – The Chaz 2.0 Oct 26 '11 at 14:23
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6@GM2001: Please stop editing titles to contain messages like "AWESOME" or " REOPEN !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!...please!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!". This is not the first time you're doing it. – ShreevatsaR Oct 26 '11 at 14:36
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7@GM2001: If you edit this question any more, I will lock it, which will prevent further edits, comments, and answers. – Zev Chonoles Oct 26 '11 at 14:37
2 Answers
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These perhaps aren't particularly deep, but they are the first that come to mind.
Austin Mohr
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I think one should not confuse "important with "deep". The facts that $\sqrt{2}$ is irrational, that there is no surjective map $X\to2^X$, or that there are an infinity of primes, are certainly important or even "fundamental", but their proofs are so simple that one cannot call them "deep". A theorem is "deep" when its proof is really hard and, above all, requires a theory that transcends the realm the problem is formulated in. Consider, e.g., Gauss' theorem about which regular $n$-gons can be constructed with ruler and compass.
Christian Blatter
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4I think the theorem that $\sqrt{2}$ is irrational would have met your criterion for depth at the time it was discovered. Imagine that the Greeks saw $\sqrt{2}$ as the length of the diagonal of a unit square (rather than as the positive solution to $x^2=2$). Then to show the irrationality of $\sqrt{2}$, we need to transcend geometry to go to number theory, where we have available the fundamental theorem of arithmetic. – Srivatsan Oct 23 '11 at 15:49
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3To add on Srivatsan's comment, the proof of $|P(X)|>|X|$ while seemingly trivial nowadays required the development of an entire new field in mathematics. I'd say this qualifies as pretty deep. – Asaf Karagila Oct 23 '11 at 19:35