[FOM] Only one proof

Giovanni Sambin sambin at math.unipd.it
Mon Aug 31 12:30:52 EDT 2009

There is a nice example of such a theorem in logic, and that is 
Solovay's theorem proving so-called arithmetical completeness of the 
modal logic of provability GL.
My knowledge of the literature is not up to date (after G. Boolos, /The 
Logic of Provability/. Cambridge U. P. 1993), but as far as I know no 
proof can avoid the central argument in Solovay's original (and very 
smart) proof.
Giovanni S.

joeshipman at aol.com wrote:
> Almost all the important theorems of mathematics, over time, acquire 
> multiple proofs.  There are many reasons for this; but I am interested 
> in important theorems which, long after they are discovered, have 
> "essentially" only one proof. (Only important theorems, because they 
> are the ones which one would expect to be revisited enough that other 
> proofs would be found.)
> The best candidates I have are Dirichlet's 1837 theorem that every 
> arithmetic progression with no common factor contains infinitely many 
> primes, and the 1960 Feit-Thompson theorem that every group with odd 
> order is solvable.
> Can anyone think of other examples of comparable significance, or 
> explain what is special about these theorems, or argue that one of 
> these is not special because an essentially different proof than the 
> original one has been found?
> -- JS
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