[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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