[FOM] Only one proof

[Robert Lubarsky]

The first example below is not an example. For an essentially different
proof of Con(CH) other than its truth in L, force over any model of ZF to
collapse the continuum to be aleph_1.

Bob Lubarsky


-----Original Message-----
From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of
William Tait
Sent: Monday, August 31, 2009 12:46 PM
To: Foundations of Mathematics
Cc: William Tait
Subject: Re: [FOM] Only one proof

Two examples in set theory are Goedel's proof of the consistency of CH  
relative to ZF using inner models and Cohen's proof of the consistency  
of Not-CH using forcing.

There seems to be something different about these example in  
comparison with yours. [Qua examples, they seem less interesting.]  
What is it?

Bill Tait

On Aug 29, 2009, at 9:37 PM, 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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