[FOM] Only one proof
joeshipman at aol.com
Mon Aug 31 16:40:19 EDT 2009
Although there is something essentially common to all proofs of the
Godel and Cohen results, there is a wide variety of approaches to them
and the details and context differ greatly.
Much more is "common" to all proofs I know of the Dirichlet theorem --
they really go through the same or analogous detailed steps, except
there are a couple of variations in the treatment of the vanishing of
the character L-functions (you can either multiply them together to get
a zeta function and show it has a pole at 0, which requires more
analysis but handles them in a unified way; or you can treat the real
and complex characters separately, using the fact that if a complex
character is 0 then at least two are to simplify the analysis at the
cost of requiring a separate proof and a new idea to get the
non-vanishing of real characters).
I've now found a 1948 paper by Selberg (published in the April 1949
Annals of Mathematics 50/2 pp.297-304) which has an "elementary" proof
which avoids compex numbers, though it still involves nonvanishing of
real "characters" and also depends on Selberg's "elementary" proof of
the Prime Number Theorem). So it still depends on Dirichlet to some
extent and it took 111 years to find!
From: William Tait <williamtait at mac.com>
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?
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