FOM: Poincare on Cantor's Theorem
cxm7 at po.cwru.edu
Mon Feb 12 10:15:17 EST 2001
kanovei at wmwap1.math.uni-wuppertal.de (Kanovei) wrote:
>What Cantor's theorem is really dependent on is
>the assumption that P(N) (the continuum) *already exists*
>to the moment of writing the proof and, say, will not
>gain new elements until the proof is finished.
That is exactly how Poincare saw the argument, as of 1912 (the year he
died). He considered it a brilliant insight, and an entirely valid proof,
with a false conclusion--thus a reductio of the assumption.
More precisely, he believed the continuum exists alright, but we cannot
regard all its elements as "given" at any point--because we can list all
the reals we are given (an infinite list since it will contain 1, 2, 3, ...
for example), and Cantor has shown any such list "gives" us a new real number.
He never clearly said what "given" really means. He was clear that he
liked the diagonal argument.
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