FOM: Cantor'sTheorem & Paradoxes & Continuum Hypothesis

[Robert Tragesser]

The logical moves in the "diagonal argument"
in the RAA parts of the
usual proof of Cantor's Theorem
(that the cardinal number of the set of all
subsets of a set is greater than the
cardinal of the set) exploit a variation
on the trick yielding Russell's Paradox,
the Barber Paradox, etc.  Crudely:
exploiting logical negation to define
or construct an item d in terms of a set D
in a way that d is different from each
member of the set D and yet from another
point of view on the definition/construction
of d and D, d is also a member of D.
I'm curious about the provenance and
evolution of this style of argument. The
diagonal construction and RAA in the case
of the proof by diagonalization of decimal
expansions that the reals are nondenumerable
seems like an argument one would naturally
come to.  But the like argument in the
case of the proof of Cantor's Theorem seems
much trickier and it would be hard to explain
how one thought of it except through trying
to adapt the first diagonalization argument to
the issue of the cardinality of the set of
all subsets of a set.
This suggests that argument supporting
Russell's Paradox, etc., had its origin
in the adapting of the first diagonalization
argument to the needs of a proof of Cantor's
Theorem?

Isn't it fair to say that that proof of
Cantor's Theorem is a capital example of
the sort of purely logical, nonconstructive
proof which motivated Brouwer's churlish
observations about the logical?
Is it right to say that a constructive proof
of Cantor's Theorem would provide an answer to
the truth of the Generalized Continuum Hypothesis?
(I mean of course a constructive proof within
Cantor-Zermelo set theories.)
A "yes" here would be good for making dramatically
a padagogical point about the _virtu_ of constructive
proofs.

robert tragesser
westbrook, connecticut



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