FOM: Final remarks on Cantor, Hilbert, and CH

[jshipman@bloomberg.net]

Is CH inherently vague or indefinite?  I don't think so.  Is it incapable of
being decided by us?  Well, I hope not, but I won't rule out the possibility --
my feeling, though, is that our understanding of sets of real numbers,
countable ordinals, and the totalities of same is continuing to improve and I
see no reason to give up trying to decide CH.  We know this will require new
axioms, but this is not such a big deal, it has happened before more than once.
Zermelo analyzed his proof of the W.O.T. and isolated a new principle, the Axiom
of Choice, which became generally accepted.  Godel analyzed Hilbert's "proof" of
CH and isolated a new principle, the Axiom of Constructibility, which was NOT
generally accepted (this is the only reason Hilbert gets quote marks and Zermelo
doesn't).  The process of mathematical investigation eventually leads to
publishable results nowadays only when there is ultimately a ZFC-formalizable
proof, but that's OK because it can be a proof that A->B rather than a proof of
B, where A can be a new axiom.  In the case of CH promising approaches have been
suggested by Freiling (symmetry), Woodin (Pmax), and Friedman (Transfer).-Joe S.