FOM: JShipman on CH &elementary proof issue
Robert S Tragesser
RTragesser at compuserve.com
Thu Dec 11 12:13:28 EST 1997
Joe Shipman remarked that the independence of CH shows that we
can't have an elementary proof/disproof of CH.
But this ignores (doesn't it?) the issue that ZF/ZFC do(es) capture
all that might be compellingly thought to pertain (elementally) to pure
Though (playing off Sol Feferman's broodings over the issue of new
axioms), this that follows is the likliest scenario (isn't it?):::
Someone succeeds in giving a perfectly compelling proof/disproof of
CH which is patently nonelementary (in the sense that it strikingly draws
on techniques/ideas clearly removed from the domain of pure sets).
Then someone else gives a deep analysis of this proof, finding
that hidden in it is a principle which can be adequately expressed in the
language of pure sets (say, ZF). Now this axiom is added to ZF/ZFC, and
a proof/disproof can now be given of CH. I think we'd say that this was an
elementary proof, and that the nonelementary proof helped us to find our
way to the missing aspect of pure sets.
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