[FOM] shipman's challenge: the best defense
joeshipman at aol.com
Tue Jan 8 18:29:38 EST 2008
Would you regard any of the standard renderings of "the set of real
numbers is uncountable" in the language of set theory as sufficiently
faithful that a formal machine-checkable ZFC-proof of it has a claim to
be considered as "a proof that the set of real numbers is uncountable"?
Or do you think that all such renderings of the statement in the
language of set theory fail to formalize "the set of real numbers is
uncountable" sufficiently faithfully for a ZFC-proof of the
set-theoretic sentence to settle the question of whether the real
numbers are uncountable?
If the latter, do you think that any other formal system could
accomplish what ZFC doesn't here?
From: Gabriel Stolzenberg <gstolzen at math.bu.edu>
Having said this, I invite Joe to think about the "adequacy" of
a formalization of the informal statement, "The set of real numbers
is uncountable." The claim that it is a "faithful representation"
of the informal statement might well make a mathematician (e.g., me)
uncomfortable. I believe this is a familiar point.
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