FOM: more on the "vagueness" of CH
neilt at hums62.cohums.ohio-state.edu
Tue Dec 2 08:58:44 EST 1997
Torkel says that there is an open and indeterminate character to the
reals and mappings between them. This is supposed to make CH vague,
whereas a claim like GC, about the much more determinate natural
numbers, is not vague.
But this misses the philosophical point of my earlier mailing about
Fefereman's "gut feeling" that CH is "inherently vague". My challenge
was to have someone show why the *component expressions* in CH had
indefinite *senses*. I was pointing out that one cannot have one's
cake and eat it. If those expressions have definite enough senses
for at least *some* sentences involving them to have definite
truth-values, then why should it be the case that a sentence like CH,
involving no new expressions, is itself suddenly "vague" or
"indefinite" in sense, and thereby deprived of a determinate
This is what still has to be explained.
Note also that appealing to the nice, determinate natural numbers will
not do if the intention is to provide a foil for the supposedly
messier business with sets. We already know from Harvey's results that
some very simple statements about natural numbers are equivalent to
the consistency of very large cardinals.
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