FOM: Feferman on inherent vagueness of CH
Jerry Seligman
pytms at ccunix.ccu.edu.tw
Tue Dec 2 19:58:01 EST 1997
(1) Vagueness can come from quantifiers as well as from predicates.
Consider the
the all-too-friendly inhabitants of Smalltown, Nowhere County:
Everyone greets everyone else at least once a year.
Everyone greets everyone else at least once a week.
The first may have a determinate truth value because everyone in the whole
county sees everyone else at least once a year - at the Nowhere County
Fair, perhaps. But we may suppose that the domain of quantification is
vague, there
being no census in Smalltown, and the second statement lacks a determinate
truth value.
The vagueness of CH, if any, lies in the domain of quantification, namely
sets of sets of naturals numbers, sets of reals, the universe of sets, or
whatever. (It depends on how you state CH.)
(2) This does not address Neil Tennant's point, since it only shows that
an expression without vague parts can lack a determinate reference. What
Neil is worried about, if I understand him correctly, is the claim that
such expressions lack a definite *sense*. The conviction that they have
sense is emphasized in FOM because some statements about possibly vague
collections (such as the sets of reals) have a determinate truth value, and
we can prove what it is. There is
a strong intuition that if you can prove it, it must make sense.
(3) Nonetheless, I think we should resist the idea that the sense of a
statement be linked to its future provability from new axioms (see Maddy's
post). The consequence that some such statements have a sense without our
knowing it, and others are not, puts sense too far from understanding for
my taste. The sense of CH is given as much by its utility in proving other
statements, and its wider role in reasoning about the independence of
axioms and other foundational matters as by the possibility of its being
proved from new axioms. An expression that can be usefully employed in
this way must make sense, in some sense, even if it does not have a
determinate reference (truth value).
(4) The last point is probably unfair to Penelope Maddy, who only suggested
that we take the lack of future provability as a gloss on "inherent vagueness",
not as a semantic thesis about the sense of mathematical statements. Following
this suggestion would be to understand "inherent vagueness" as a matter of
reference (albeit future-historical reference) rather than sense, and this
could make Feferman's claim compatible with Tennant's conviction that CH
has a sense. This may not be what Feferman wants.
(5) Sense does not always determine reference, even in mathematics. Isn't
that a lesson of FOM?
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