FOM: more on the "vagueness" of CH

Torkel Franzen torkel at
Wed Dec 3 04:21:00 EST 1997

  Neil says:

  >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*.

  What is indefinite (on the view at issue) is the range of the quantifiers:
"for every set of reals there is a mapping...". 

  >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

  The indefinitess of such quantifiers is no obstacle to many statements
about sets having perfectly determinate truth-values, in the sense of
being logically decided by  principles about sets that are evident to
us even given the indefiniteness of the assumed realm of sets.

  >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.

  I don't see what your point is here. People who don't think there is
any definite (or perhaps even meaningful) question about the existence
of large cardinals take a very different view of statements about the
consistency of assuming the existence of large cardinals.

  >On the other hand, if "our picture of the world of sets" means our
  >intuitive source of axioms current and future, then the claim that no
  >principles will ever appear from this source to settle CH is simply
  >the claim that CH is absolutely undecidable.

  In not wanting to describe CH as "absolutely undecidable", Feferman
presumably had in mind that the phrase "absolutely undecidable"
suggests that there is some fact of the matter, that the "decision" is
not just a matter of coming down on one side of the fence or the
other, but of finding something out, such as whether or not CH is in
agreement with our current or potential understanding of the world of
sets. The contrast between "inherently vague" and "absolutely
undecidable" emphasizes the view that there simply isn't anything to
find out, that there isn't anything in our understanding of sets that
settles the matter one way or the other, even implicitly and as a matter
of strong plausibility rather than proof.

  Of course nobody can claim that it is impossible for human beings to
arrive at a decision regarding CH. For example, through the powerful
influence of Cantor's ghost we might all become convinced tomorrow
that CH is true, for no reason that we would now consider compelling.
The idea that there is "inherently" nothing in our understanding of
the world of sets that settles CH is necessarily a problematic one since
we can't pretend to any profound theoretical understanding of just
what our understanding of the world of sets amounts to. I don't think
it's very rewarding to try to squeeze any precise or theoretical
content out of this notion of "inherent vagueness". Its main thrust
is pragmatical: there is no reason to believe that our mathematical
intuition settles every question, and CH seems a case in point.

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