FOM: Feferman on inherent vagueness of CH

[Torkel Franzen]

  Neil says:

  >It seems that some pretty *simple* sentences might be in the same boat as
  >CH---I have in mind here something like Goldbach's Conjecture. I
  >cannot believe that GC is 'inherently vague'; nor can I believe that
  >CH is. Why should the undecided status of GC be any different from
  >that of CH?

  The obvious reply to your question is that "mapping between sets of
reals" is an open and indeterminate notion in a way that "natural
number" is not. We have good reason to assume that this widespread
distinction is the basis of the general attitude behind Feferman's
view that CH is "inherently vague".

  This open and indeterminate character of the notions involved in CH
does not in itself rule out the possibility that some convincing
principle (i.e. one that appeals to us as being in accordance with our
general understanding or picture of the world of sets, as do the
axioms of ZFC) will eventually turn out to settle CH, but to suspect
CH of being "inherently vague" is, I would assume, to suspect that no
such principle will ever appear, that there isn't anything in our
picture of the world of sets that settles the matter. It is of course
problematic just what "inherent" means here, since the notion of an
implicit but well-defined content of our picture of the world of sets
is a problematic one.

  >There is a way out of all these difficulties about definite 
  >sense and determinate truth-value: one can be an 
  >intuitionist.

  The existence of problems that are not settled by any known evident
principles isn't more problematic in classical than in intuitionistic
mathematics. After all, there is no reason why we should insist that
every question has a non-arbitrary answer.

  >But the intuitionist refuses to say of any 
  >sentence that it has a definite truth-value without being able 
  >to say what that truth-value is.

  The intuitionists I know don't hesitate to say of many sentences
that they have a definite truth-value even though they can no more
say what that truth-value is than they can dismantle the moon and
eat it. For example, they say of any statement of the form "e is a prime",
where e is some arithmetical expression, that it has a definite truth-value.