FOM: Further comments on CH and "inherent vagueness"

[Torkel Franzen]

  John Steel and Neil Tennant, if I understand them correctly, are
bothered by the lack of principled or intrinsic distinction between
statements like CH which (on the view at issue) are "inherently
vague", and other statements that are not. I would like to try to
clarify the matter.

  The term "inherently vague" is no doubt unfortunate in that
vagueness in any strict sense is hardly at issue. So let me use the
term "essentially indefinite" to express my understanding of the term
"inherently vague" in this context. That is, the view at issue is that
(i) there is nothing in our understanding of the world of sets that
settles CH one way or the other, and (ii) there is no other fact of
the matter as regards the truth or falsity of CH than that which is
explicit or implicit in our understanding of the world of sets.

  This doesn't imply that CH is in any way deficient in meaning. There
isn't any distinction in kind of meaning between CH and other
statements of set theory. All statements of set theory refer to a
fictitious world of sets. Some such statements are logically decided
by principles evident to us, i.e. by descriptions of the world of sets
that we find intellectually agreeable. Others are known not to be
decided by any such principles currently formulated. Mathematicians,
naturally enough, tend not to work on deciding such statements, since
deciding them is known to require the introduction of new principles,
which is not an ordinary mathematical activity. The view that CH is
essentially indefinite holds that no principle deciding it will ever
be found, that our view of the world of sets simply isn't definite
enough to decide it. This is a pragmatic attitude rather than a
theoretical opinion (because of the dearth of theory in this
connection). There is no reason why everybody should share it.  John
Steel suggests that

    Whichever way it
    turns out, true, false, meaningless, or some blend of the three, I suspect
    the solution will involve some philosophical/conceptual analysis of what
    it is to be a solution.

  Indeed, long before the independence results, Hardy said of the continuum
problem that

     This again appears to be a mathematical question; one would
     suppose that, if a proof were found, its
     kernel would lie in some sharp and characteristically
     mathematical idea. But the question lies much
     nearer to the borderline of logic, and a mathematician interested
     in the problem is likely to hold logical
     and even philosophical views of his own.

  The difference in attitude between Feferman and Steel is that
Feferman doesn't think there is, by now, much hope of finding any
solution of the problem that people will agree on (any more than they
now agree that V=L |- CH settles the matter). But this is no reason,
even from Feferman's point of view, why others should drop the
problem.

   Neil Tennant comments further on the matter of consistency statement:

   >The point is this: such a Harvey-sentence H about natural
   >numbers (that nice, concrete, supposedly well-behaved realm concerning
   >which even those who think CH vague nevertheless think all claims are
   >definitely meaningful) can be *established as true* only by *assuming
   >the existence* of large cardinals!  (By Godel's second incompleteness
   >theorem, if you have a system S proving H, and H proves Con(BigGuy),
   >then S must assume EvenBiggerGuy!)

  This isn't strictly speaking correct. S doesn't have to assume
anything at all about infinite sets to (formally) prove Con(BigGuy). But
the basic point certainly remains: arithmetical statements decided by
strong assumptions in set theory are not necessarily decided by any less
"metaphysical" arguments that could ever be devised. This is a basic
fact for the philosophy of mathematics, but not one that obviously
requires us to take the statements of set theory to refer to any
well-defined realm of sets. Why shouldn't statements about an ill-defined
realm of sets have consequences for a well-defined realm of natural
numbers?

   >What would happen, then, if some such number-statement H were
   >equivalent to ExistsGuySoBig that CH gets decided?  Would we
   >contrapose a la Feferman and say "Hey, Harvey's innocent-looking
   >statement about numbers must be inherently vague (since CH is)!"?

  It's unclear what sort of situation is envisaged here, since
no arithmetical sentence consistent with ZFC decides CH in ZFC
(or implies any axiom of infinity).

   But this apart, the response to the general tenor of the remarks quoted
is that any convincing argument by which CH is settled would show that
CH is not essentially indefinite.


  Torkel Franzen