FOM: Reply to Franzen on Cantor and Hilbert

[Torkel Franzen]

  Joe Shipman says:

  >I would say, contra Torkel, that there is indeed a sense in which a question
  >"*essentially* incapable of being decided" is meaningless.

  Well, my "in any sense" was of course too strong, as such phrases
almost always are. However, the view that CH is "essentially
indefinite" does not imply that the problem of the truth or falsity of
CH is mathematically meaningless, only that it doesn't have any
solution. It doesn't even imply that looking for a solution is
pointless, since all sorts of interesting stuff may result from such a
search. Also, the view that CH is "essentially indefinite" isn't
one that can sensibly be invoked as a basis for suggesting what others
should or shouldn't be doing. It's not as though there is any
theoretical basis for regarding working on the problem as a waste of
time, the way there is a theoretical basis for discouraging people
from looking for a decision procedure for predicate logic. So I don't
really see any opposition between your continuing to look for a
solution to the continuum problem and others regarding the problem as
having no solution. It's a pragmatic difference. (A militant general
doctrine to the effect that set theory is senseless or otherwise
inappropriate as mathematics would be a different matter.)

  Again, the view that CH is "essentially indefinite" doesn't have to
be based on any view that statements in set theory in general have
this character, or on any a priori analysis of CH. On the contrary, we
may take the view that we have a shared understanding of the
(fictitious) world of sets that leads us to a number of non-arbitrary
principles, on which we agree in virtue of this understanding. It's
just that our understanding or intuition doesn't necessarily cover
every aspect of this fictitious world, and in the case of CH,
experience, including independence results, suggests that there isn't
anything in our understanding that can decide it in any way that will
strike set theory enthusiasts in general as convincing and
non-arbitrary.