FOM: CH, GCH, and determinacy

Stewart Shapiro shapiro+ at
Wed Dec 3 16:30:33 EST 1997

I just got on this most enjoyable list.  I am not quite used to the water,
but will start splashing anyway.

The threads on CH are most interesting.  It does not seem to be a matter of
vagueness, as that is understood in most of the philosophical literature.
Are there borderline cases of natural numbers or of sets of natural
numbers?  The issue concerns some sort of determinateness.

I presume that those opposed to the determinateness of CH are no fans of
(standard) second-order logic.  There is a second-order sentence (with no
non-logical terminology) that is a logical truth if and only if the CH
holds, and there is a sentence that is a logical truth if and only if the
CH is false.  Examples are in my book somewhere.   (I don't have a copy
here.) I presume that the anti-determinateness-of-CH folks hold that the
second-order quantifiers are "inherently vague" or otherwise indeterminate.
 After all, the second-order quantifiers are (close enough to) quantifiers
over sets.

Does anyone entertain the view that CH is determinate (i.e., has a fixed,
determinate truth value) but that GCH is not?  Kreisel once pointed out
that CH is not (semantically) independent of (say) second-order ZFC, but
that the GCH may be independent.  (Technically, (under standard
assumptions) it is independent of ZF whether the GCH is independent of
second-order ZF).

Incidentally, there is a second-order sentence that is logically true if
and only if the GCH holds.

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