FOM: more on inherent vagueness

Neil Tennant neilt at
Tue Dec 2 22:42:26 EST 1997

Penny Maddy underscores Torkel's formulation of what is meant by CH being
"inherently vague": it is to "suspect that no...principle will ever appear" that would settle CH, that "there isn't anything in our picture of the world of
sets that settles the matter".

Care is needed here. If "our picture of the world of sets" means our current
axiomatization of set theory, then one is saying no more than that CH is
independent of that axiomatization. That doesn't clarify its status as
"inherently vague".

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. But the whole point of Feferman's classifiaction of
CH as "inherently vague" was to *contrast that* with absolute undecidability.
The latter, presumably, is interesting only for statements that are *not* vague!

So we are *still* begging for an explication of "inherent vagueness" that
both makes the notion different from that of absolute undecidability, *and*
avoids my earlier compositionality objections.

Neil Tennant

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