FOM: true but not provable
William Tait
wwtx at midway.uchicago.edu
Mon Oct 30 18:42:24 EST 2000
Whether or not
(1) A is true but not provable
i.e.
(1) A and {A is not provable)
is meaningful or not for a mathematical proposition A, it is
certainly a useless proposition. To be warranted in asserting (1), we
must be warranted in particular in asserting A (which, unlike `A is
provable', is a mathematical proposition), i.e. we must have a proof
of A, thereby refuting (1).
Torkel Franzen wrote (Oct 27)
> As for your claim that it is "wrong and basically meaningless" to
>say that there are true but unprovable sentences, I don't think I
>understand it. You agree that it is a mathematical theorem that there
>are true arithmetical sentences not provable in T, for any consistent
>extension T of PA, but you seem to be saying that we must not take
>this theorem at face value, but regard it from some purely formal
>standpoint. Why is that?
But clearly when whoever wrote that it is wrong and basically
meaningless to say that there are true but unprovable sentences, he
did not intend the proof predicate to refer to some particular formal
system. For the extension T of PA in question, assuming it partially
formalizes our mathematics, we can say that the Goedel sentence G for
it is true, i.e. assert G, because we can prove it in the
second-order extension of T. (Here I am assuming, only for
simplicity, that T is first-order.) So in the intended sense of
provability, G is provable.
The question is: what does `A is unprovable' mean? I suggested
recently that, in the case of A = CH or any other presently
undecidable sentence, it is best understood as a sociological
prediction about what axioms we might in the future adopt.
Bill Tait
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