[FOM] Hilbert and conservativeness
William Tait
wwtx at earthlink.net
Sun Aug 28 22:16:45 EDT 2005
On Aug 27, 2005, at 10:19 AM, Aatu Koskensilta wrote:
> At several places Hilbert notes that a finitistic consistency proof
> for
> 'ideal' mathematics implies that 'ideal' mathematics is conservative
> over finitistic mathematics w.r.t. finitistically meaningful ('real')
> statements.
>
> What I'm wondering is how Hilbert knew this. Did he note, as we might
> do today, that if T_1 |- Cons(T_2) then every Pi_1 sentence
> provable in
> T_2 is provable in T_1 (provided the theories meet the relevant
> conditions); or did he simply believe that finitistic mathematics is
> complete and hence any consistent theory extending it is conservative?
I think that the finitist statements that he was referring to are
quantifier-free *sentences*, say, of primitive recursive arithmetic---
in any case, sentences which, if true, are provable by a computation.
With this understanding, his his assertion is completely justified.
It is easy to misunderstand him about this, since he does speak of
finitist proof of Pi_1 sentences---such as consistency statements.
Bill Tait
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