[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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