[FOM] Hilbert and conservativeness
Aatu Koskensilta
aatu.koskensilta at xortec.fi
Sat Aug 27 11:19:10 EDT 2005
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?
Aatu Koskensilta (aatu.koskensilta at xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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