[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') 

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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