[FOM] Iterating under Con(T)
aatu.koskensilta at xortec.fi
Sun Mar 12 11:37:31 EST 2006
On Mar 10, 2006, at 11:56 PM, Alasdair Urquhart wrote:
> Shoenfield showed the existence of a recursive progression
> that is complete for true arithmetic:
> "On a restricted omega rule", Bulletin of the Polish Academy
> of Sciences, 7 (1962), 405-407.
I don't have access to this paper, but I don't think Shoenfield
establishes the existence of a recursive progression that is complete
for true arithmetic. Rather, Shoenfield shows that the restricted omega
rule suffices to prove all truths of arithmetic, where the restricted
omega-rule says that if you have a recursive enumeration of proofs of
A(0), A(1), ... you can infer AxA(x). The existence of a recursive
progression in which all arithmetical truths are provable was proved
later by Feferman, using Shoenfield's result.
As already pointed out, Torkel Franzén's Inexhaustibility covers all
the gory details in a very readable fashion, although it mainly focuses
on the espistemologically more interesting autonomous progressions.
Aatu Koskensilta (aatu.koskensilta at xortec.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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