[FOM] Iterating under Con(T)

[Aatu Koskensilta]

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