[FOM] Iterating under Con(T)

[Richard Zach]

This was actually the topic of Turing's paper on ordinal logics, for
which he received his Princeton PhD under Church.  Turing's result is
that you get all true Pi_1 sentences if you iterate to omega + 1.

A very accessible introduction to transfinite progressions of theories
is in Franzén's /Inexhaustibility/, LNL 16,2004.

-RZ

On Fri, 2006-03-10 at 14:46 -0500, Richard Heck wrote:
> joeshipman at aol.com wrote:
> > Heck:
> >> To make the question more precise, define the following sequence of
> >> theories by transfinite recursion:
> >> T_0 = PA
> >> T_{k+1} = T_k + Con(T_k)
> >> T_l = \cup_{k<l} T_k, for l a limit
> > ...[S]houldn't you be closing each step under logical implication? 
> > Otherwise your T_i are not "theories".

> Charles Parsons suggested that the answer to a properly formulated 
> version of this question was that you get all true Pi-1 sentences and 
> that the right place to look was Feferman's "Transfinite Recursive 
> Progressions of Axiomatic Theories". Both suggestions seem right.
> 
> Richard Heck
>