[FOM] Iterating under Con(T)

Richard Zach rzach at ucalgary.ca
Sat Mar 11 13:51:56 EST 2006

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.


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

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