[FOM] Iterating under Con(T)

[Richard Heck]

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".
Yes, if by a "theory" one means a deductively closed set of formulae. I 
tend to prefer the usage on which a theory is an arbitrary set of 
formulae, though I know that some other people (perhaps even most other 
people) use "theory" the other way.
> This is not really a definition. Eventually there will be a limit 
> ordinal k for which there is no canonical statement Con(T_k), so you 
> won't be able to define T_(k+1).
Yes, I see, but let me get this more precisely. Is the worry (i) that 
there will be a limit ordinal such that the set of "axioms" of the 
theory at that stage is not even arithmetic? so that, not only is there 
no "canonical" consistency statement, there isn't a consistency 
statement at all? Or is the worry (ii) that, while there may be a 
formula T_k(x) true of the Goedel numbers of the axioms, the formula 
won't be of an appropriate form? or that we won't be able to prove in an 
appropriate setting that it is true of just those numbers? Or is there 
some other worry I've missed? (Sorry if I'm seeming dense. I know a lot 
about some things, and very little about other things. This is one of 
those other things.)

If the worry is (ii), then I guess it seemed to me that it can be 
finessed somehow. Maybe not. Even if it can, though, I understand that 
we may eventually run into (i), as well, and so there may not be a fixed 
point.

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

-- 
Richard G Heck Jr
rgheck at brown.edu
http://bobjweil.com/heck/