[FOM] Is Con_Q provable in Q?

[joeshipman@aol.com]

Pavel Pudlak writes "the statement that a theory T is consistent is very strong even for a fairly weak T. For example, the consistency of Robinson's arithmetic Q, which has no induction axioms at all, is not provable in Bounded Arithmetic."
 
More of Pudlak's exploration of these matters can be found at this link:
 
http://www.math.cas.cz/~pudlak/bottom1.ps
 
Here's a more general version of Avron's question:
 
Is there ANY consistent theory T in the language of arithmetic (addition and multiplication) for which it makes sense to say that T proves Con(T)?  
 
The theory need not be complex enough to code proofs or anything like that, but we must fix in advance a uniform way of going from (axiomatizations of) theories to their consistency statements.
 
 
-- Joseph Shipman
 
 
-----Original Message-----
From: Arnon Avron <aa at tau.ac.il>
To: fom at cs.nyu.edu
Sent: Sat, 28 May 2005 10:20:23 +0300
Subject: [FOM] Is Con_Q provable in Q?



In all texts I know about Godel second incompleteness theorem (the
theorem about consistency proofs), it is proved for theories which
are "strong enough", where "strong enough" means: consistent axiomatic
extensions of PRA. My intuition, which is not very reliable, tells me
that the theorem should apply also to weaker theories, for example: to
Robinson's arithmetics Q. Can anybody give me references to works 
which might be relevant to this question?

Thanks

Arnon Avron

School of CS
Tel-Aviv University
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