[FOM] Is Con_Q provable in Q?

[Torkel Franzen]

Arnon Avron writes:
>
>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?

 A.Bezboruah and J.C.Shepherdson, JSL Vol.41 No. 2 (1976) prove that
Q does not prove its own consistency (choosing a suitable formalization
of "Q is consistent"). They comment that "We are extremely grateful
to Kreisel for his patient and painstaking criticism and his
attempt (even if not altogether successful) to steer us to areas where
more significant versions of Gödel's second theorem might be proved."