[FOM] Is Con_Q provable in Q?
praatika at mappi.helsinki.fi
Sun May 29 05:38:51 EDT 2005
Arnon Avron <aa at tau.ac.il> wrote:
> 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?
The problem with Q is that in the standard proof of the second
incompleteness theorem, one has to establish that the three derivability
conditions hold, and that requires induction which Q does not have.
However, there is a version of the second theorem for Q:
A. Bezboruah & John C. Shepherdson: "Godel's Second Incompleteness Theorem
for Q", Journal of Symbolic Logic, Volume 41 (1976), 503-512.
All the Best
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