[FOM] Is Con_Q provable in Q?

praatika@mappi.helsinki.fi 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


Panu Raatikainen
Helsinki Collegium for Advanced Studies
P.O. Box 4
FIN-00014 University of Helsinki
Tel:  +358-(0)9-191 23437
Mobile:  +358-(0)40-840 0789
Fax:  +358-(0)9-191 24509
Email: panu.raatikainen at helsinki.fi

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