[FOM] A question about \omega-consistent theories

pax0@seznam.cz pax0 at seznam.cz
Mon Apr 27 15:11:38 EDT 2009

 Arnon Avron wrote:
>  Can anybody give me an example of a (not necessarily recursive
>  or r.e.)  \omega-consistent theory in the language of PA which
>  proves all true quantifiers-free sentences in this language,
>  but also some sentence which is  not true (in the standard model of PA)?
>

Just take the theory
PA+not omega-con(PA),
where omega-con is the formalized property of being omega consistent.

This theory proves all true Q.F. sentences (by Sigma_1-completeness of PA),
but omega(PA) is presumably true.

Jan Pax