[FOM] A question concerning incompleteness
andrerog
andrerog at ifi.uio.no
Fri Jun 6 05:34:08 EDT 2014
Dear Arnon Avron
If you do not require an element without predecessors in Robinson
arithmetic
I believe you get a theory with finite models. The integers modulo some
prime p
ought to be an example. One can find finite (and hence recursive)
theories that
characterise each such finite model up to isomorphism.
Theories that characterise models up to isomorphism are complete.
Sincerely
André Rognes
On 2014-06-03 18:40, aa at tau.ac.il wrote:
> Dear Fomers,
>
> I have a question:
>
> It is well-known that no consistent axiomatic extension
> of Robinson's system Q or Shoenfield's system N
> (from his great book "Mathematical Logic") can be complete.
>
> Does this theorem remain true if we delete from Q
> the axiom that states that every number different than 0
> has a predecessor, or if we delete from N the axiom
> that states that the relation < on the natural numbers
> is linear?
>
> Thanks,
>
> Arnon Avron
>
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