[FOM] A question concerning incompleteness

[andrerog]

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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