[FOM] A question concerning incompleteness

[Arnon Avron]

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