[FOM] Re: Indispensability of the natural numbers

[Timothy Y. Chow]

> Roughly speaking, as I understand it, the usual picture of the
> "standard" natural numbers (closed under all primitive recursive
> functions) is to the "true" natural numbers (given by rules R1, R2, and
> R3) as the "nonstandard" model theoretic natural numbers are to the
> "standard" natural numbers.

Can you elaborate on this?  Is there a way of forming a clear picture of
an entity which satisfies R1, R2, R3 but is not closed under all primitive
recursive functions?  Or alternatively, is there a theory that has as a
theorem "There exist N1 and N2 that both satisfy R1, R2, R3, but N1 is
closed under primitive recursive functions while N2 is not"?

Tim