[FOM] Re: a definable nonstandard model of the reals

[Dave Marker]

Dear Phil:
Thanks for your thoughtful message. The point that bothers me is
that even once we know that we can expand No to make it a model
of nonstandard analysis, why should the expansion be definable?
We know it is unique up to automorphism, but I don't see why
it would be canonical.

It seems to me that the argument is that
i) there is a unique "saturated" class model with the desired ordertype
ii) the reduct to the field language will be isomorphic to No

Thus we can expand No to make it into such a model (and this model
is unique up to isomorphism).
Is this right?

My question is whether there is a natural or canonical to equip No
with the additional structure that makes it the desired model of
nonstandard analysis. A simpler version of this question: how do
we define "integer" so that we get a model of the theory of the
real field with a predicate for the standard integers?

Best Wishes
Dave