[FOM] Non-standard order types

[Dana Scott]

I have to submit that anyone who ever thought about models of PA would
be able to prove that the order type of a non-standard model is of
type
              omega + (omega* + omega) x d,

where d is a dense ordering without endpoints.  In the countable case,
of course, d will have the type eta of the rationals.

Let a < b be non-standard and infinitely far apart.  Let

         c = (a+b)/2  or  c = (a+b+1)/2,

according as a+b is even or not.  Then c is (more or less) midway
between a and b, and not only is  a < c < b but they are infinitely
far apart.  That proves density.

Then using 2 a and a/2 or (a+1)/2, we note these elements are both
infinitely distant from a.  That proves d has no first or last
element. 

I don't think a long historical investigation will come up with much,
when a proof like this is so very elementary.  (Of course, one needs
to understand first that there CAN BE non-standard models.  And I
guess Skolem probably gets the credit here.)

Turning to ordered FIELDS, I asked a very long time ago: Which ordered
sets are the orderings of ordered fields.  I do not think anyone ever
answered this.  Did they?