[FOM] History of nonstandard models of PA

[Alasdair Urquhart]

Dana Scott rightly observes that we presently
consider the characterization of the order
types of countable nonstandard models a very easy, if
not trivial result.  However, because we consider
it trivial now does not mean that it was trivial
eighty years ago.

It is not difficult to illustrate this kind of thing.
Look at Goedel's review of Skolem 1933a (Collected Works,
Volume 1, p. 379).  Goedel observes that the impossibility
of characterizing the natural numbers by an "axiom system"
follows from his 1931 incompleteness theorem.  Quite 
surprisingly, he does NOT observe that Skolem proves
the stronger result that the set of all true arithemtical
sentences does not characterize N.  He also fails
to observe that Skolem's result is an easy consequence
of the compactness theorem.  Robert Vaught has
interesting comments on this in his introduction to
the reviews in the Collected Works Volume 1.

It was surprising to me to find how few publications
on nonstandard models of arithmetic there were
prior to 1960.