[FOM] Order type of non-standard models

[Alasdair Urquhart]

I went to the library last week and got hold
of Thoralf Skolem's "Selected Works in Logic",
edited by Fenstad.  

My memory of the reference to order types of
non-standard countable models of PA was not
entirely correct, but not entirely wrong, either.

In his paper of 1934 proving the existence of 
countable nonstandard models for first-order
arithmetic, Skolem simply remarks that they are
not isomorphic to the standard model, because 
they are of a higher order type.

"The content of Proposition 4 can be expressed
as follows: Finitely or countably many
sentences with variables restricted
to individuals, that hold for a sequence of
type omega, cannot distinguish this sequence from
certain sequences of a higher order type"
			(Selected Works, p. 364).  

Hence, Skolem certainly does not state the result on
order types.  However, there is an "Editorial Note"
at the end of the paper that reads (in part) as
follows:

"The following (more specialized and independent)
results that were obtained in the years 1927/28 in 
the seminar conducted by Herr A. Tarski (Warsaw
University), have an immediate connection with the
above results of Herr Th. Skolem.

1.  Herr A. Tarski has pointed out in this connection,
that the set of arithmetical sentences (= sentences
with only individual variables), in which as constant
symbols (apart from logical symbols, such as the symbol
for identity) the symbol for the ordering relation 
"<" occurs, and that is satisfied by the order type
omega is identical with the corresponding set for
the order type   omega + (omega* + omega).tau
(tau an arbitrary order type)."
			(Selected Works, p. 366)

This isn't the same as the result we are talking about,
but it is clearly closely related.  I think it is
pretty obvious that Tarski was a referee for Skolem's paper!

However, I can't find a reference to the result in
Tarski's papers before 1950, so Martin Davis is probably 
right when he attributes the result to Henkin's thesis.

The result is given in the last paragraph of Henkin's 
paper "Completeness in the Theory of Types" (JSL Vol. 15, 
pp. 81-91), based on his Princeton thesis of 1947, 
written under Alonzo Church:

"A detailed investigation of these numerical structures
is beyond the scope of the present paper.  As an example,
however, we quote one simple result:  Every non-standard
denumerable model for the Peano axioms has the order
type omega + (omega* + omega) eta, where eta
is the type of the irrationals."  

I think that perhaps the first proof published in a journal
was given by John G. Kemeny in "Undecidable problems of elementary
number theory", Mathematische Annalen, Vol. 135 (1958), pp. 160-169.
The review of this paper by Skolem in the JSL Vol. 23, p. 359,
simply says that it is a "previously known theorem," but
doesn't give an attribution.  

That is as far as my historical research extends.  
Apologies if my translations from the German are bad.

Alasdair Urquhart