FOM: correction of dates re Skolem

[Neil Tennant]

In an earlier posting I included some mistaken dates of publication
for two of Skolem's papers on non-standard models for arithmetic.
They were published in 1933 and 1934.
Both Harvey Friedman and Sol Feferman a/have pointed out that Skolem's
method was a speic/special case of an ultraproduct construction, which
will in general give semantic compactness directly, without needing to
go via a completeness theorem.

Now here is a foundational question for logicians on this list:

Is there any proof of existence of non-standard models for full arithmetic
that uses methods and assumptions too weak to prove either completeness
(on sets of sentences) or compactness?

Neil Tennant