FOM: non-standard models

Neil Tennant neilt at
Wed Dec 3 00:56:58 EST 1997

I'd like to thank Professors Tait, Felsher and Machover for their helpful
references to the Skolem writings. 
I think the 1922 remark about relativities in simpler axiom systems than ZF
does not clearly show that he had in mind back then the possibility of a
model for Th(N) not isomorphic to N. Rather, it seems (from the immediately
precedin g context of the 1922 paper) that he would have been thinking,
rather, of a theory like that of the reals (at first r/order) whose intended
model is uncountable, but for which the Lowenheim method that Skolem improved
on would furnish a countable model, contrary to the theoretician's real
intentions (if you will excuse the poor pun!).
In 1929 Skolem had non-standard models for finite sets of arithmetical sentences. The 1930 paper extended that to countably infinite sets (such as Th(N) itself).
It was remarkable that this was done independently of any (completeness or)
compactness theorem for first-order logic.
But, given that, I find it even more remarkable that no-one had pointed out
even earlier than Skolem's first discovery that non-standard models would
exists *if* first-order logic turned out to be complete. Perhaps this is
because the concepts needed for the statement of the completeness theorem
were really only sharpened sufficiently in the very work in which Godel
proved completeness in 1929.

Neil Tennant

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