FOM: Nonstandard models; Skolem

[Moshe' Machover]

Neil Tenant writes:

>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!).

This may well be the better interpretation. However, I wish to point out
that NT's reference to Z*F* here is ironic; for it is in this very paper
that Skolem proposes his version of Replacement, and it is this version
(rather than Fraenkel's, which is somewhat vague, to coin a phrase :-) )
that we actually use today.

> 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.

There may be another, more psychological reason. To most logicians
nonstandard models were a great embarassment. They preferrred to look the
other way. Note that even after 1929 few people actually seriously studied
NS models. Their study started seriously only with Henkin (1949) and took
off gradually during the 1950s. Only Robinson, from 1960, taught us to love
these monsters and put them to good use. But Skolem was in a special
position. While being a top-rate logician and set-theorist, his
constructivist inclinations made him critical of the attempt to reduce
mathematics, and in particular arithmetic, to set theory or logic. So he
had a kind of Schadenfreude in pointing out that nonstandard models exist.
He relished them for ideological (philosophical) reasons. This is clear
from the whole tenor of his 1922 paper.


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