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Stephen G Simpson simpson
Wed Dec 3 13:48:39 EST 1997

On Dec 3,  0:56, Neil Tennant wrote:

	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
}-- End of excerpt from Neil Tennant

It is interesting that sometimes logicians have lucked out with axiomatizing
something when they didn't quite know what they were talking about but
felt there was surely something (even something important) that they were
talking about.  Other times, e.g., set theory, they did not luck out with
the strategy of first the axioms, then figure out what it is you are 
axiomatizing.  (-8 John

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