FOM: archives; an unsigned message on lucky axioms

Stephen G Simpson simpson at
Thu Dec 4 10:50:14 EST 1997

Two remarks from your moderator:

1. I would like to remind everyone that FOM archives and individual
postings are available on my web page at

2. Neil Tennant writes:
 > Someone else (was it Steve?) sent in an unsigned message with no
 > sender's name in the header, ...

Sorry, this mixup was caused by my attempt as moderator to edit out
some superfluous mail headers.  The message was actually from John
Case <case at>.  The text of the message appears below.

-- Steve Simpson
FOM moderator

 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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