FOM: archives; an unsigned message on lucky axioms
Stephen G Simpson
simpson at math.psu.edu
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
http://www.math.psu.edu/simpson/Foundations.html
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 cis.udel.edu>. 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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