From: utzoo!decvax!yale-com!leichter
Newsgroups: net.math
Title: Re: Loweheim Skolem
Article-I.D.: yale-com.776
Posted: Wed Feb  2 09:48:31 1983
Received: Thu Feb  3 03:42:33 1983
References: watmath.4443

You're ignoring the assumptions of Lowenheim-Skolem:  "Every theory WITH
FIRST ORDER AXIOMS ... ".  There does not exist a theory with first-order
axioms that defines a complete ordered field, for example.  (Proof:  One
shows that all complete ordered fields are isomorphic to the reals, and
that the reals are uncountable.  QED)  It IS possible to build countable
models of the reals that have the same first-order properties; that's one
kind of "non-standard" analysis.

For those who are wondering what this is about:  A first order statement is
one in which one can have objects and sets of those objects, but not sets
of sets of those objects.  (This is informal but I don't remember the formal
definitions any more.)  What you consider the "objects" to be is arbitrary,
but if you want a set of first-order axioms, you have to make a consistent
choice once and for all, such that all the variables in ALL your axioms
represent either objects, or sets of those objects - but nothing nested deeper.
The problem with the "complete ordered field" is stating the "every set with
an upper bound has a least upper bound" axiom.
							-- Jerry
						decvax!yale-comix!leichter