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