🚀 go-pugleaf

In article <COLUMBUS.92Nov19151402@strident.think.com> columbus@strident.think.com (Michael Weiss) writes:
>Quite true: the simplest "upward" L-S theorem asserts that if a theory with
>a countable language has an infinite model, then it has models of arbitrary
>cardinality kappa.  This is easy to see from the completeness theorem: just
>add kappa constants to the language, and axioms stating that they are all
>distinct; you can easily show from the hypothesis that this extended theory
>is consistent, so it has a model.

In its proper generality: For any theory T, either there exists an
integer such that all models of T are of cardinality less than that
integer, or T has models of all infinite cardinalities greater than or
equal to the cardinality of its language.

Just so that people don't think first order logic is a total wimp, it
should be pointed out that a first order theory *can* pin down
everything about an infinite model *except* its cardinality.  One says
that T is *categorical* in power b, or b-categorical, when all models
of T of cardinality b are isomorphic.  A theory may or may not be
categorical in power aleph_0, and independently may or may not be
categorical in a given uncountable power.  But if it is categorical in
some uncountable power then it is categorical in all uncountable powers
(Morley's theorem).  Hence only four kinds of theories are possible.
All four can happen.
-- 
Vaughan Pratt	  		A fallacy is worth a thousand steps.