clarification on cardinality of FOL theories

During the last tele-con there was some discussion on uncountability and
First-Order Logic Theories.

According to a number of text-books I have on my shelves, a first-order
logic language can itself be of any cardinality, and theories in that
language can have models of any cardinality as well (unless they have an
explicit axiom stating they are finite). By Loewenheim-Skolem, if the
language is countable, theories written using the language will also
have countable models. And whether or not a theory is countably or
finitely axiomatisable seems independent of other cardinality
considerations involving the language, etc. 

So it would be appreciated to hear clarifications of what was troubling
the participants with regard to the cardinality of the suggested system
of hierarchies of classes of classes, which appears to be fully
First-Order, not uncountable as a language, and given to finite
axiomatization. Was the concern related to the capabilities of various
known reasoning systems? And if so, where does the question of
uncountablity enter?

Received on Thursday, 5 September 2002 13:51:49 UTC