Re: clarification on cardinality of FOL theories

On September 5, Ziv Hellman writes:
> 
> 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?

The problem is in the specification of sequences in Pat's latest
document. This says that IR contains all sequences of elements over
IR. This would mean that IR is not countable. This seems at odds with
the fact that, as you rightly point out, a first order theory should
have a countable model.

Ian

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Received on Friday, 6 September 2002 05:35:52 UTC