[SCL] Identity and Horrocks sentences

[Chris Menzel]

On Mon, Jun 02, 2003 at 12:05:30PM -0500, Pat Hayes wrote:
> >Not sure what you mean.  Upward and downward L-Sk hold for both FOL and
> >FOL=.
> 
> Upward doesn't: (forall (?x ?y)(?x = ?y)) doesn't have a countable 
> model in FOL=, does in plain FOL. Countable means cardinality 
> aleph-0, right?

That's not upward LST.  There are several ways to state the theorem, but
the simplest is that if a theory has an *infinite* model of cardinality
k, then it has models of every cardinality larger than k.

-chris