[SCL] Identity and Horrocks sentences

[pat hayes]

>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.

OK, I stand corrected. Sorry. However, whatever it is called, the 
fact remains that if a set of sentences is consistent in FOL then it 
is satisfied in a countable model (trivially; just make aleph-0 
copies of the items in the universe), whereas the same is not true 
for FOL+=.

Pat
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