Hi David,
Here are some thoughts from someone who is far from a mathematician and
who is more a database practictioner than a database theorist, so take
them with whatever amount of salt you see fit.
When I worked with sets during the Dutch equivalent of highschool, I
usually had to use a two-part notation. I can't replicate the symbols
here and I don't recall all the correct names, but it consisted of a
definition of a domain and a listing or formula to define the values. So
you could have a set that was defined as a subset of the domain of
positive integers consisting of the numbers 2, 4, and 7; but you could
also have a set defined as a subset of the domain of real values
consisting of the numbers 2, 4, and 7.
Later, after highschool, I started to see a simplified notation for sets
that exposes only the values in the set but not the domain.
Are the two sets above equal? I guess that you could defend both answers
here - the sets have the same members, but not the same domain
definition. I also guess that the notation used can sometimes be an
indication of how the answer would be in any give UoD.
For a general answer, I'm tempted to say that there have to be two
equality operators for set arithmetic, one looking at the values of the
set members only, the other also looking at the domain.
Anyway, whatever you favor as an answer to the question of equality of
the two sets above - once you've chosen an answer, the answer to
equality of two empty sets logicallly follows.
