FOM: RE: Axiom of Extensionality

[Cristian Cocos]

> There is a set of unicorns, which is empty.

The fact that the set of unicorns is empty is an empirical fact, not an
analytic truth. You are trying to (illegitimately) draw in matters that
belong to the factual realm (like the existence/non-existence of
unicorns, or the existence of Alice, Bob and/or Carol, or of people who
live in Spencer Walk etc.) into a discussion about/of analytic truths
(like, presumably, the axiom of extensionality).

> If there were unicorns, would this be a different set?  But 
> then how could both be the set "of unicorns"?

Actually they are not. This is an abuse of language. They are, say, sets
"unicorns_a" and "unicorns_b", where the former is
the-set-of-unicorns-in-possible-world-a and the latter
...-in-possible-world-b. Two different sets; it is just that referring
to each as "set of unicorns" is often more convenient (as opposed to the
infinitely more cumbersome "the-set-of-unicorns-in-possible-world-a"
etc.), but can cause confusion.

> Or take the set {Alice, Bob, Carol}.  Does this continue to 
> exist even if they cease to?  Is it that it currently has the 
> number 3, but the number 0 if they do not exist?

How can the set {Alice, Bob, Carol} have number 0? Again, the problem
has to be regarded beyond contingent factual matters like the existence
of Alice, Bob and/or Carol. Its number is 3 regardless of such facts.

Cristian Cocos
UWO