🚀 go-pugleaf

Mike Oliver <oliver@math.ucla.edu> wrote in message news:<3BF86980.CC10AFAF@math.ucla.edu>...
> John wrote:
> 
> > Unfortunately, the meaning of "we will restrict extensionality to
> > sets" is hardly clear.  It could mean equi-membered things have to be
> > equal IF they are sets; it could mean equi-membered things have to be
> > equal ONLY IF they are sets; and it could mean equi-membered things
> > have to be equal IF AND ONLY IF they are sets.
> 
> This paragraph doesn't make any sense to me.  The only possible
> sense of "restrict extensionality to sets" is, if you know two
> things are sets and have the same members, you can infer they're
> equal.  No assertion at all is made about things that have the
> same members but are *not* sets.

If rationality is restricted to humans, only humans are rational.  If
bachelorhood, is restricted to adult males, only adult males are
bachelors.  Likewise, if the axiom of extensionality is restricted to
sets, only sets are subject to the axiom of extensionality.

Have you some reason why "extensionality is restricted to sets" should
not be understood in the usual way?

> Obviously they *can* be  equal, even if they're not sets -- a thing is
> equal to itself, whether it's a set or not.

Restricting extensionality to sets doesn't keep atoms from being
identical.  It just means extensionality doesn't apply to them. Which
should come as no surprise.  After all, does extensionality apply to
clubs or toes?

John