Re: Equal Sets and Identical Sets

On Mar 11, 12:28 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:

For '=' as defined in Z set theory (with a suitably
adjusted axiom of extensionality), we PROVE that in Z set theory '='
is such that the identity axioms are satisfied. That is provable since
Z set theory has a finite number of predicate symbols (in general, for
a language with an infinite number of predicate symbols, this would
not be provable).

That's all correct.

That is to say that we prove BOTH directions of the
Leibniz principle: the identity of indsicernibles and the
indiscerniblity of identicals. Then an interpretation must map to {<x
x> | xeU}, which is the identity relation on the universe.

Again, as I mentioned in another post, that part might be off the
mark. I have to admit that last night I was not able to finish proving
that '=' as defined (not primitive) in set theory with suitably
adjusted axiom of extensionality does entail that '=' is interpreted
as the identity relation on the universe. I hope later to give more of
the details.

MoeBlee
.

Loading