Re: Equal Sets and Identical Sets

MoeBlee <jazzmobe@xxxxxxxxxxx> writes:

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.

why would you expect that to be the case?
That's more than asking that the defined equality satisfies
the axioms for identity, after all.


MoeBlee

--
Alan Smaill
.

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