Re: Axiom of Pairing, Scheme of Replacement from others

Here's that discussion I mentioned:

http://en.wikipedia.org/wiki/Talk:Axiom_of_empty_set

It seems I was wrong as to which author is involved, but the page under
discussion is one you quoted from also.

From the page being discussed:

"That said, any axiom of set theory or logic that implies the existence
of any set will imply the existence of the empty set, if one has the
axiom schema of separation. However, if separation is derived as a
theorem schema from the axiom schema of replacement (as is sometimes
done), then that derivation requires the axiom of empty set. So it
could not be used to eliminate the axiom of empty set."

In the discussion you'll see where the author went off course:

He says, "Some versions of First-order logic implicitly assume that
something exists. However, I consider them to be wrong [...]"

In other words, he's not using the standard first order logic of formal
Z and ZF set theories. Using just standard predicate logic, we can
derive the existence of an empty set from the axiom schema of
separation alone or from the axiom schema of replacement alone. But the
author of the page demurs from using standard predicate logic and
thereby declares that the axiom schema of separation does not entail
the existence of an empty set. In other words, he's making a claim
about set theory based on his OWN view ("I consider to be wrong) of
what logic should be used (and that is not even mentioned on the page,
but only in the discussion about the page). Lousy. An example from
Wikipedia.

MoeBlee

.

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