Axioms in Tao's Analysis seem different from those in MathWorld?

Question

In Tao's Analysis the chapter on set theory introduces axioms such as "all sets are objects" and "there exists an empty set", but then when I look up the ZFC axioms he's building up to they're very different: http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html

Is there a reason for this difference?

Answer 1

At the bottom of the MathWorld site it says:

Unfortunately, there seems to be some disagreement in the literature about just what axioms constitute "Zermelo set theory." ... Enderton (1977) includes the axioms of choice and foundation, but does not include the axiom of replacement. It includes an Axiom of the empty set, which can be gotten from (6) and (3), via $\exists X(X=X)$ and $\emptyset=\{u:u \not=u\}$.

You can derive the existence of an empty set from the existence of any set using the Subset Axiom, and that's why some formulations don't have 'the empty set exists' as a separate axiom, but other formulations find it elementary enough to warrant a separate axiom.

As far as sets being objects goes: That doesn't really translate into an explicit axiom, but rather is recognized by the fact that we can quantifiy over them.