Couple of questions on the Axiom of Extensionality

Question

I understand that the axiom of Extensionality says that two sets are equal iff they have the same elements. Which is clear enough. But take a look the definition for this axiom given by mathworld(http://mathworld.wolfram.com/AxiomofExtensionality.html) which says:

$$\forall u(u \ \epsilon A \equiv u\ \epsilon B)\Rightarrow A=B$$

This would mean that the vacuous case in which in $u \ \epsilon A \equiv u\ \epsilon B$ can be false yet $A = B$ can still hold. How can this be? Paul Halmos mentions the following example which I think brings out this point(or atleast in the way I understand it).

The example is if $x$ and $A$ are human beings we will write that $x \ \epsilon A$ whenver x is an ancestor of A. (Ancestors are parents, grandparents,grand-grandparents and so on). By the analogue of the axiom of Extensionality we would say that if two humans are equal then they will have the same ancestors(only if part ; which is true) and also if two humans have the same ancestors then they are equal(the if part; and this is false).

I believe he says its false because two siblings can have the same ancestors yet are not the same person.

Can explain what that example illustrates and how the definition holds in the vacuous case as given by mathworld? Thank you very much.

Here is a link to Halmos' book. The example is on page 3.

https://books.google.com/books?id=x6cZBQ9qtgoC&printsec=frontcover#v=onepage&q&f=false

Answer 1

No. Just because something isn't mentioned in the axiom does not mean that it is necessarily not true.

When formalizing logic to include equality, something which is the modern standard in mathematics, one of the axioms included is the following schema:

If $x=y$, then for every formula, $\varphi(x,p_1,\ldots,p_n)\leftrightarrow\varphi(y,p_1,\ldots,p_n)$.

This means that if $A=B$, then the formula $u\in x$ is true for $A$ if and only it it is true for $B$. So the converse of the axiom holds by the axioms of logic.

It might be worth mentioning that in some older treatments of set theory, equality wasn't standard as part of the logic, and was defined as the equivalence given in the axiom of extensionality. Namely, we would define $A=B$ as a shorthand for $\forall u(u\in A\leftrightarrow u\in B)$.