Am I correct in understanding Axiom of Extension? Halmos mentions the following: > **Axiom of extension:** Two sets are equal **_if and only_** [emphasized] if they have the same elements. My understanding of Axiom of Extension as presented above is as following: Axiom of Extension is **independent** of the Axioms of Equality in first-order logic with equality. A set might (or must _always_?) also have an _intension_ which determines its _extension_. So two equal sets (by Equality Axioms) **logically implies** that they must have the **same** intension _as well as_ extension, _doesn’t it?_ Now, Axiom of Extension makes a **_logically unprovable_** remark that a set’s extension is all that matters. Please fix my understanding. * * * Also, is “if and only if” required in Halmos’ statement? Cuz according to Jech, > If $X$ and $Y$ have the same elements, then $X=Y$ … > The converse … is an axiom of predicate calculus. I’ve never come across this axiom. So help!

> According to Jech, the converse of the Extensionalty Axiom "is an axiom of predicate calculus."

If the underlying logic is predicate calculus **with** equality, we have the _substitution axiom for formulas_ :

> $x = y → (\varphi → \varphi')$,

where $\varphi'$ is obtained by replacing any number of free occurrences of $x$ in $\varphi$ with $y$.

Thus, considering the formula $(z \in x)$ as $\varphi$, we have :

> $x=y \to (z \in x \to z \in y)$.