I am learning Set Theory from the book Naive Set Theory by Halmos as part of my course. The first chapter is on the Axiom of Extension. I understand what it is but what I don't understand is why it has the word "extension" in the title for this axiom. I can't understand how equality has the same meaning as extension. Please can someone clarify this.
The axiom states that two sets are equal if they have the same elements, i.e. they are equal in "extension" (scope, content), as opposed to equality in "intension" (meaning, concept). For example, the set of black US presidents is currently equal in extension to the set containing Barack Obama as a single element, but they are different in intension. The axiom of extension means that the set theory only deals with the content of sets, not with the concepts used to form them.
I like to think of it as: Two sets $A$ and $B$ are equal if and only if $B$ is an extension of $A$, and $A$ is an extension of $B$. That is, two sets are equal iff $A \subset B$ and $B \subset A$.