TITLE: What does "extension" mean in the Axiom of extension
QUESTION [13 upvotes]: 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.

REPLY [2 votes]: 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 $B \subset A$.