# What does “extension” mean in the Axiom of extension

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 name is axiom of extensionality. – Asaf Karagila Jun 29 '14 at 23:28
• See Axiom of extensionality : the role in the theory is simple; its historical background is more complex : see this post. In "ancient times" a property or predicate has an extension, formed by all the objects which satisfy the predicate. Thus, in modern set-theory, the meaning of extensionality in the corrisponding axiom is " what is relevant for the 'identity' of a set are only its elements, i.e. its *extension". – Mauro ALLEGRANZA Jun 30 '14 at 6:22
• Related article: en.wikipedia.org/wiki/Extensionality – Jack Jan 31 '17 at 17:19
• @AsafKaragila It’s called “axiom of extension” in the book. books.google.it/… – egreg Sep 25 '17 at 7:58
• @egreg: I used to hear how great is this book. But as time progresses forward, I grow to dislike it more and more. – Asaf Karagila Sep 25 '17 at 9:08

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 $B \subset A$.