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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.

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    $\begingroup$ The name is axiom of extensionality. $\endgroup$
    – Asaf Karagila
    Jun 29, 2014 at 23:28
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    $\begingroup$ 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". $\endgroup$ Jun 30, 2014 at 6:22
  • $\begingroup$ Related article: en.wikipedia.org/wiki/Extensionality $\endgroup$
    – user9464
    Jan 31, 2017 at 17:19
  • $\begingroup$ @AsafKaragila It’s called “axiom of extension” in the book. books.google.it/… $\endgroup$
    – egreg
    Sep 25, 2017 at 7:58
  • $\begingroup$ @egreg: I used to hear how great is this book. But as time progresses forward, I grow to dislike it more and more. $\endgroup$
    – Asaf Karagila
    Sep 25, 2017 at 9:08

2 Answers 2

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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.

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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 $B \subset A$.

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    $\begingroup$ The term extension is a technical term in logic. $\endgroup$
    – Asaf Karagila
    Sep 25, 2017 at 8:06
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    $\begingroup$ @AsafKaragila I was just trying to provide some context and another way of using the word extension to give intuition to the OP. $\endgroup$
    – user5826
    Sep 25, 2017 at 16:34

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