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Halmos's Naive Set Theory explains the "extensionality" in "axiom of extensionality" as:

Every set is determined by its extension.

and that's it. What is a set's extension, then? Intuitively it seems to me that an (or maybe "the", who knows?) extension of $X$ is a set that contains all of $X$'s elements, and possibly some other ones too. Am I right? If not, could you please explain what an extension of a set is?

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Let $M$ denote a set equipped with a binary relation $R$. Then given $p \in M$, there is a corresponding predicate $$E_p : M \rightarrow \{\mathrm{true},\mathrm{false}\}$$

defined as follows.

$$E_p(x) \iff R(x,p)$$

We call $E_p$ the extension of $p$. The Axiom of Extensionality just says that every $p \in M$ is determined by its extension $E_p$. More precisely, $M$ satisfies the the axiom of extensionality iff for all $p,q \in M,$ we have that if $E_p = E_q$, then $p=q$.

Thinking of the elements of $M$ as being "sets" in some sense or another and $R$ as denoting membership, the definition of $E_p$ just says that $E_p(x)$ is true iff $x$ is an element of $p$, and false otherwise. Hence in this language, the Axiom of Extensionality says that a set is determined by its elements.

Of course, $M$ doesn't have to be a model of ZFC for this to make sense, and the elements of $M$ don't have to be set-like. Thinking of $(M,R)$ as a digraph, extensionality just says that if $p$ and $q$ have precisely the same vertexes immediately below them, then they're equal.

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  • $\begingroup$ The first paragraph will be miles better if you use $E$ for the binary relation on $M$. $\endgroup$ – Asaf Karagila May 12 '15 at 15:36
  • $\begingroup$ @AsafKaragila, nice suggestion. $\endgroup$ – goblin May 12 '15 at 15:41
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The term extension has an ancient logico-philosophical tradition.

See Port Royal Logic and :

I call the extension of an idea the subjects to which this idea applies.

Thus, from a modern set-theoretic perpective, the extension of a set $X$ is the "collection" of all those objects that are elements of $X$, i.e. $X$ itself.

If so, the set notion is based on "extensionality", i.e. all that matters in defining the "identity card" of a set are its members.

Conclusion, two sets that have exactly the same elements are "the same" set.

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This has nothing to so with the idea of "extending" a set (i.e., making it bigger). Rather this refers to how far the set extends or its "scope" - in short: its elements. In other words: If two sets $A$ and $B$ have exactly the same elements, then they are in fact the same set.

The extension of a concept is the totality of all things falling under the concept.

The instension of a concept on the other hand is the collection of all properties that all examples of the concept factually share.

Several different intensional definitions can be extensionally equivalent, for example "the morning star", "the evening star" bothrefer to the same single object, the planet Venus. Or "bipedal non-feathered beings" and "rational beings" both descibe the concept of humans (at least in the usual universe of discourse; e.g., a Greek mythologist would require that we add the adjective "mortal" to both descriptions).

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The Axiom of Extensionality in Set Theory states that a set is completely determined by its elements, or, in other words, that two sets are equal (i.e. the same set) if they contain the same elements.

In this sense, the "extension" of a set is simply its content, which is all you need to identify it.

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I don't know what exactly Halmos meant by it. How it was described in my set theory class: Formally, the axiom of extensionality says two sets are equal if and only if they have exactly the same members, i.e. $$x=y \iff \forall z (z \in x \iff z \in y)$$

Its called extensionality as opposed to 'intensionality'. Extensionally means every last detail down to how the nuts and bolts are made, intensionality means just having the same defining qualities. For instance, according to extensionality, the morning star and the evening star are the same...they are names for Venus. Intensionally, they are different, they refer to different aspects, etc.

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