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Here, Terence Tao writes:

it means that one cannot, after all, make equality of sets just a definition, it has to be an axiom.

Isn't this just a matter of denotation? Why can't one call $A=B\iff \forall x(x\in A\Leftrightarrow x\in B)$ a definition?

I think in the book Mathematical Thinking: Problem-Solving and Proofs it is called a definition rather than an axiom, and I don't think there is something false with it.

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In modern mathematics, equality is part of the underlying logic. So equality is not defined, two things are just equal when they are the same.

However, in set theory, we want to cast equality in terms of $\in$, rather than just take it "as granted". Here comes the Axiom of Extensionality, which tells us that if two sets are not equal, then this can be identified by an atomic formula using the $\in$ relation.

In old enough texts, however, equality was not considered as a part of the underlying logic, and there equality was defined using the Axiom of Extensionality (when it comes to set theory, anyway).

To reiterate, the point of the Axiom is to connect between the two relations, $=$ and $\in$; and in sufficiently old texts, this was in fact taken as the definition of equality.

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    Thanks. But my point is: even if one has these axioms of equality and takes them for granted, why can't one *call* the axiom of regularity a definition? Isn't it a little too fastidious to say that one isn't allowed to do so? – 2017-02-01
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    Axioms *are* definitions. Equality is defined using axioms of *logic*. Sets are defined using the axioms of *set theory*. Vector spaces are defined, well you get where I'm going with this. – 2017-02-01
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    @Asaf: I think sweeping assertions about what is and is not part of the underlying logic of "modern" mathematics are entertaining but unwise. People doing constructive analysis will disagree with your notion of equality as much today as they did in 1967 when Bishop wrote his well-known book on the subject. – 2017-02-01
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    @Rob: I think that for rather naive questions, somewhat naive answers can be much more pedagogically helpful, as compared to nitpicking answers that start dealing with issues one might not be able to fully appreciate yet. – 2017-02-01
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    I don't think this question (which isn't particularly naive) deserves a naive answer. You have just (in effect) told the OP that Tao's point of view is right and that his or her alternative point of view is not worthy of an explanation. What you are dismissing as nitpicking is the point of the original question! – 2017-02-01
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    @Rob: Is there a real difference between "axiom" and "definition" in modern mathematics? I don't think so. I don't even think that constructivists would say so. I do think that a naive question would be of someone still holding the "high school view" that an axiom "is a given" and a definition "is defined" and there is some sort of a distinction between the two. I do agree, that some things need to be clarified, but this is not necessarily the best course of action when one first comes across these sort of questions. Sometimes it's better to first see the big picture, and only then the details – 2017-02-01
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    @Asad: duh? there is a real difference between primitive **notions** and defined **notions** in mathematics. Your "big picture" on this question is smaller than the OP's because you insist that equality must alway be provided as a primitive notion. – 2017-02-01
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The status of the law of extensionality: $$ A = B \iff \forall x(x \in A \iff x \in B) \tag{EXT} $$ depends on how you formalise first-order logic:

  • if (as is common) you treat $=$ as a logical symbol, then $=$ is "predefined" as denoting a given notion of equality in the domain of discourse, and you have to view (EXT) as an axiom. This is how Tao is viewing things.
  • otherwise (as is not uncommon $\ddot{\smile}$), you treat $=$ as non-logical symbol, then (EXT) serves as definition of that symbol.

In my view, this is a minor matter of terminology and technical detail. But others will disagree.

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    I wouldn't call it a minor matter of terminology, but I would consider it a technical detail for most mathematicians' purposes. That said, in general, the formulation linking the logical notion of equality to a specific formula has a rather profound impact on the theory as a whole, whereas the latter is, as you say, basically just a definition and has no impact on the rest of the theory. – 2017-02-02
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It's actually the axiom of extensionality of ZF theory https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory#1._Axiom_of_extensionality It's like in the essence of the set concept, like number axioms. They're just true.

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    Of course one can call it an axiom. I know that. My question is about why one can't call it a "definition". I referred to a book which calls it "definition" and I also referred to someone who says that one can't call it a "definition". I want to know what to believe, and this "answer" doesn't help at all with that. – 2017-02-01