TITLE: Why isn't the axiom of extensionality considered a definition of equality?
QUESTION [10 upvotes]: This question was essentially asked and responded to here, but I don't feel it was really answered: 
Is the axiom of extensionality merely called an "axiom" by convention, or is there a clear distinction between axioms and definitions that forbids it from being considered a definition of set equality?
The answer in the linked post seems to address what would happen if the axiom were rejected altogether, but doesn't discuss whether there is good reason to consider it an axiom rather than a definition. If axioms and definitions are just two different words for the same thing, then that's fine—but I get the feeling I'm missing something. Do we simply call a definition an axiom if it defines a particularly foundational concept (like equality)?

REPLY [7 votes]: If you're asking about why the axiom of extensionality is needed in set theory specifically, the answer isn't about $=$ but rather about $\in$.
Remember that a priori, "$\in$" is just some binary relation symbol. Structures in the language $\{\in\}$ are essentially just directed graphs where there is at most one edge between any two vertices. Thought of this way, extensionality says:

If $a, b$ are vertices, and every vertex with an edge to $a$ also has an edge to $b$ and vice versa, then $a=b$.

Obviously this is not true in general! E.g. consider a graph with fifteen vertices and no edges. Basically, the point is that the semantics for first-order logic does understand what "$=$" means; however, it doesn't understand what "$\in$" is supposed to mean until we add some axioms.