Axiom of Extensionality - Why not called equality?

Question

Axiom of Extensionality, I understand that if two sets have exactly the same members they are equal. However why it is called Extensionality? Why not equality?

Answer 1

"Extension" refers to a set being defined by its content, as opposed to "intension" which is a term to say it is defined by some form of specification.

Let $A = \{x \text{ in } \mathbb{R} \text{ such that} -1 \leq x \leq +1\}$

Let $B = \{y \text{ in } \mathbb{R} \text{ such that } y = \sin(x) \text{ for some } x \text{ in } \mathbb{R}\}$.

The sets have different intensions (specifications), but contain the same elements and therefore have the same extensions. So (according to the "Axiom of Extension") $A = B$.

Answer 2

Because it says a set is determined by its extension.