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PROFESSOR: Let's take a quick look at the axioms of Zermelo-Frankel Set Theory With Choice. So the axioms of ZFC define the standard theory of sets, which is now accepted by most mathematicians as a reliable and simple basis for developing and justifying all of mathematics. Among the axioms, maybe a simple want to understand and really the motivation for this short video is twofold. One is practice with writing predicate formulas, and the other is to think a little bit more about self application. So one of the basic axioms of set theory is called extensionality, which is capturing the idea that a set is determined by its members. So let's consider the assertion that two sets x and y have the same elements, which we could write as a predicate formula in set theory as for all x, x is a member of y, if and only if x is a member of z. Now we could use this is a definition of equality. It's what we mean by y and z are equal. But we don't really need to even in...