Lesson 1

Let \(X\) be the set of all sets. Then \(2^{X} \subseteq X\), which contradicts Cantor's theorem. Here begin the paradoxes. Specifically, this one is known as Russell's paradox: if \(Z\) is the set of all sets that are not elements of themselves, then will \(Z\) be an element of itself? Variations of this paradox: the barber who can shave only those who cannot shave themselves; reflexive/self-applying adjectives.

A solution: allow only sets that do not lead to paradoxes. Naive set theory by Cantor: only sets encountered in nature. Zermelo–Fraenkel axioms (ZFC): sets are derived only from axioms.