This is advanced material. It is a lot of fun to know and think about, but you will not be tested in EE342 about any of the following.
Once you get familiar with set comprehension, consider the following (Russell’s paradox): \(A\) is the set of all sets that are not members of themselves, specifically
\[A = \{ Z: Z \notin Z \}.\]The problem now is: is \(A\) a member of \(A\)? Well, if so, \(A\notin A\) (by the definition of the set). But if not, \(A\notin A\), and the definition above implies \(A\in A\). It is akin to saying “This sentence is a lie” (think about it).
This is a paradox where we cannot have either the statement or its negation to be true. This cuts quite deep not just into math but also philosophy. A really nice (but some may say unfairly exaggerated in some aspects) account on the beginnings of set theory in a period of transformation in math is in the wonderful graphic novel, Logicomix.
A small note about Russell’s paradox so I don’t leave you hanging. The common version of axioms in set theory in vogue is the Zermelo-Fraenkel version with the axiom of choice (abbreviated as the ZFC), which explicitly restricts the generality of set comprehension statements. We will see the Axiom of Choice in one of the optional experiences in the next module