I'm reading a book on the Philosophy of Mathematics, and the author gave a "5-line proof" of Fermat's Last Theorem as a way to introduce the topic of inconsistency in set theory and logic. The author acknowledges that this is not a real proof of the theorem, but the way it was presented implies that it was supposed to look somewhat convincing. I, however, have absolutely no idea how the proof given even remotely relates to FLT, and would greatly appreciate it if someone could make the connection for me. Below is an almost verbatim excerpt from the book.
Theorem: There are no positive integers $x$, $y$, and $z$, and integer $n>2$, such that $x^n + y^n = z^n$.
Proof. Let $R$ stand for the Russell set, the set of all things that are not members of themselves: $R= \{x : x \notin x\}$. It is straightforward to show that this set is both a member of itself and not a member of itself: $R \in R$ and $R \notin R$. So since $R \in R$, it follows that $R \in R$ or FLT. But since $R \notin R$, by disjunctive syllogism, FLT. End.
Let $S$ be a statement which is both true and false. Since $S$ is true, $S \lor T$ is true for any statement $T.$ Since $S$ is false and $S \lor T$ is true, $T$ must be true. Thus $T$ is a true statement, i.e. every statement is true.
This is the idea that the "proof" is trying to get across. The statment that $R \in R$ is both true and false, and so using the above arguement, you can prove that anything is true, in particular FLT. Similarly, you can prove that any statement is false.