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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.

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    $\begingroup$ Notably, the paradox of this set is called Russell's paradox, which can be avoided by making any definition of a set need to be based off of some pre-existing set somehow, so you can't just refer to "the set of all sets such that..." $\endgroup$ Jun 26, 2016 at 22:39
  • $\begingroup$ To be honest, I too didn't understand how it relates to FLT. Can someone explain what the link is ? $\endgroup$
    – Saikat
    Jun 27, 2016 at 6:58
  • $\begingroup$ @user230452 Basically it's saying: If we accept this axiom, then we have a statement that is both true and false; this invalidates the whole theory. Why? Because False statements imply everything, in particular they imply FLT (but also riemann hypothesis, 4*8 = 1, etc.) $\endgroup$
    – Ant
    Jun 27, 2016 at 8:15
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    $\begingroup$ @Ant I'm sorry, how do false statements imply everything ? I understood how the paradox is that $R$ is both a part of itself, and not, but not how that is linked to FLT. $\endgroup$
    – Saikat
    Jun 27, 2016 at 10:15

2 Answers 2

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The point is you could equally replace "FLT is true" with "FLT is false" or "My head is made of cheese". By introducing $R$, the author makes the system inconsistent and once a system is inconsistent, every statement in it is both true and false. This is called the principle of explosion.

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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.

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