Wrong Math: Prove anything is true, using Naive set theory

Let’s find something to prove, something bizarre, something wrong, something we know is not true. How about, let’s prove that the only even number… is 11!

Before we start our proof there’s a couple things we need to define. The first, is the notion of disjunctive syllogism. In propositional logic, the rule of disjunctive syllogism says that for propositions P and Q: $((P\vee Q)\wedge \sim P)\longrightarrow Q$

To show a simple example of this, consider the following:

The car is green or the car is red.

The car is not green.

Therefore, the car is red.

Now, the second thing we need for our proof is what we call a Russell set $\mathcal{R}$. We define it to be $\mathcal{R}=\{x:x\notin x\}$

Within Naive set theory, this paradoxical set named after Bertrand Russell, brought the foundations of mathematics into question. It is easily seen that not only is $\mathcal{R}\in\mathcal{R}$, but also $\mathcal{R}\notin\mathcal{R}$. So lets prove that the only even number is 11:

PROOF: Since $\mathcal{R}\in\mathcal{R}$, then it follows that $\mathcal{R}\in\mathcal{R}$ or that the only even number is 11. Well, $\mathcal{R}\notin\mathcal{R}$, so by disjunctive syllogism… the only even number is 11! $\Box$

Hopefully this has been entertaining. Now to prove the Riemann Hypothesis…