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:

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 . We define it to be

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 , but also . So lets prove that the only even number is 11:

PROOF: Since , then it follows that or that the only even number is 11. Well, , so by disjunctive syllogism… the only even number is 11!

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

This week, I show some simple yet amazing examples of functors between categories.

In this short lecture, I show some simple examples of how monoids relate to category theory.

Here is a compilation of pdfs that I have written, more precisely, I have taken multiple books in each field and have combined them each into a comprehensive study guide. Each pdf consists of the bulk/basics of each field, with proofs and examples being omitted. They are mostly complete, but will always be a work-in-progress. Whenever I commit to knowing/remembering something forever, I organize the material in such a way so that it is easy to engrain the information in my head… these books accomplish that.

Algebraic Number Theory

Analytic Number Theory

Category Theory

Modern Algebra

Complex Analysis

Real Analysis

I cannot proceed without giving credit to those authors, whom I have learned most from, and whose material I have chosen to use in my references. They are: Tenenbaum, Montgomery, Vaughan, Apostle, Ireland, Rosen, Titchmarsh, Jarvis, Marcus, Alaca, Lang, Ash, Samuel, Neukirch, Mac Lane, Leinster, Awodey, Hungerford, Dummit, Foote, Palka, Stein, Bass, and Royden.