The first math skill you learned as a child was probably counting. The natural numbers: 1, 2, 3, and so on. The second was arithmetic. 2+2, 8 times 5, and so on. These both probably come naturally to you now, but it’s unlikely that your kindergarten teacher ever explained set theory and the successor function while teaching you how to count. That’s what I hope to do here: show you the rigorous foundations of the mathematics you use every day without even thinking.

In ye olden times, the proof of 2+2=4 probably went something like this:

“I have two apples…

Divisibility is a ubiquitous concept. Everyone who’s attended elementary school knows what it means for one number to be divisible by another. This relatively simple concept forms the basis of many fascinating theorems, proofs, and objects in mathematics. From the prime numbers to identifying every finite abelian group up to isomorphism. However, the definition of divisibility you learned in school can be extended to create weirder and less well-understood ideas of what it means for a number to be divisible by another.

The generalization of divisibility we’ll be discussing in this article is called k-ary divisibility. k-ary divisibility extends the…

Cardinals are the numbers that let us describe how many elements are in a set. For normal people, the only cardinals that matter are the natural numbers (i.e. 0, 1, 2, etc.) and perhaps the first infinity, but for mathematicians, there are many, many more cardinals to play with. While there are literally countless cardinal numbers, it helps to begin with some small and familiar ones before we work our way up to larger numbers.

The smallest of all cardinal numbers are the natural numbers. These are used constantly in everyday life; you have two legs, the world has 7.8…