Linear Orderings: The Architecture Underneath the Numbers [Linear Orderings Part 2]
You've spent your whole life using the number line, and you've probably never thought to ask: what exactly makes it a line? It's not the individual numbers — it's the relationship between them. The fact that $3 < 5 < 7$, that between any two rationals there is another, that the integers have gaps. These are not properties of the numbers themselves but of the ordering structure they come equipped with. The study of these ordering structures — independently of what the elements actually are — is order theory. And it is one of the most beautiful corners of mathematics I have come across. A linear ordering on a set $L$ is a relation $<$ satisfying: Transitivity: if $x < y$ and $y < z$, then $x < z$. Totality: for any distinct $x, y \in L$, either $x < y$ or $y < x$. Irreflexivity: $x \not< x$ for all $x$. Think of it as a ranking of elements on a line — everyone has a place, no two people share a place (unless they're the same person), a...
