Set Theory and Beyond

I've already told you what is an ordering, but let me give a quick reminder. A ( strict) ordering  on some set $X$ is a a binary relation, usually denoted by some variation on the symbol $<$, which is  irreflexive,  that is for all $x\in X$, it is not  the case that $x<x$, transitive, that is for all $x,y,z\in X$, if $x<y$ and $y<z$, then $x<z$. One very important example is the standard ordering on natural numbers. In that case, we have that $X=\omega$ and $<=\in$. To understand the second thing, recall that in Set Theory, natural numbers are coded as $0=\mathrm{\varnothing }$, $1=\{0\}$, $2=\{0,1\}$, $3=\{0,1,2\}$, and so on. Hence, for example, $1<3$ because $$1\in \{0,1,2\}=3.$$ In addition to the irreflexivity and transitivity, this relation satisfies some other important properties. One of them is being total, that is for all $x,y\in X$, one of the options is true: $x<y$ or $x=y$ or $x>y$. The orderings which...