Wellorderings
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 \(0=\emptyset\), \(1=\{0\}\), \(2=\{0,1\}\), \(3=\{0,1,2\}\), 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...