 This is the first time we talk about the transfinite and ordinals in proof theory. Ordinals have become very important in advanced proof theory. The concept of an ordinal is a generalization of that of a natural number. The latter are used for counting finitely many things whereas ordinals can also “count” infinitely many things. It is derived from the concept of an ordering \(\prec\) of a set X which arranges the objects of X in order, one after another, in such a way that for every predicate P on X there is always a first element of X with respect to \(\prec\) that satisfies P if there is at least one object in X satisfying P. Such an ordering is called a well-ordering of X. Certainly the usual less-than relation on \(\bbN\) is a well-ordering. Here every number \(\ne 0\) is the successor of another number. If one orders the natural numbers \(&gt;0\) in the usual way but declares that 0 is bigger than every number \(\ne 0\) one arrives at another ordering of \(\bbN\). Let’s call it \(\prec\). \(\prec\) is also a well-ordering of \(\bbN\). This time 1 is the least number with respect to \(\prec\). However, 0 plays a unique role. There are infinitely many numbers \(\prec 0\) and there is no number \(m\prec 0\) such that 0 is the next number after m. Such numbers are are called limit numbers (with respect to \(\prec\)).