In one of the fundamental papers in the genesis of set theory, Cantor (1883a) isolated the notion of a well-ordering on a collection as one of the central conceptual pillars on which number is built. Cantor took the view that the notion of a counting number must be based on an underlying ordering of the set of things being counted, an ordering in which there is a first element counted, and, following any collection of elements counted, there must be a next element counted, assuming that there are elements still uncounted. This kind of ordering he called a ‘well-ordering’, which we now define as a total-ordering with an extra condition, namely that any subset has a least element in the ordering. Cantor recognised that each distinct well-ordering of the elements gives rise to a distinct counting number, what he originally called an ‘Anzahl [enumeral]’, later an ‘Ordnungszahl [ordinal number]’, numbers which are conceptually quite different from cardinal numbers or powers, meant to express just the size of collections.[13] This distinction is hard to perceive at first sight. Before Cantor and the rise of the modern theory of transfinite numbers, the standard counting numbers were the ordinary finite numbers.[14] And, crucially, for finite collections, it turns out that any two orderings of the same underlying elements, which are certainly well-orderings in Cantor's sense, are order-isomorphic, i.e., not essentially  distinct.[15]  This means that one can in effect identify a number arrived at by counting (an ordinal number) with the cardinal number of the collection counted. Thus, the ordinary natural numbers appear in two guises, and it is possible to determine the size of a finite collection directly by counting it. Cantor observed that this ceases to be the case in rather dramatic fashion once one considers infinite collections; here, the same elements can give rise to a large variety of distinct well-orderings.