Numbers: The Ordinal Approach
At the end of the previous chapter we saw that it was a demerit of the cardinal approach that it treated each number individually. Each number was a possible answer to the question ‘How many?’ and was intimately connected to the things of type a that were F, which it was the number of. We had the picture of fishing in a sea of sets or extensions and netting a shoal of equinumerous ones. The ordinal approach, which was pioneered by Dedekind, taken up by Cantor, and was adopted to some extent by Peano, is not concerned with separate individual numbers, but with whole sequences of numbers. These are specified by properties of order, either by the order in which we count them or as explicit ordinals, ‘first’, ‘second’, ‘third’, etc. The protagonists of the ordinal approach are primarily counting men, but it is instructive to recall the etymology of the explicit ordinals, where we noted that in English, German, Latin and Greek, the word for ‘first’ is superlative in form. So, likewise, are the words ‘next’ and nächst and ‘last’. In Latin and Greek alter and words for second, are comparative in form, as are also the English words, ‘former’ and ‘latter’. From a logical point of view, too, orderings are specified by transitive irreflexive relations, of which comparatives are the standard exemplars.1 The ordinals are a special sort of linear ordering, with superlative properties, that is to say that every subset of them has a least member, which implies that every member has a next member.