Ordinal numbers

Abstract

We learn numbers and counting as a process of succession. ‘Eleven’ has little real meaning to us except as ‘the number after ten’. In this chapter, we use this process of succession to define the natural numbers - to do God’s work, in Kronecker’s phrase - starting from nothing (more precisely, the empty set) and progressing from one number to the next. As succession is the defining characteristic of natural numbers, so induction is the key proof technique. We can use it to define the arithmetic operations and to prove their basic properties.

We have learned to pass with such facility from cardinal to ordinal number that the two aspects appear to us as one. To determine the plurality of a collection, that is, its cardinal number, we do not bother anymore to find a model collection with which we can match it - we count it... The operations of arithmetic are based on the tacit assumption that we can always pass from any number to its successor, and this is the essence of the ordinal concept.

And so matching by itself is incapable of creating an art of reckoning. Without our ability to arrange things in ordered succession little progress could have been made. Correspondence and succession... are woven into the very fabric of our number system.

Tobias Dantzig, Number: The Language of Science [12]