In the section of Numerical Order we note that: For the properties of the order relations to be true we require that a set cannot be placed in one-to-one correspondence with a proper subset of itself (since this would mean we could have s < s). Sets with this very sensible property are called finite sets.
Sets without this property are therefore nonfinite (or infinite). If we allow the concept of a 'set of all numbers' without recognising an upper bound, u, then this 'set' would be nonfinite. This is shown by the one-to-one correspondence that matches each number with its immediate successor, that is, it matches the set {0, 1, 2, ... } with its proper subset {1, 2, 3, ...}.
The validity of the method of mathematical induction for the nonfinite 'set of all numbers' cannot be proved without assuming the well-ordering principle, that any set of numbers has a least member. This is obviously true in any finite set, but it cannot be proven in a nonfinite set, except in terms of other equivalent assumptions.
The belief that allowing the concept of nonfinite sets and numbers is at least unnecessary, or at most that it is totally meaningless, is finitism. The finitist argues that in all practical applications in which infinity is used, such as in the methods of differential and integral calculus, as customarily presented, it will be found that the concept is no longer present in the final results, as applied, and that in fact it can be eliminated from the argument. The approach taken here is a radically sceptical one. Infinity is defined and some consequences of its acceptance are indicated, but nonfinite systems are not developed in detail.