A number s is said to be less than (or smaller than) a number t if a set of size s can be matched to a proper subset of a set of size t, and we write this relation s < t. We also express the converse relation that t is greater than (or larger than) than s, written t > s. The relations < and > are strict order relations, i.e. they are irreflexive, asymmetric and transitive.
We also abbreviate ‘s is less than or equal to t’ as s < t, and ‘t is greater than or equal to s’ as t > s. The relations < and > are non-strict order relations, i.e. they are reflexive, antisymmetric and transitive.
We can continue to define symbols for numbers for as far as we can count, and theoretically this can 'go on for ever': but there are in fact practical bounds on this. The larger the numbers the more digits we have to handle accurately. It is therefore advisable to recognise an upper bound u beyond which we will not go, just as we have the lower bound 0. In doing arithmetic in computers for instance there are limitations on memory, and numbers can only be stored up to a certain number of digits. For any number n we have 0 < n and n < u. (This type of statement is often abbreviated to 0 < n < u.)
The smallest number greater than n, if it exists, is called its successor (or immediate successor), and the greatest number smaller than n, if it exists, is its predecessor (or immediate predecessor). Every number except u has a unique successor. Every number except 0 has a unique predecessor.
A general way to prove that some statement about numbers is true for all the numbers 0 to u is mathematical induction in which we first prove the statement for 0, then we prove that if it is true for any number n it must also be true for its successor. It follows that it is true for all cases.
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.