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.
Nonfinite sets and numbers are now discussed in a separate secion in Infinity.