ABSTRACT
The finitely definable reals (decimal fractions) are enumerable, that is, they can be ordered so that there is a first one, a second, a third, . . . , and in general after each number there is a next. Their definitions can, for example, be grouped according to the number of words used, in ascending order, and ordered alphabetically within each group. Then, in terms of the enumeration of the numbers defined, it is possible to define another real number which is not already included, by means of a procedure due to Cantor which is known as ‘diagonalization’. The new number is defined by saying that its nth digit differs from the nth digit of the nth number by replacing it with the next digit up, or by ‘0’ if it is ‘9’. So if, for example, the 23rd digit of the 23rd number in the enumeration is ‘7’ then it is replaced by ‘8’. Since this number differs from each of the numbers in the enumeration at one decimal place, it must be distinct from each of them.
