Consider the serial arrangement of the rationals in their order of magnitude.
A real number is a class (a, say) of rational numbers which satisfies the condition that it is the same as the class of those rationals each of which precedes at least one member of a.
Thus, consider the class of rationals less than 2r; any member of this class precedes some other members of the class - thus 1/2 precedes 4/3, 3/2 and so on; also the class of predecessors of predecessors of 2 2.
Note that the class of rationals less than or equal tò 2r is not a real number.
For example, the class of rationals whose squares are less than 2r satisfies the definition of a real number; it is the real number A I 2.
Now, owing to the necessary inexactness of measurement, it is impossible to discriminate directly whether any kind of continuous physical quantity possesses the compactness of the series of rationals or the continuity of the series of real numbers.