It is easy to prove that the series of segments of any series is Dedekindian. For, given any set of segments, their boundary will be their logical sum, i.e. the class of all those terms that belong to at least one segment of the set.[18]
[18]For a fuller treatment of the subject of segments and Dedekindian relations, see Principia Mathematica, vol. II. * 210-214. For a fuller treatment of real numbers, see ibid., vol. III. * 310 ff., and Principles of Mathematics, chaps. XXXIII. and XXXIV.
The above definition of real numbers is an example of "construction" as against "postulation," of which we had another example in the definition of cardinal numbers. The great advantage of this method is that it requires no new assumptions, but enables us to proceed deductively from the original apparatus of logic.
There is no difficulty in defining addition and multiplication for real numbers as above defined. Given two real numbers
and
, each being a class of ratios, take any member of
and any member of