On the Equivalence of Three Fundamental Definitions of Irrational Number

The arithmetical theory of irrational numbers has been developed in three main forms, of which the first was given by Weierstrass in his Lectures on Analytical Functions, the second is that of Cantor and the third that due to Dedekind. Of these three theories, it has been shown that those of Cantor and Dedekind are fundamentally identical and it has been established that whereas the theory of Dedekind operates with the whole aggregate of rational numbers, the other operates with sequences selected out of that aggregate. We here propose to establish the fundamental identity of all the three theories of irrational number by establishing the same for the theories of Weierstrass and Cantor and to show that the entities used in the former definition are but particular types of the entities used in the latter.