Transfinite numbers.

It can be said without fear of serious contradiction that among the notions of mathematics, none imposed so difficult a task in formulating a definition as that of infinite applied in the theory of sets. Indeed, the whole subject of theory of functions is so preeminently based upon a proper understanding of the infinite that it is not surprising that early writers in analysis, having no clear concept of the infinite, were led to erroneous results. It was Dedekind, who perceiving mathematics as a free creation of the human mind and consequently independent of spatio-temporal relations, set about rendering analysis rigorous. He was immediately followed by George Cantor whose notion of power of sets led to extremely significant results. Indeed, this notion may be placed alongside differentiation in originality and breadth of scope. He created as one of the nineteenth-century mathematicians said “a heaven from which there was no escape”. Dedekind and Cantor were not concerned with the nebulous philosophy which the notion of infinite had previously inspired, and were what we now term Formalists. It is this point of view which will be adopted in the development of the theory of transfinite numbers. Mindful, however, of the fact that no dogmatic attitude exists but that there exists at least a compelling if not convincing counter-dogma, there will be found a short essay on Intuitionism, the counter point of view further on in this work. [...