ABSTRACT
It has always been held to be an open question whether the continuum is composed of elements; and even when it has been allowed to contain elements, it has been often alleged to be not composed of these. This latter view was maintained even by so stout a supporter of elements in everything as Leibniz.† But all these views are only possible in regard to such continua as those of space and time. The arithmetical continuum is an object selected by definition, consisting of elements in virtue of the definition, and known to be embodied in at least one instance, namely the segments of the rational numbers. I shall maintain in Part VI that spaces afford other instances of the arithmetical continuum. The chief reason for the elaborate and paradoxical theories of space and time and their continuity, which have been constructed by philosophers, has been the supposed contradictions in a continuum composed of elements. The thesis of the present chapter is, that Cantor’s continuum is free from contradictions. This thesis, as is evident, must be firmly established, before we can allow the possibility that spatio-temporal continuity may be of Cantor’s kind. In this argument, I shall assume, as proved the thesis of the preceding chapter, that the continuity to be discussed does not involve the admission of actual infinitesimals.
