 Cantor was led to the paradoxes by having introduced the “conceptual sphere” of the transfinite numbers. Each transfinite ordinal is the order type of the set of its predecessors; e.g., ω is the order type of \(\{0, 1, 2, 3, \ldots\}\), and \(\omega+2\) is the order type of \(\{0, 1, 2, 3, \ldots, \omega, \omega +1\}\). Thus, to each initial segment of the series of ordinals, there corresponds an immediately greater ordinal. Now, the “whole series” of all transfinite ordinals would form a well-ordered set, and to it there would correspond a new ordinal number. This is unacceptable, for this ordinal \(o\) would have to be greater than all members of the “whole series”, and in particular \(o &lt; o\). This is usually called the Burali-Forti paradox, or paradox of the ordinals (although Burali-Forti himself failed to formulate it clearly, see Moore &amp; Garciadiego 1981).