the one hand as an intrepid supporter of actual infinity, he nonetheless considered Bolzano’s concept to be insufficient and erroneous.

> The genuine-infinite as we encounter it, for example in the case of well-defined point sets, …has found its most decisive defender in a philosopher and mathematician of our century with a most acute mind, Bernhard Bolzano, who has developed his views relevant to the subject especially in the beautiful and substantial essay, *Paradoxes of the Infinite.* …Bolzano is perhaps the only one who assigns the genuine-infinite numbers their rightful place. …However, the actual way in which he deals with them, without being able to advance any kind of real definition of them, is something about which I am not at all in agreement with him, and I regard for example Sections 29-33 of that book as *unfounded and erroneous*. The author lacks both the general concept of cardinality and the precise concept of number-of-elements for a real conceptual grasp of determinate infinite numbers. (Cantor 1976, p. 78).

This view of Cantor’s persists until today. For example, in his book *Labyrinth of Thought* José Ferreiros says of Bolzano

> He proposed to base mathematics on notions similar to the set. 