ABSTRACT
Aristotle was not willing to accept as legitimate the culmination of this process: the infinite set of all natural numbers. This would be a "completed" or "actual" infinity, and Aristotle declared that such were illegitimate. Aristotle's views heavily influenced the scholastic religious philosophers of the twelfth century, particularly Thomas Aquinas. The limit processes of the calculus that became so important for mathematics in the eighteenth and nineteenth centuries exemplified potential infinity. After the middle of the nineteenth century mathematical problems that arose quite naturally out of current concerns seemed to call for the use of completed infinities in their precise formulation. Among the mathematicians who were coping with this situation, it was only Georg Cantor who, flying in the face of Gauss's warning, accepted the challenge to create a profound and coherent mathematical theory of the actual infinite. Frege was supportive of Cantor's embrace of the actual infinite, recognizing its importance for the future of mathematics.
