Abstract
This chapter presents Cantor's views on the foundations of mathematics. If one tries to reconstruct the genesis of Cantor's ideas, one must seek a motivation for why he began to investigate the powers of sets in terms of one-to-one correspondences. Cantor mentioned that the relation α <β establishes a well-ordering in any set of ordinals. A key problem for Cantor was to establish a connection between transfinite ordinals and cardinals. He did so by applying his restraint principle—Hemmungsprinzip. This principle consists in the demand to create a new number by means of the two generating principles only if the totality of all the preceding numbers has the power of a well-determined number class that already completely exists. Cantor's Platonistic ontological foundation of mathematics had important consequences for his views about the infinite. It is the reason why he was not disquieted by the antinomies, which he did not regard as antinomies at all.
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