A standard interpretation

Abstract

The popular interpretation of the origins of the set-theoretic ‘paradoxes’ is now almost a classic story, and it is to this account that I now turn along with some criticisms. The term popular is not used in a derogatory way: it means that this interpretation is the best known and accepted by the general public. This standard interpretation is simply a summary of the account given by mathematicians and professional historians of mathematics, among others, in general encyclopedias, philosophical and mathematical encyclopedias and dictionaries, textbooks of mathematics and textbooks on the history and philosophy of mathematics. The following quotations are representative of the prevailing chronicle:

• There remained logical difficulties in the theory of transfinite numbers and paradoxes appeared, such as those of Burali-Forti and Russell. This again led to different schools of thought on the foundations of mathematics.

• The crisis was brought about by the discovery of paradoxes or antinomies in the fringe of Cantor’s general theory of sets. Since so much of mathematics is permeated with set concepts and, for that matter, can actually be made to rest upon set theory as a foundation, the discovery of paradoxes in set theory naturally cast [sic] into doubt the validity of the whole foundational structure of mathematics.

My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer. How do I know this? Because I have studied it from all sides for many years; because I have examined all objections which have ever been made against the infinite numbers; and above all, because I have followed its roots, so to speak, to the first infallible cause of all created things.

Georg Cantor.