New Infinitary Mathematics

A rethinking of Cantor and infinitary mathematics by the creator of Vop?nka's principle.
 
The dominant current of twentieth-century mathematics relies on Georg Cantor’s classical theory of infinite sets, which in turn relies on the assumption of the existence of the set of all natural numbers, the only justification for which—a theological justification—is usually concealed and pushed into the background.

This book surveys the theological background, emergence, and development of classical set theory, warning us about the dangers implicit in the construction of set theory, and presents an argument about the absurdity of the assumption of the existence of the set of all natural numbers. It instead proposes and develops a new infinitary mathematics driven by a cautious effort to transcend the horizon bounding the ancient geometric world and mathematics prior to set theory, while allowing mathematics to correspond more closely to the real world surrounding us. Finally, it discusses real numbers and demonstrates how, within a new infinitary mathematics, calculus can be rehabilitated in its original form employing infinitesimals.

About the Authors

Petr Vopenka (1935-2015) was a Czech mathematician and philosopher. In addition to teaching math and logic at Charles University, Jan Evangelista Purkyne University, and the University of West Bohemia, he also served as the Czech minister of education in the early 1990s. In mathematics, he is perhaps best known for establishing Vopenka's principle. Alena Vencovska is a Czech mathematician. Hana Moravcova is a Czech translator. Roland Andrew Letham translates from Czech. Vaclav Paris is a Czech translator.