Chapter 46 - Georg Cantor, paper on the ‘Foundations of A General Set Theory’ (1883)

Abstract

In the revolutionary monograph titled Foundations of a General Set Theory, Georg Cantor set out the earliest detailed version of his transfinite set theory, including a theory of transfinite ordinal numbers and their arithmetic; and a defense of the theory on historical and philosophical grounds. Cantor sought to do away with these restrictions, and to establish the uniqueness theorem on the most general terms possible. Additionally, among the most immediate results of Cantor's new theory of transfinite ordinal numbers were applications in real and complex analysis. Cantor mentioned only the algebraic numbers, not the set of all real numbers. Cantor had finally come to the realization that his “infinite symbols” were not just indices for derived sets of the second species, but could be regarded as actual transfinite numbers that were just as real mathematically as the finite natural numbers Cantor's theory of transfinite numbers has been further extended into the realm of inaccessible cardinals and a host of other highly refined theories of transfinite numbers.