2010 Reprint of 1915 Edition. Georg Ferdinand Ludwig Philipp Cantor was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. In fact, Cantor's theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware. In 1895-97 Cantor fully propounded his view of continuity and the infinite, including infinite ordinals and cardinals, in his best known work, Contributions to the Founding of the Theory of Transfinite Numbers . This work contains his conception of transfinite numbers, to which he was led by his demonstration that an infinite set may be placed in a one-to-one correspondence with one of its subsets.

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"In it, Jourdain outlines the contributions of many of Cantor's forerunners--including Fourier, Dirichlet, Cauchy, Weierstrass, Riemann, Dedekind, and Hankel--and then further contextualizes Cantor's groundbreaking theory by recounting and examining his earlier work. In this volume, Cantor addresses: -the addition and multiplication of powers, -the exponentiation of powers, -the finite cardinal numbers, -the smallest transfinite cardinal number aleph-zero, -addition and multiplication of ordinal types -well-ordered aggregates the ordinal numbers of well-ordered aggregates, -and much more. German mathematician GEORG CANTOR (1845-1918) is best remembered for formulating set theory. His work was considered controversial at the time, but today he is widely recognized for his important contributions to the field of mathematics."

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