THE THEORY OF SETS AND INFINITE SETS - THE FINAL PAPER

CANTOR, GEORG.

Über unendliche, lineare Punktmannichfaltigkeiten.

Leipzig, B.G. Teubner, 1884. 8vo. Bound in a nice contemporary half calf with five raised gilt bands. Red leather title label with gilt lettering to spine. All edged gilt. In "Mathematische Annalen", Band 23, 1884. Entire volume offered. Corners with wear, otherwise a very fine and clean copy. Pp. 453-488. [Entire volume: IV, 598, (2) pp.].

First printing of Cantor's seminal sixth paper in the landmark series consisting of a total of six papers which together constitute the foundation Theory of Sets (Mengenlehre) and Transfinite Set Theory. Cantor here introduces his new Set Theory with which he created an entirely new field of mathematical research and is widely regarded as being one of the most important mathematical conquests in the 19th century.

"Cantor published a sequel in the following year as a sixth in the series of papers on the Punktmannigfaltigkeitslehre (The present paper). Though it did not bear the title of its predecessor, its sections were continuously numbered, 15 through 19; it was clearly meant to be taken as a continuation of the earlier 14 sections of the "Grundlagen" itself. In searching for a still more comprehensive analysis of continuity, and in the hope of establishing his continuum hypothesis, he focused chiefly upon the properties of perfect sets and introduced as well an accompanying theory of content" (Dauben, P. 111)

Hilbert spread Cantor's ideas in Germany and praised Cantor's transfinite arithmetic as "the most astonishing product of mathematical thought, one of the most beautiful realizations of human activity in the domain of the purely intelligible". He is famously quoted for saying "No one shall expel us from the paradise which Cantor created for us". Bertrand Russel described Cantor's work as "probably the greatest of which the age can boast".

"The major achievement of the "Grundlagen" was its presentation of the transfinite ordinal numbers as a direct extension of the real numbers. Cantor admitted that his new ideas might seem strange, even controversial, but he had reached a point in his study of the continuum where the new numbers were indispensable for further progress. 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." (Grattan-Guinness, Landmark Writings in Western Mathematics, Pp. 604-5).

Dauben: (Cantor)1884a.

Order-nr.: 47159