THE THEORY OF SETS AND INFINITE SETS - THE SECOND PAPER

CANTOR, GEORG.

Über unendliche, lineare Punktmannichfaltigkeiten.

Leipzig, B.G. Teubner, 1880. 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 17, 1880. Entire volume offered. Corners with wear, otherwise a very fine and clean copy. Pp. 355-358. [Entire volume: IV, 576 pp.].

First printing of Cantor's important second paper of 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's second paper of 1880 was brief. It continued the bricklaying work of the article of 1879, and it too sought to reformulate old ideas in the context of linear point sets. It also introduced for the first time an embryonic form of Cantor's boldest and most original discovery: the transfinite numbers. As a preliminary to their description, however, Cantor introduced several definitions. He also pointed out that first species sets could be completely characterized by their derived sets." (Dauben, P. 80)


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)1880d.

Order-nr.: 47160