Contributions to the Founding of the Theory of Transfinite Numbers
"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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absolutely infinite aggre aggregate F Aleph-one analytic functions arithmetic belong Bolzano Cantor Cauchy commutative law condition consequently contained continuum corresponding elements Crelle's Journal defined denote derivatives determined e-number element m0 elements of F equal equation equivalent everywhere dense F and G finite cardinal number finite number following theorem formula fundamental series geometrical Georg Cantor given Hankel imaging infinite aggregate infinity integers interval inversely irrational numbers limiting element logical lowest element Math mathematics memoir numerical magnitude order of magnitude order of precedence ordinal type point-aggregate principle proof proved rank rational numbers real numbers relations of precedence Riemann second kind second number-class segment of F similar segment simply infinite series simply ordered aggregate supposition theory of aggregates theory of functions theory of transfinite totality transfinite aggregate transfinite cardinal number transfinite numbers trigonometrical series values Weierstrass well-ordered aggregate whole number zero
Page v - This volume contains a translation of the two very important memoirs of Georg Cantor on transfinite numbers which appeared in 1895 and 1897. These memoirs are the final and logically purified statement of many of the most important results of the long series of memoirs begun by Cantor in 1870. A very full historical account of this work and the work of others which led up to it is given in the introduction, and the notes at the end contain indications of the progress made in the theory of transfinite...