Contributions to the Founding of the Theory of Transfinite Numbers - Tapa blanda

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1915 edition. Excerpt: ... of two aggregates which are similar to one another an imaging of the one on the other." The addition and multiplication of ordinal types, and the fundamental laws about them, were then dealt with much as in the memoir of 1895 which is translated below. The rest of the paper was devoted to a consideration of problems about -ple finite types. In 1888, Cantor, who had arrived at a very clear notion that the essential part of the concept of number lay in the unitary concept that we form, gave some interesting criticisms on the essays of Helmholtz and Kronecker, which appeared in 1887, on the concept of number. Both the authors referred to started with the last and most unessential feature in our treatment of ordinal numbers: the words or other signs that we use to'represent these numbers. In 1887, Cantor gave a more detailed proof of the non-existence of actually infinitely small magnitudes. This proof was referred to in advance in the Grundlagen, and was later put into a more rigorous form by Peano. We have already referred to the researches of Cantor on point-aggregates published in 1883 and later; the only other paper besides those already dealt with that was published by Cantor on an important question in trie theory of transfinite numbers was one ipublished in 1892. In this paper we can see the origins of the conception of ' covering" (Belegung) denned in the memoir of 1895 translated below. In the terminology introduced in this memoir, we can say that the paper of 1892 contains a proof that 2, when exponentiated by a transfinite cardinal number, gives rise to a cardinal number which is greater than the cardinal number first mentioned. The introduction of the concept of ' covering" is the most striking advance in...