This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meeting of the Association for Symbolic Logic at Haifa, in the Massachusetts Institute of Technology logic seminar, and to the Paris Logic Group. The author would like to express his thanks to the various organizers, as well as his gratitude to the Hebrew University of Jerusalem for its hospitality during the preparation of this article in the autumn of 1995.

this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs. membership distinction, a distinction only clarified at the turn of this century, remarkable though this may seem. Russell runs with this distinction, but is quickly caught on the horns of his well-known paradox, an early expression of our motif. The motif becomes fully manifest through the study of functions f :

set theory' [12] as his source, and another refers to Barrow `Theories of everything' [2]. One contents himself with references to two earlier unpublished papers of his own. Others give no source. For definiteness let me write down a proof, not in Cantor's words, which contains all the points we shall need to comment on. (1) We claim first that for every map f from the set {1, 2, . . . } of positive integers to the open unit interval (0, 1) of the real numbers, there is some real number which is in (0, 1) but not in the image of f. (2) Assume that f is a map from the set of positive integers to (0, 1). (3) Write 0 . a n1 a n2 a n3 . . . for the decimal expansion of f(n), where each a ni is a numeral between 0 and 9. (Where it applies, we choose the expansion which is eventually 0, not that which is eventually 9.) (4) For each positive integer n, let b n be 5 if a nn #= 5, and 4 otherwise. (5) Let b be the real number whose decimal expansion is 0 . b 1 b 2 b 3 . . . . (6...

Abstract. Cantor’s algebraic calculation of the power of the continuum contains an easily repairable error related to Cantor own way of defining the addition of cardinal numbers. The appropriate correction is suggested. 1. The exponentiation of powers Cantor’s most significant contribution to the theory of transfinite numbers is, without a doubt, Beiträge zur Begründung der transfiniten Mengelehre 1. A memory of more than 70 pages divided into two parts which appeared in the Mathematische Annalen in the years 1895 and 1897 respectively ([4], [5]). Beiträge’s first six epigraphs are devoted to found the arithmetics of cardinals. Cantor begins by defining the concept of set and the union of disjoint sets, after which he proposes the following definition of power or cardinal number ([6], p. 86): We call by the name ”power ” or ”cardinal number ” of [the set] M the general concept which, by means of our active faculty of thought, arises from the set M when we make abstraction of the nature of its various elements m and of the order in which they are give.