Traditional logic and the early history of sets, 1854?1908

Abstract

6.1. Pre-Russellian logicism.6.2. Effect of the antinomies. I. INTRODUCTIONA long-standing tradition considers Georg CANTOR as the 'creator' or founder of set theory.1 This view, correct when restricted to transfinite set theory, stands in need of revision within the context of a general account of the emergence of sets in mathematics. Such an account should face the broad questions: how did sets become important in the different branches of mathematics?how did they emerge as basic mathematical objects, as a fundamental language for mathematics? 2 Simultaneously, one should try to establish how and why some mathematicians began to think of sets as the foundation for mathematics. In fact, within the context of the early history of sets during the second half of the nineteenth century, it is convenient to differentiate three aspects of our picture of set theory: the language of sets, the theory of sets, and the idea that sets form the foundation of mathematics. These aspects are so intertwined today that it may seem artificial to distinguish them, but in my opinion the distinction is useful, and even necessary, if we want to clarify the history of sets. During the second half of the nineteenth century, the interaction between the three aspects was quite complex.Along these lines, a discussion of the work of Bernhard RIEMANN (1826-66) and Richard DEDEKIND (1831DEDEKIND ( -1916 forms the core of this paper. In my view, both mathematicians offer the most clear examples of the use of set language in mathematics, and the elaboration of foundational views based on the notion of set, prior to the emergence of CANTOR's work. This is why they should figure prominently in a general account of the emergence of sets. The triad RIEMANN/DEDEKIND/CANTOR constitutes a quite closed net offering basic clues for a satisfactory explanation of the ascent of sets. (In a detailed description, subsidiary authors, like GAUSS, DIRICHLET, WEIERSTRASS, and HANKEL, would also be of great importance.) RIEMANN, DEDEKIND, and CANTOR give the main examples of set-theoretical and set-1 For early and authoritative instances, cf. the dedication of HAUSDORFF 1914, ZERMELO's words in the preface to CANTOR 1932, and several passages of HILBERT 1930. But see ZERMELO's crucial paper on the axiomatization of set theory, where he said that it had been "created by CANTOR and DEDEKIND" (ZERMELO 1908a, 200).2 Around 1870, the language of sets was being applied in geometry, elementary arithmetic, algebra, and even complex function theory. During the last two decades, much has been done to consider these developments, at least for the cases of algebra and real analysis. For the emergence of point-sets within real analysis cf. HAWKINS 1975, DAUBEN 1979, COOKE 1992. DEDEKIND's structural view of algebra is studied in DUGAC 1976and 1981, EDWARDS 1980, SCHARLAU 1981, EDWARDS, NEUMANN & PURKERT 1982. A wealth of material on both topics can be found in MOORE 1982, and an attempt to synthesize the foregoing studies in FERREIRÓS 1993a.Viewed in the light of such prev...

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