![]() ![]() If d itself were a value of F, say d = F ( b ) for some b ∈ x, then we would have the paradigmatic contradiction: b ∈ d exactly when b ∉ d. The concepts here are fundamental: Taking infinite collections as unitary totalities, a set is countable if it is in one-to-one correspondence with the set of natural numbers consisting of those members a of x that do not belong to their corresponding subset F ( a ). Set theory was born on that day in December 1873 when Cantor established that the continuum is not countable. The last section describes the modern investigation of relative consistency in terms of forcing, large cardinals, and inner models. Next will come a description of the work of Kurt G ödel on the constructible sets, work that made first-order logic central to set theory, followed by a description of the work of Paul Cohen on forcing, a method that transformed set theory into a modern, sophisticated field of mathematics. The next two parts describe the subsequent transmutation of the notion of set through axiomatization, a process to be associated largely with Ernst Zermelo. The first part describes the groundbreaking work of Georg Cantor on infinite sets analyzed in terms of power, transfinite numbers, and well-orderings. In what follows, set theory is presented as both a historical as well as an epistemological phenomenon, driven forward by mathematical problems, arguments, and procedures. That set theory is both a field of mathematics and serves as a foundation for mathematics emerged early in this development. The subject was then developed as the logical distinction was being clarified between "falling under a concept," to be transmuted in set theory to " x ∈ y ", x is a member of y, and subordination or inclusion, to be transmuted in set theory to " x ⊆ y ", x is a subset of y. Set theory arose in mathematics in the late nineteenth century as a theory of infinite collections and soon became intertwined with the development of analytic philosophy and mathematical logic. With this reduction in play, modern set theory has become an autonomous and sophisticated research field of mathematics, enormously successful at the continuing development of its historical heritage as well as at analyzing strong propositions and gauging their consistency strength. ![]() Part of its larger significance is that mathematics can be reduced to set theory, with sets doing the work of mathematical objects and their collections and set-theoretic axioms providing the basis for mathematical proofs. Set theory is a mathematical theory of collections, "sets," and collecting, as governed by axioms. ![]()
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