Zermelo’s Axiom of Choice: Its Origins, Development, and Influence
This book grew out of my interest in what is common to three disciplines: mathematics, philosophy, and history. The origins of Zermelo's Axiom of Choice, as well as the controversy that it engendered, certainly lie in that intersection. Since the time of Aristotle, mathematics has been concerned alternately with its assumptions and with the objects, such as number and space, about which those assumptions were made. In the historical context of Zermelo's Axiom, I have explored both the vagaries and the fertility of this alternating concern. Though Zermelo's research has provided the focus for this book, much of it is devoted to the problems from which his work originated and to the later developments which, directly or indirectly, he inspired. A few remarks about format are in order. In this book a publication is indicated by a date after a name; so Hilbert 1926, 178 refers to page 178 of an article written by Hilbert, published in 1926, and listed in the bibliography.
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Zermelo's Axiom of Choice: Its Origins, Development, and Influence
Gregory H. Moore
Limited preview - 2012
aleph Aleph Theorem algebra arbitrary choices argument Assumption Axiom of Choice Axiom of Separation axiomatization Baire property Baire's Bernstein Borel Cantor cardinal number consistency Continuum Hypothesis countable sets Countable Union Theorem Dedekind Dedekind-finite deduced defined definition demonstration denumerable Denumerable Axiom denumerable subset disjoint domain element equipollent established existence finite number finite set first-order logic Fraenkel function f Gödel Hadamard Hausdorff Hilbert implies independence infinite cardinal infinite set infinitely many arbitrary infinity Jourdain König later Lebesgue Lemma letter Levi limit point mathematicians mathematics maximal principles measure model of ZF Mostowski Multiplicative Axiom non-measurable set notion obtained order-types ordinal Partition Principle Peano Poincaré postulate Prime Ideal Theorem proposition proved published real functions real numbers required the Axiom result Russell Russell's Schoenflies second number-class sequence set theory Sierpiński Skolem Tarski topological transfinite Trichotomy Trichotomy of Cardinals uncountable well-ordered set Well-Ordering Principle Well-Ordering Theorem Whitehead Zermelo's Axiom Zermelo's proof Zermelo's system