Axiom of Choice
AC, the axiom of choice, because of its non-constructive character, is the most controversial mathematical axiom, shunned by some, used indiscriminately by others. This treatise shows paradigmatically that: Disasters happen without AC: Many fundamental ma
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Horst Herrlich
Axiom of Choice
1876
1 23
Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris
1876
Horst Herrlich
Axiom of Choice
ABC
Author Horst Herrlich Department of Mathematics University of Bremen P.O. Box 33 04 40 28334 Bremen Germany e-mail: [email protected]
Library of Congress Control Number: 2006921740 Mathematics Subject Classification (2000): 03E25, 03E60, 03E65, 05C15, 06B10, 08B30, 18A40, 26A03, 28A20, 46A22, 54B10, 54B30, 54C35, 54D20, 54D30, 91A35 ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN-10 3-540-30989-6 Springer Berlin Heidelberg New York ISBN-13 978-3-540-30989-5 Springer Berlin Heidelberg New York DOI 10.1007/11601562
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SPIN: 11601562
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Dedicated in friendship to George, Gerhard, and Lamar
It is a peculiar fact that all the transfinite axioms are deducible from a single one, the axiom of choice, — the most challenged axiom in the mathematical literature. D. Hilbert (1926)
It is the great and ancient problem of existence that underlies the whole controversy about the axiom of choice. W. Sierpi´ nski (1958)
Wie die mathematische Analysis gewissermaßen eine einzige Symphonie des Unendlichen ist. D. Hilbert (1926)
VI
Preface
Zermelo’s proof, and especially the Axiom of Choice on which it was based, created a furor in the international mathematical community. ... The Axiom of Choice has easily the most tortured history of all the set–theoretic axioms. Penelope Maddy (Believing the axioms I)1
Of course not, but I am told it works even if you don’t believe in it. Niels Bohr (when asked whether he really believed a horseshoe hanging over his door would bring him luck).2 Without question, the Axiom of Choice, AC (which states that for every family of non–empty sets the associated product is non–empty3 ), is the most controversial axiom in mathematics. Constructi
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