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Lorenz J. Halbeisen
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Some of my lectures
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G ZARLINO 著
L.J. Halbeisen, Combinatorial Set Theory, Springer Monographs in Mathematics,
DOI 10.1007/978-1-4471-2173-2_5, c Springer-Verlag London Limited 2012
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P126
NOTES
The Axiom of Choice. Fraenkel writes in [26, p. 56 f.] that the Axiom of Choice is
probably the most interesting and, in spite of its late appearance, the most discussed
axiom of Mathematics, second only to Euclid’s axiom of parallels which was introduced
more than two thousand years ago. We would also like to mention a different
view to choice functions, namely the view of Peano. In 1890, Peano published a
proof in which he was constrained to choose a single element from each set in a certain
infinite sequence A1,A2, . . . of infinite subsets of R. In that proof, he remarked
carefully (cf. [73, p. 210]): But as one cannot apply infinitely many times an arbitrary
rule by which one assigns to a class A an individual of this class, a determinate
rule is stated here, by which, under suitable hypotheses, one assigns to each class
A an individual of this class. To obtain his rule, he employed least upper bounds.
According to Moore [66, p. 76], Peano was the first mathematician who?while
accepting infinite collections?categorically rejected the use of infinitely many arbitrary
choices.
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