Definitions
from The American Heritage® Dictionary of the English Language, 5th Edition.
- noun An axiom of set theory asserting that for a nonempty collection A of nonempty sets, there exists a function that chooses one member from each of the sets including A.
Etymologies
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Examples
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An interesting application of the Axiom of Choice is the Banach-Tarski Paradox that states that the unit ball can be partitioned into a finite number of disjoint sets which then can be rearranged to form two unit balls.
Set Theory Jech, Thomas 2002
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This is a bit nerdy, and hard to follow, but the Axiom of Choice is a math question that roughly goes something like:
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This is a bit nerdy, and hard to follow, but the Axiom of Choice is a math question that roughly goes something like:
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The principle of set theory known as the Axiom of Choice has been hailed as “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” (Fraenkel, Bar-Hillel & Levy 1973,
The Axiom of Choice Bell, John L. 2008
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Q: What's sour and equivalent to the Axiom of Choice?
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Non-constructive methods (such as non-constructive existence proofs) and non-constructive axioms (such as the Axiom of Choice).
Platonism in the Philosophy of Mathematics Linnebo, Øystein 2009
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Yet it was meeting former Axiom of Choice member Loga Ramin Torkian and producer Carmen Rizzo that really made me fall in love with the music she was producing.
Derek Beres: Global Beat Fusion: Six Degrees of the Middle East 2009
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Different set theorists have different views on controversial subjects such as non-well-founded sets, the Continuum Hypothesis, the Axiom of Choice, the set/(proper -) class distinction, etc.
Impossible Worlds Berto, Francesco 2009
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The reasons for accepting the Axiom of Choice were in the end purely mathematical.
Naturalism in the Philosophy of Mathematics Paseau, Alexander 2008
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For example, the Axiom of Choice allows the taking of maximal ideals in rings and other structures; it entails the kinds of maximality principles that even the analysts were already using; it simplifies transfinite arithmetic; and despite its suspicious abstractness it turns out to be equivalent to
Naturalism in the Philosophy of Mathematics Paseau, Alexander 2008
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