History of AxiomOfChoice

	Revision 9 . . February 8, 2001 4:57 am by AyeSpy
	Revision 8 . . (edit) February 8, 2001 3:26 am by LarrySanger [* Did just a little more reformatting]
	Revision 7 . . February 8, 2001 3:25 am by LarrySanger [* Did just a little reformatting]
	Revision 6 . . February 8, 2001 3:17 am by (logged).bomis.com
	Revision 5 . . February 8, 2001 3:17 am by (logged).bomis.com
	Revision 4 . . February 8, 2001 3:16 am by (logged).bomis.com
	Revision 3 . . February 8, 2001 3:15 am by (logged).bomis.com
	Revision 2 . . February 8, 2001 3:14 am by (logged).bomis.com
	Revision 1 . . February 8, 2001 3:13 am by (logged).bomis.com

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Difference (from prior major revision) (no other diffs)

Changed: 1c1

The AxiomOfChoice was formulated about a century ago by ErnstZermelo?, and was quite contraversial at the time. It states the following:

The AxiomOfChoice was formulated about a century ago by ErnstZermelo?, and was quite controversial at the time. It states the following:

Changed: 19c19

And therein lies the crux of the AxiomOfChoice. All it states is that there is some function f that can choose an element out of each set in the collection. It gives you no indication about how the function would be defined, it simply mandates its existance.

And therein lies the crux of the AxiomOfChoice. All it states is that there is some function f that can choose an element out of each set in the collection. It gives you no indication about how the function would be defined, it simply mandates its existence.

Changed: 23c23

One of the reasons that some mathematicians don't particularly like the AxiomOfChoice is the fact that it implies the existance of some bizarre counter-intuitive objects. An example of this is the BanachTarskiParadoxicalDecomposition which amounts to saying that it is possible to "carve-up" the closed unit sphere into finitely many pieces, and using only rotation and translation, reform the pieces into 2 spheres each with the same volume as the original. Note that the "proof" given in the BanachTarskiParadoxicalDecomposition is an existance proof only, it does not tell you how to carve up the unit sphere to make this happen, it simply tells you that it can be done.

One of the reasons that some mathematicians don't particularly like the AxiomOfChoice is the fact that it implies the existence of some bizarre counter-intuitive objects. An example of this is the BanachTarskiParadoxicalDecomposition which amounts to saying that it is possible to "carve-up" the closed unit sphere into finitely many pieces, and using only rotation and translation, reform the pieces into 2 spheres each with the same volume as the original. Note that the "proof" given in the BanachTarskiParadoxicalDecomposition is an existence proof only, it does not tell you how to carve up the unit sphere to make this happen, it simply tells you that it can be done.