History of Axiom of choice

	Revision 15 . . December 17, 2001 4:13 am by Hagedis
	Revision 14 . . (edit) December 17, 2001 3:58 am by AxelBoldt [link]
	Revision 13 . . (edit) December 17, 2001 3:21 am by Matthew Woodcraft [/Talk]
	Revision 12 . . (edit) September 30, 2001 11:44 pm by Zundark [fix links]

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

Changed: 12,13c12,13

:2. Let X be the collection of all subsets of the natural numbers {0, 1, 2, 3, ... }. :: Then f can be the function that chooses the smallest element in each set.

:2. Let X be the collection of all non-empty subsets of the natural numbers {0, 1, 2, 3, ... }. :: Then f can be the function that chooses the smallest element in each set.

Changed: 16c16

:: Then f can be the function that chooses the midpoint of each interval.

:: Then f can be the function that chooses the midpoint of each interval.

Changed: 18,19c18,19

:4. Let X be the collection of all nonempty subsets of the reals. :: Now we have a problem. There is no obvious definition of f that will guarantee you success.

:4. Let X be the collection of all nonempty subsets of the reals. :: Now we have a problem. There is no obvious definition of f that will guarantee you success, because the other axioms of ZF set theory do not well-order the real numbers.

Removed: 39,40d38

The axiom of choice is hard to use as is. Usually, an equivalent to the axiom of choice called Zorn's Lemma is used in real life.

Removed: 42,44d39

On the other hand, axiom of choice also allows one to prove that the solid unit sphere can be broken up into five (infinitely complicated) pieces, shuffled around (with no scaling/shearing etc., just rotation and translation), and put back together to make two solid spheres, each identical to the original. This is the so-called Banach-Tarski Paradox.