Axiom of Choice

选择公理是个有趣的东西。罗素的比方很逗。

http://www.math.vanderbilt.edu/~schectex/ccc/choice.html

Bertrand Russell was more famous for his work in philosophy and political activism, but he was also an accomplished mathematician. His book Introduction to Mathematical Philosophy includes some discussion of AC. Here is my paraphrasing of part of what he said:

To choose one sock from each of infinitely many pairs of socks requires the Axiom of Choice, but for shoes the Axiom is not needed.

下面这个事情足够神奇，

Perhaps the most bizarre is the Banach-Tarski Paradox: It is possible to take the 3-dimensional closed unit ball,
B = {(x,y,z) : x2 + y2 + z2 < 1}
and partition it into finitely many pieces, and move those pieces in rigid motions (i.e., rotations and translations, with pieces permitted to move through one another) and reassemble them to form two copies of B.