Hurkyl said:IIRC, the proof uses surgery theory.
One of the main things to emphasize about the construction is that its intermediate steps involve sets that are not measurable. All of the clever work is done with sets for which you cannot define volume, so there isn't any reason to expect that you have the original volume when you're done.
And, incidentally, the big point about the construction is that it only uses three pieces. It's a trivial exercise to prove that you can rearrange all of the points in one sphere to form two spheres of the same size, if you do it one point at a time.
The Axiom of Choice is a fundamental mathematical principle that states, given any collection of non-empty sets, it is possible to choose a single element from each set. It is important because it allows for the creation of infinite collections and the development of other mathematical concepts such as cardinality and topology.
The Axiom of Choice cannot be proven within the standard framework of set theory, known as Zermelo-Fraenkel set theory. It is considered to be an assumption, or an axiom, that is consistent with other axioms and is necessary for the development of certain areas of mathematics.
Yes, there have been various criticisms and controversies surrounding the Axiom of Choice since its introduction in the late 19th century. Some mathematicians have argued that it leads to counterintuitive results, such as the Banach-Tarski paradox, where a solid ball can be divided into a finite number of pieces and reassembled to form two identical copies of the original ball.
No, the Axiom of Choice is not necessary for all areas of mathematics. In fact, there are some areas of mathematics, such as constructive mathematics, which reject the Axiom of Choice and instead rely on more limited principles of choice. However, the Axiom of Choice is still widely accepted and used in many areas of mathematics.
There are alternative axioms that have been proposed as replacements for the Axiom of Choice, such as the Axiom of Determinacy and the Axiom of Boundedness. However, these axioms have their own limitations and are not widely accepted as replacements for the Axiom of Choice.