Confused about the Banach-Tarski's theorem

[O'Nero Samuel]

"In 1924, Stefan Banach and Alfred Tarski*** proved that it is possible to cut one

sphere into five pieces that can be recombined to give two spheres, each the size of the

original.

Take any two sets not extending to infinity and containing

a solid sphere each; then it is always possible to dissect one into the other with a finite

number of cuts. In particular it is possible to dissect a pea into the Earth, or vice versa"...Motion Mountain.

What is the applicable implication of this? And does anyone has a link or blog that could show the mathematics that was used?

[D H]

In particular it is possible to dissect a pea into the Earth, or vice versa

No, it isn't. The way in which the Banach-Tarski theorem dissects a sphere involves disjoint points. There's a problem here: Matter is quantized. The best way to look at the Banach-Tarski theorem is that it is yet another demonstration of “The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?”

 

Nerdy math joke: What's an anagram of Banach-Tarski?

[mathematic]

No, it isn't. The way in which the Banach-Tarski theorem dissects a sphere involves disjoint points. There's a problem here: Matter is quantized. The best way to look at the Banach-Tarski theorem is that it is yet another demonstration of "The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"

 

Nerdy math joke: What's an anagram of Banach-Tarski?

 

Axiom of choice, well-ordering theorem, and Zorn's lemma are equivalent.

 

 

 

Banach-Tarski is unphysical, since it involves a decomposition into non-measurable sets.

[D H]

The best way to look at the Banach-Tarski theorem is that it is yet another demonstration of “The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?”

Axiom of choice, well-ordering theorem, and Zorn's lemma are equivalent.

Of course they are. That statement about the axiom of choice, the well-ordering principle, and Zorn's lemma is a joke by Jerry Bona, and a pretty good one.

 

 

Banach-Tarski is unphysical, since it involves a decomposition into non-measurable sets.

Exactly.

[mathematic]

Of course they are. That statement about the axiom of choice, the well-ordering principle, and Zorn's lemma is a joke by Jerry Bona, and a pretty good one.

 

 

 

Who is Jerry Bona?