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A. Makelov

February 28, 2014

Posted by on *`What’s an anagram of Banach-Tarski?’*

*`Banach-Tarski Banach-Tarski.’*

Banach and Tarski

The Banach-Tarski theorem says the following:

Theorem 1[Banach-Tarski theorem] Given a solid ball in 3-dimensional Euclidean space , we can partition it into a finite number of pieces, so that we can rearrange them to get two solid balls congruent to the first ball.

This clearly goes against people’s intuitions about *volume*. Often, it is noted that the Banach-Tarski theorem is a consequence of the axiom of choice; this is inherent, since it is in fact *equivalent* to the axiom of choice. In one of its most strikingly obvious formulations the latter says:

[Axiom of choice] The product of a collection of nonempty sets is nonempty.

That is, for any collection of sets , where the index set can be an arbitrary set, one can find an indexed family such that for all .

**So, we have these two things which are equivalent, and one is completely, obviously true, whereas the other is completely, obviously false. OK.**

What do these two things have to do with each other? Here’s the beginning of an answer. Our intuitions about concepts like volume and area are formalized in mathematics through what is called a measure. Here’s a definition that summarizes the **intuitively desirable** properties of such a measure:

Definition 2Ameasureon is a non-negative, translation-invariant, countably additive function that assigns to each parallelepiped its volume. That is to say,

- .
- disjoint,
- .

We would then hope to be able to build up complicated sets from many parallelepipeds. Or something like that in any case.

Why do we allow only countably many sets in (2)? Well, the (only!) alternative is to allow at least uncountably many sets, which would then imply that the measure of the entire is equal to the sum of the measures of the points; but points should have zero volume! So our condition (3) is actually pretty liberal.

However, it turns out that in the above definition we wanted too much, i.e. **it is inconsistent**:

Theorem 3There exists a subset for which doesn’t exist.

*Proof:* Consider the equivalence relation on given by

By the **axiom of choice**, we can pick an indexed collection of representatives for each equivalence class; call the set underlying this collection of representatives . To really convince yourself that this is *indeed a set *(because that’s tricky business), you should play around with the axioms of Zermelo-Fraenkel set theory and the definition of an indexed collection. We shall show that doesn’t exist.

Assume the opposite, and consider, for each , the sets

Observe that the are disjoint, for otherwise we would have for and , and hence , contradicting the fact that are in different equivalence classes.

On the other hand, . Thus, by the properties of measure, we have

The first inequality gives us , whereas the second gives us , thus the contradiction with the assumption that exists.

This is where our intuition about volume breaks: it’s impossible to formalize it so that it works for *all* sets. Now, it’s kind of clear that at least one of the pieces in the decomposition of the ball in the Banach-Tarski paradox has to be similar to the set above, and that is where `volume conservation’ fails.

What people do to define the measure in a consistent way is to be very careful about the sets for which the measure applies. This leads to the ideas of -algebras and Lebesgue measure, which are the established formalisms of measure theory. There still exist sets that are not Lebesgue-measurable (the one constructed above is an example), but this is no longer an inconsistency of the theory; it’s a `weirdness’ of math.

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Hi, thanks for the interesting article. How do you compose your articles? I have tried a couple of different ways to compose mathematically-oriented blog articles and the results are always kind of ugly. I’d be grateful for any info about the set-up you use.

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Sure, I’d be happy to share some info. I’m using a slightly modified version of Luca Trevisan’s latex2wp (which is described here: http://lucatrevisan.wordpress.com/latex-to-wordpress/ ); my modifications were mainly meant to make my current preambles and macros be used, instead of Luca’s default ones; I wrote some short Python code to do that, and it can be found here: https://github.com/amakelov/learn/tree/master/wordpress

When I have latex2wp configured, I write my posts in my latex editor (there’s a blog article about it here: https://amakelov.wordpress.com/2014/04/19/my-current-latex-configuration-a-relatively-easy-way-to-learn-how-to-take-latex-notes-in-real-time/ and then use latex2wp to convert them to wp-friendly html.

Finally, I think the choice of the theme of the blog has a huge influence on the appearance; I’ve managed to find the one I’m currently using, which gives me a kind-of-OK look that I’m reasonably happy with. But I’m also considering switching to blogging on my own website via github pages and using MathJax for typesetting math.

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famozno! 😀

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I don’t understand this step: “The first inequality gives us mu(A)>0, whereas the second gives mu(A) = 0. I’m probably just missing something simple. Could you explain this?

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Why are you saying that Banach-Tarsky is equivalent to AC? I can’t find a proof of BT => AC, could you point me to one?

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