Banach–Tarski Paradox

This is a fabulous result, both positive and negative, for many reasons. But one of the things people don't realise is that there is a reason why it's interesting mathematically and not just a gimmick.

In Classical Euclidean Geometry there are five axioms, and while the first four seem clear and obvious, the fifth seems a little contrived. So for centuries people tried to prove that the fifth was unnecessary and could be proven from the other four.

These attempts all failed, and we can show that they must fail, because there are systems that satisfy the first four, but do not satisfy the fifth. Hence the fifth cannot be a consequence of the first four. Such systems are (for obvious reasons) called Non-Euclidean Geometries.

So we can use explicit examples to demonstrate that certain proofs are impossible, and the Banach-Tarski Theorem is a result that proves that a "Measure"[0] cannot have all four obviously desirable characteristics.

For more information, here's a blog post[1] I wrote some time ago:

https://www.solipsys.co.uk/new/ThePointOfTheBanachTarskiTheo...

It's intended to be readable, but the topic is inherently complex, so it may need more than one read through. If you're interested.

[0] Technical term for a function that takes an object and returns a concept of its size. For lines it's length, for planar objects it's area, for 3D objects it's volume, and so on.

[1] In case people want to discuss that separately I've submitted it as a separate post here: https://news.ycombinator.com/item?id=40798224

There are three ways to resolve this paradox:

1. Accept that our intuition about volumes is off when dealing with point clouds so weird that they cannot actually be described, but require the axiom of choice to concoct them.

2. Reject the axiom of choice and adopt the axiom of determinacy. This axiom restores our intuition about volumes to all subsets of Euclidean space, at the expense of which sets can be formed. (That said, the axiom of determinacy allows other sets to be formed which are not possible with the axiom of choice, so it wouldn't be correct to state that the axiom of determinacy causes the set-theoretic universe to shrink.)

3. Keep logic and set theory as it is, but employ locales instead of topological or metric spaces. Locales are an alternative formalization of the intuitive notion of spaces. For many purposes, there are little differences between locales and more traditional sorts of spaces. But, crucially, a locale can be nontrivial even if it does not contain any points. Locale-theoretically, the five pieces appearing in the Banach–Tarski paradox have a nontrivial overlap (even though no points are contained in the overlapping regions), hence you wouldn't expect the volumes to add up.

I tried to give a varied account on the axiom of choice at the Chaos Communication Congress once, the slides are here: https://www.speicherleck.de/iblech/stuff/37c3-axiom-of-choic...

What's an anagram of Banach-Tarski?

Banach-Tarski Banach-Tarski!

The gorgeous book "Discrete groups, expanding graphs and invariant measures" by A. Lubotzky investigates the structures that give rise to measure-theoretic phenomena like the B-T paradox. It is a graduate level monograph, but I recommend it wholeheartedly. It illustrates how the study of some paradoxes from the early 20th Century led to amazing and highly applicable mathematics like expander graphs and the spectral theory of non-commutative groups.

Mandatory watch, by Vsauce: https://www.youtube.com/watch?v=s86-Z-CbaHA

Can someone explain this more simply?

If you cut up the sphere's surface into pieces, the combined surface area will remain the same. If you then reassemble them in a different configuration into two spheres both the same size as the original, the surface area will be twice as much.

I don't see how that could be true. What am I missing here?

ETA: thanks for all the explanations. The most succinct answer seems to be because it assumes the surface is made of infinitely many points, and infinity breaks math. 2*inf = inf.

One more reason why it makes no sense to treat infinity as a number.

This must be the most unintuitive result of all of mathematics. Its very interesting what a seemingly simple axiom like the axiom of choice can lead to -- simple as in 'even a 9-year old can understand it', the consequences are rather enormous and not simple at all.

"Tame topology is the name for the largely programmatic quest for a refoundation of topology and geometry that avoids ‘pathological’ objects like space-filling curves or counter-intuitive results like the Banach-Tarski paradox that occur in the traditional approach."

https://ncatlab.org/nlab/show/tame+topology

There's a short proof of a 2D version of the paradox on page 684 of the Princeton Companion to Mathematics: https://sites.math.rutgers.edu/~zeilberg/akherim/PCM.pdf

What's the use of this paradox? Does it have any practical implementation?

If I remember the paper correctly, it also uses a metric that isn't just the usual euclidean one.

A. K. Dewdney did a Computer Recreations on this -

"A matter fabricator provides matter for thought" on the hub - DOI:10.2307/24987222 ( https://www.jstor.org/stable/24987222 ) [Early April 1989]

It made quite an impression as a kid. Even 30 years later I think about it every now and again.