Hi I would like to understand Banach-Tarski paradox, but well... my knowledge with set theory is very very limited. I know what a union is, I will tell you how to partition a simple set, know couple of things about sets being disjoint and well, that will be it.

ANd every idea of proof (take away fancy Vsauce video on youtube which doesnt do it for me) relies heavily on knowledge about sets.

The main question here. Can you explain to me in plain english what is a Paradoxical set and what is meant by equidecomposable sets? Maybe give me a relatively simple example of each?

I tried looking for definitions online, but they all require some set theory background, most of the time I don't even understand the symbols presented, leave alone the whole strings of them.

Thanks for anything on this


Imagine a square, a rhomboid and a circle: Square, Rhomboid and Circle And you ask: "Could I cut the rhomboid in such a way that I am able to put the cut parts together so that I get the square on the left?"

Of course you can! Just cut one of the the "corner pieces" and put it on the other side:

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And now you ask: "Could I cut the square in such a way that I am able to put the cut parts together so that I get the circle on the right?"

And you think about it and realize that independently of how you cut the square into pieces, as long as you are making finitely many cuts, you won't be able to rearrange them into a perfect circle.

If you then think of the square, the rhomboid and the circle as sets of points, you could say that the square and the romboid would be equidecomposable, but the square and the circle not.

Creating an analogy for paradoxical sets is harder, but I'll give it a try:

Imagine you have a sphere made of a mystical, special material. You put the sphere into a grinder and grind it up into really fine powder. It now happens that this special material of which the sphere is made of allows you to let the powder which comes out of the grinder fall into two molds which have the exact same shape and size as the original sphere. To you surprise you observe that at the end of the grinding process, the two molds are filled up to the rim with powder, leaving you with two complete spheres, which is obviously paradoxical.

Math explanation:

In relation to this analogies I want to try to explain this two terms with sets:

Two sets $A$ and $B$ are called equidecomposable if you can split up $A$ into a finite number of subsets, move this subsets around - but in a "rigid fashion", which means that you can not "deform" the subsets, like changing the relative positions of the elements of such a subset or bending the corner piece of the rhomboid so that it gets curved - and be left with the set $B$.

A paradoxical set is a set $A$ which you can split up into two subsets $A_1$ and $A_2$, "transform" each of the (infinite) elements of the subsets each in such a way that at the end of the transformation, each of the subsets are equal to your original set $A$ (this sounds crazy and total unintuitive, which is the reason why its called "paradoxical").


In the context of Banach-Tarski, two subsets $A,B$ of $\Bbb R^3$ (or other suitable spaces where "movements" make sense) are called equidecompoable if there exist finitely many subsets $C_1,\ldots, C_n$ and orientation-preserving movements (sequences of rotations and translations) $T_1,\ldots, T_n$ such that $A$ is the disjoint union of $C_1,\ldots, C_n$ and $B$ is the disjoint union of $T_1(C_1),\ldots, T_n(C_n)$. In other words, you can partition $A$ into finitely many pieces, move these around and obtain a partition of $B$.

A paradoxical set is one that is equidecomposable to two disjoint copies of itself. For example, the Banach-Taski theorem shows that the unit sphere in $\Bbb R^3$ is a paradoxical set. Usually, the theorem is proved wit rotations alone, no translations. This makes little difference, exept that paradoxical is better defined as: A set $A$ is paradoxical if it is the disjoint union of two subset such that each of these two subsets is equidecomposable with $A$.

  • $\begingroup$ soooo by moving the subsets around you dont mean that a for a stupid example subsets {3,5},{7,9} are moved and now it is {7,9},(3,5} but we perform rotation on them, so order stays the same, but subsets now are upside down, or something similiar? I think that was the main misconception in my thinking. I thought you literally change the order of subsets, which didnt make sense to me because order doesnt matter in sets in general. Thank you :) $\endgroup$ – Scavenger23 Dec 3 '15 at 21:36
  • $\begingroup$ @KuderaSebastian: you have to have an infinite number of points involved so that arguments about the number of points don't matter. It is similar to te fact that I can partition $\Bbb N$ into the odd and even numbers, each of which is equinumerous with $Bbb N$ This is perhaps not so surprising because I have "stretched out" the intervals between the number. In the Banach-Tarski case, the geometric relationships between the points in each piece are maintained. The pieces have to be such that you can't define a volume, or that will shoot you down. $\endgroup$ – Ross Millikan Dec 3 '15 at 21:45

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