The Point Of The Banach Tarski Theorem 


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The Point Of The

Added after seeing some discussion ...
This is not intended to explain why the BanachTarski theorem is true, nor to give a proof, nor to talk about what the pieces look like, etc. No ... the purpose of this article is to explain why the result is relevant. In short, the BanachTarski Theorem tells us what is and is not possible in measure theory. It tells us that things we would in general want to be true of a measure are mutually inconsistent. Now read on ... 
Which is a perfectly reasonable reaction. Indeed, if you then have the BanachTarski theorem explained to you, you most likely will still go: "Huh?" It's a perplexing result, some even call it a paradox, but the fact that it's so odd actually masks the fact that it's a truly important result, with some deep implications.
So I'm going to explain here why it's important, and what some of the implications are. The hope is that even if you do know the result you will find this interesting, largely because in the shock of seeing what the result says, you've never been shown why it's interesting.
So for those of you who don't know the result, here it is in simple, nontechnical terms:
Apart from using the Axiom Of Choice. 
So in part because it's so surprising, and shocking, and nonsensical, you might think it's in a deadend of mathematics and of no real use or interest. That's what this article is intending to address.
To do so, let's consider the idea of measurement. We can talk about the length of a line, the area of a polygon (or other shape), the volume of a lump, or whatever. The development of the concept of accurate measurement goes back millennia, and is critical in the development of commerce, engineering, and so many other things. So we're going to look at the concept of measuring something, and see what we can say about it mathematically.
To do that we'll talk about a function called a "measure". One of the problems in maths is we use ordinary words in a technical sense, so it's a bit dangerous to call this thing a "measure," but I'll try to use that word only ever in its technical sense.
And fairly obviously we'll need a different function, or measure, when we're in one dimension as compared with two dimensions, or more, so we'll talk about a measure for each dimension.
So a "measure" is a function that takes a set and returns a number that somehow represents the length, area, volume, whatever. 
Let's start writing these down. If we are working in $n$ dimensions and we have a measure, $\mu$ which is defined on subsets of $\mathbb{R}^n,$ we expect that:
The last one we can repeat over and over to get the idea of the measure being finitely additive. In fact, we'd really like to extend that to being countably additive, so that:
But there are other things we expect to be true of a measure. We expect that if we take the measure of something, move the something around, and then take the measure again, we get the same answer. Moving something around should not change its size. In mathematics the idea of moving something around is captured by the idea of what we call an isometry. An isometry is a function that doesn't change distances, so if we have an isometry $\tau$ and apply it to a set $A$, none of the distances between points in $A$ will change, so we can think of $\tau(A)$ as being $A$ moved somewhere else (and maybe flipped over to give a mirror image).
So for any set $A,$ and any isometry $\tau,$ then we expect of a measure $\mu$ that:
So far:
Outline of the
The idea is:

If $\mu(V)\;=\;0$ then when we add them all up we get zero, which would mean the measure of $[0,1)$ is zero.  If $\mu(V)\;\ne\;0$ then when we add infinitely many together we definitely don't get a finite answer. 
Which is wrong.  Which is wrong. 
So what do we do?
Well, what we have to do is relax one of our requirements, and make it weaker. The obvious thing that people want to try is to reduce the power of the additivity rule. So our requirements become:
As it happens, for $\mathbb{R}^1$ we can do this. Even more, for $\mathbb{R}^2$ we can do this! But for $\mathbb{R}^3,$ we can't.
How do we know? Because of the BanachTarski theorem.
The BanachTarski theorem says that if $B$ is the unit ball in $\mathbb{R}^3$, there exist pairwise disjoint sets $\{A_i\}_{i=1}^n$ and isometries $\{\tau_i\}_{i=1}^n$ and $\tau$ such that:

It's not enough in three dimensions.
And that's what the BanachTarski theorem is really all about.
There's more we can say about this. Do we want or need every set to have a measure? We can define some really weird sets  should it always make sense for them to have a concept of length/area/volume?
There is one thing that could save us, and that's the Axiom Of Choice. Or rather, denying the Axiom Of Choice. In each case, showing that in $\mathbb{R}^1$ there's an unmeasurable set, and showing that in $\mathbb{R}^3$ we can have a paradoxical decomposition, we need to use the Axiom Of Choice. So maybe we should choose not to believe in the Axiom Of Choice.
But that's another discussion ... you can find some of it here:
Sieve Of Eratosthenes In Python  :  Photocopy A Mirror ... 
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