After watching the Vsauce video many times I still can not seem to fully grasp it.
In: Mathematics
Let’s just talk about the circumstance surrounding it. People want to define concept of length/area/volume on *arbitrary* sets, in a consistent manner. That means assigning a non-negative number to each set, telling them how big they are. Consistent here mean their value match up when you add them, and that when you move them in a geometric manner the value don’t change. Of course, the definition varies as what constitute “add” and “move them in a geometric manner” mean varies. Depends on the precise definition, this requirement can be too strong that it’s impossible.
It was discovered very early that it’s impossible to do this on one-dimensional line when “add” is define to allow infinite adding. The method of proving invoke axiom of choice, a controversial axiom that nonetheless was most often used by the very people who care about defining concept of volume (analyst). Because of this, the requirement was relaxed to just finite adding.
Now up to 2 dimension. Now if you had read Euclid, you know that Euclid never *define* the concept of area. He simply cut polygons and reform the pieces into other polygon, and declare this means they have the same area. This is known as scissor congruence. Building on Euclid’s result, a theorem by Wallace–Bolyai–Gerwien theorem state that scissor congruence is equivalent to area. Upgrading this further, Banach proved that, when “adding” mean finite adding, and “geometric movement” means isometries (rotation, translation), it is possible to define the concept of area on every sets (this also use axiom of choice).
A note on axiom of choice: it’s a non-constructive axiom, so there are no ways to even know what specific object it produces, nor how to compute them.
However, a different result (due to von Neumann if I remembered correctly), showed that if “geometric movement” also include shearing, this is no longer possible. This put further restriction onward, if people want to define volume.
But the genesis of the problem is already here at 2-dimension, it’s just not quite enough to cause trouble. Let me explain the seed of this issue in dimension, but first an analogy. Remember the Hilbert’s hotel paradox? You have an infinite hotel that is full, how do you put a new guest in? Just shift everyone by one room, leaving a room empty.
How’s that related to what we are doing? The issue here is, it is possible in 2 dimensions to find 2 isometries that are very incongruence. They acts in extremely independent manner. Let’s call them T and S. Write TS to means move by S first, then move by T. TST would means move by T, then S, then T. The key facts here is this, if you give me 2 arbitrary strings of letter T and S, if the strings are different they are always different operations. So TSST is different from SSTTS.
This situation allow us to recreate the Hilbert’s hotel paradox. An orbit of a point is the set of all points that that point move to after applying a sequence of T and S. This sets can fit into 2 copies of itself, not 1. How? One copy is what happen after you move the set by T, another is when you move by S. The 2 copies are guaranteed to not collide, since you know different operations always give different result!!!
(for a visual depiction of this, if you know binary tree from computer science, imagine you have an infinite complete binary tree, then your 2 subtrees are each identical to the whole tree).
This paradoxical situation suggests that things must go wrong in 2-dimension, and it does if you allow shearing. Alas, thanks to the scissor congruence theorem above, it doesn’t if you only allow isometries. Basically, if you try to apply this paradox naively as above, you just get a scattered set of points, which logically should have area 0 anyway.
Also note the importance of having highly incongruent isometries. This paradoxical situation doesn’t happen in dimension 1: if T and S are 2 isometries, then TTSS is always equal SSTT; this is because TT and SS are always translation, so it doesn’t matter which one you do first the final outcome is the same. This cause collision if you try to construct the paradox. A similar problem happen to dimension 2 as well. Even though we have incongruent isometries, we have to use both rotation and translation together to achieve that, because each alone have the commuting issue just like in 1 dimension. But if you want to assemble a circle back to itself, you expect translation to be irrelevant.
Now let’s go to dimension 3. This is when things get real problematic. If you read Euclid, Euclid never define “volume” either. But here, he didn’t even bother with scissor congruence, basically push the entire issue under the rug. This lacks of rigor bother people, and Hilbert published, as one of his famous 23 problems, the question of whether any 2 polyhedron with the same volume can be cut and reformed into each other. This turned out to be the first one solved, by his student. And the answer is no, another things need to be considered, called the Dehn invariant (Dehn is the student’s name). To add onto this issue, rotations alone can now produce highly incongruent isometries.
This should spell trouble for dimension 3. And indeed, Banach-Tarski showed exactly why. They do that by cutting a solid sphere into 5 sets (remember, we allow arbitrary sets, so each of these sets are scattering of points), then reassemble it into 2 spheres. Even though these sets are scattering of points, if you were to give it volume, they can’t be all 0, as they add up to a sphere, so there are no avoiding the paradox this way. A sphere must have positive volume, so if you can turn 1 into 2, that means it’s impossible to define volume for all sets in a consistent manner.
Once again, this theorem use axiom of choice, so there is absolutely no ways to actually even compute, or know what points are in the sets. The idea of the proof is the same as before, we use highly incongruent isometries, but now we have a lot more rotations to use. This allows us to do a paradoxical decomposition on a spherical shell, and then improve it to a solid sphere. This put a complete stop to any hope of defining volume on all sets.
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