Banach-Tarski paradox
en.wikipedia.org
en.wikipedia.org
A: Banach-Tarski Banach-Tarski.
I did my undergraduate project on the question of finitely-additive, isometry-invariant measures that extend the Lebesgue Measure and which are defined on all possible bounded subsets of R^n.
And yes, that's relevant. There is (perhaps surprisingly) such a measure on R,and there is such a measure on R^2, but the Banach-Tarski "paradox" shows that there is no such measure on R^3 (and therefore trivially, R^n for n>3).
For contrast, we'd usually like to have a measure that's countably additive, but there is a simple construction (albeit using the axiom of choice) that shows such an object is impossible.
This has, by the way, been submitted and discussed several times in the past. In case you're interested in reading other descriptions, and previous discussions, you can find some of them here:
I suggest buying two and wearing them both. When somebody laughs at the T-Shirt, well, you do the math.
"What's an anagram of Banach-Tarski?"
A: "Banch-Tarski Banach-Tarski"
If you do give up the axiom of choice, is there a lot of cool/useful math that you'd also have to give up?
* Let X and Y be sets. Then either they have the same cardinality, or one is smaller than the other.
* Let X be an infinite set. Then there is a bijection between X and the cartesian product of X with itself, X × X.
* Tychonoff's theorem: every product of compact topological spaces is compact.
As far as Tychonoff's theorem goes, you might find this paper interesting:
While the Banach-Tarski violates your intuitions, it is probably because it is more about how there are "multiple copies" of the real numbers in the real numbers, than anything resembling a real cut.
However, to me the "basic principle" is a similar argument. The important thing (for me) was to understand this is an argument about slicing up infinities in an "Infinite comb" type way, rather than anything resembling a "cut".
Reverse mathematics (http://en.wikipedia.org/wiki/Reverse_mathematics) is a reasonably recent branch of mathematics aiming to investigate those kinds of questions.
Simplest example: you can have a product of non-empty sets be empty.
That said, it is a theorem that absolutely no theorem of number theory can possibly depend on the axiom of choice. So absolutely no "real" consequence can come from accepting or not the axiom of choice.
Consider a spherical surface S. We can construct a mapping from the point of S to the points in a finite and limited subset of R^2 (say, the square [0, 1] X [0, 1]). One way to do that is to map each point of "latitude" ρ and "longitude" θ (in degrees), to the point ((ρ+90)/180, θ/360).
Now, let's consider the rectangle [0, .5] x [0, 1]. There is obviously a 1:1 mapping from this to the original square (just halve or double one of the coordinates to change from one to the other). In an intuitive sense, there are "as many" points in half the square as in the whole square, a bit like there are as many natural numbers as there are even numbers. (This is one definition of infinite set, if I remember correctly).
Now this means that we can remap all the points from half the square to a complete sphere, and the points from the other half square to a "new" complete sphere, in fact mapping points from the original sphere S to two new identical spheres.
Is this reasoning correct, or am I missing anything (probably something obvious...)?
As a (bio)chemist, this paradox has always bugged me profusely.
Perhaps I should set aside some time someday to grasp a very basic understanding of what's going on here. :p
Once I understood this, I find it is a bit a cheat (as you might expect). The 'cuts' are not cuts in any real-world sense.
Hopefully you are happy with the idea that there are as many integers as there are both even integers or odd integers. Therefore in some sense I can "cut" the integers into two equally sized sets. This paradox does something very similar on the real numbers, making use of the form of the real numbers to avoid the "doubling" effect I got when cutting the integers.
You're right that these aren't "pieces" in the intuitive visual sense. That's why Feynman ridiculed the result, because in the initial description given to him he assumed physical pieces, not pairwise disjoint sets.
http://news.ycombinator.com/item?id=411307
There are similar paradoxical dissections of the plane:
I remember being told about the library that has all finite books that can possibly be written. Every finite string of characters appears as a book. Obviously there are infinitely many, but we won't worry too much about that.
Now we find that someone, overnight, stole all the books except for those that start with the letter "q". We're annoyed at first, but then someone suggests we just erase the initial "q" from all the remaining books.
So we do that, and lo and behold! We have the full collection again! We even now have an extra book, one that's completely blank!
It's not a paradox, but it is the way infinite things can work, and it shows how 1/26 of a collection can be the same as the whole collection. This is related to how the B-T "paradox" works, but it's very much over-simplified.
Even so, it's a useful place to get people starting to think about things in the right way.
It didn't.
Added in edit: Fascinating - Someone down-voted this comment. Here I was thinking it would be helpful that people who trust my opinion will know not to waste their time, and people who don't know me can use this comment to see if their taste in these things matches mine. But no, someone just thinks it's of negative value. Interesting. I'm always learning new things about the HN community.
Your comment is a little harsh, but totally accurate. I saw all the video and I want my 3:42 minutes back!
(Note: I remember that one of my comments initially got two or three downvotes in spite of having only accurate relevant facts (not opinions), I got a little worried but it bounce and finally that comment got like 12 points. Don't worry, be happy!)
It's like depicting the irrational numbers by coloring everything on a number line between pi and e blue.
Is it possible to do the cut-up-and-reassembly trick with a finite number of sets where the membership is computable? If so the "Banach-Tarski Algorithm" for the special case of duplicating the sphere might be helpful for understanding.
[EDITED to add: er, actually in order to prove that "without AC it can happen that all subsets of R^n are Lebesgue measurable" you need there to be an inaccessible cardinal. If you don't know what that means, ignore it. I'm mentioning it just to avoid leaving a false statement in the record.]