We use sets to model the real world. They do remarkably well, and the math we've developed to work on them includes calculus to make bridges that stay up, fluid dynamics that make aeroplanes fly, and discrete math that helps us understand routing, scheduling, and all sorts of stuff.
We use the real line to model distances. In the real world we are limited as to the accuracy we can use, but modelling those limitations is nasty. It's easier to assume that things are continuous. In its turn, we make choices that make working with these models easier, and they turn out to be amazingly useful.
But then we start poking the dusty corners. The choices we make in the development of the theory have consequences, and math is about exploring both the choices, and their consequences.
So we can choose that between every two numbers there's another number. We can choose that there's a number whose square is 2. We can choose that the sum of the reciprocals of squares : 1+1/4+1/9+1/16+1/25+1/36+ ... : is a real number.
And we can choose that there is no smallest positive number. That has a consequence. If you believe that there is no smallest positive number, then 0.99999... has to equal 1. You can't have one without the other.
So we can talk about "the length of a line." Then we can talk about the "length" of a set of points on a line. Then we start to find that these simplistic models, these obvious and natural choices, even though they are amazingly useful have some unexpected consequences.
Does that help you to understand the context?
I'd be really interested in developing an agreed dialog about this. Will you send me an email?
> I'd be really interested in developing an agreed dialog about this. Will you send me an email?
I've sent you an email, as requested.
From the sound of it, though, the answer to my puzzlement, is that it doesn't apply to this universe. In which case, I don't think it's a problem at all. I'm quite happy to imagine mathematicians doing work on geometries that don't map to the real world, for example.
Mostly, then, is this just a question of presentation? I mean, if you said "this does not apply to the real world, but to some fictitious mathematical assumption that assumes things can be divided up infinitely," I don't think anyone would have an issue with it. It sounds like it's a problem to most people because it's presented as if it's a real-world result.
Or, at least, that seems to be the impression I get, given the other comments.
What I mean is, is there's a question of misdirection here? I.e. the theorem is presented as this non-intuitive thing that can't possibly be the case in the real world, then when someone asks what would happen if you tried it in the real world, the answer is "it's assuming some things that aren't true for the real world." Because if that's the case, I'm not sure I see the paradox. Assuming a weird set of ideas, I would expect you can come up with weird answers.
Here's a question, though, because something is nagging at me. And I'm going to assume this is a universe of infinite points and no atoms (as I understand it, at least).
Let's say we have a sphere of volume 4/3 pi r^3 = 100
Now you do your cuts, but don't reassemble yet. The sphere is still the original sphere, with all the shapes it has been cut into still in virtually their original spaces.
The total volume still has to be 100, right? I mean they all still fit into the original space.
So now, you immerse it in water, in a bathtub ready to overflow, and start manipulating the pieces.
One way to look at this is to note ask what exactly it meant when you said "a sphere of volume". In the real world it's ultimately impossible to imagine spaces which don't have volume of some kind or another, but in mathematics we often study models of the world too sparse to even have "measurability". These are obviously aphysical, but it turns out that when one tries to construct reality via set theory you have to go through a zone of "funky aphysical things" before you reject them all and consider only measurable things.
BT exists just before we make that last transition and abuses the fact that the constructions we've made so far are not required to have any reasonable sense of "volume" whatsoever. All of the "infinity" bits are sort of a red herring as there's no reasonable way to think about taking physical things and cutting them into aphysical things. Instead, this all arises from taking incredibly aphysical things (raw, theoretical sets) and building them up until they feel physical.
If you lay out the constraint that "cuts" must divide measurable things into measurable things, which is a reasonable expectation we have of physical items, then BT will vanish.
I've replied to your email, but to reply here to your specific question:
> immerse it in water, in a bathtub ready
> to overflow, and start manipulating the
> pieces.
> At what point does the water level rise?
A lovely question. The answer is that the "water" isn't "water", it has to be the same infinitely fine "mush" that the ball is made of. As a result, as you move the pieces out so the "water" ends up forming non-measurable holes to fit them into, and the complement is also non-measurable. So in the same way as the balls kind of "fold out" to become two balls, so the water kind of "folds in" and takes up less space than it used to, exactly balancing the actions of the pieces.
You at exactly right. The "paradox" is that people naively assume that the Real ("Real" = math theory, "real" = CS turing-computable
and physics "real universe") numbers are a smooth continuum, but if you follow the actual definition, you discover that it is too powerful -- the Real numbers could construct physically impossible objects, which proves that real numbera aren't real. In reality, we must confine ourselves to countable sets, which has ugly asymmetries: we must distinguish a set of "nameable" numbers as more "real" than the others, but we can choose any small-enough subset of Real numbers we want, we can choose every single Real number we practically encounter, but we can't choose all of them at the same time.
I'm not sure in what sense you mean real numbers could "construct" anything, but you can make physically impossible shapes out of rational numbers too, so their cardinality has nothing to do with that.
Your final question is very interesting, it gets at the heart of the matter: the Banach-Tarski theorem shows us that in your universe of infinite points and no atoms, there does not exist a definition of volume which is consistent. So in this universe the water level in a tub cannot be defined.
The four pieces don't have measure, don't have unambiguous volume. It might help to think of them as fractals (though in fact they're even more ambiguous than that).
The only "realistic" answer I can give is that each of the four pieces would "take up" the full volume of the sphere as you moved them apart. Each of the four pieces is "shaped" like the whole sphere, but with infinitesimally small holes where the other four go. So not a single "water droplet" would ever go inside the four pieces, and while you were moving them around they'd take up the space of four of the original sphere.
How are they more ambiguous than fractals? The key structure is a shape that can be rotated onto a subset of itself.
I guess with common fractals, you camnoly scale an object onto a subset of itself. The magic of BT is that the ball-shape allows a rotation that looks like a scaling.
A fractal tends to be kind of contiguous - for most points in the fractal there's a neighbourhood of those points that's also in the fractal, and similarly for points not in the fractal. These are "dust sets", dense everywhere in the sphere and their complements also dense.
Fractals generally have a formula describing them, i.e. they are computable. Nonmeasurable sets are necessarily non-computable. This is why you will never see an accurate picture of the B-T sets, because they are not defined uniquely by any formula or algorithm (which is connected to the fact that the axiom of choice is needed to even prove these sets exist).
... I think I've managed to get my head round it. Or, at least form an initial model.
It's like the old puzzle of the hotel with infinite rooms, each one labelled with an integer.
Then, another infinite set of guests turn up. The hotel manager asks every person currently in the hotel to double their room number and go to that one. Because there are infinite even numbers, this works fine.
Then, the new set of infinite guests sleep in the odd number rooms.
If I'm right, of course, then this only works because infinity is really odd - and you couldn't get an infinity room hotel in the real world.
Short answer, yes that's a good way to think about it. The Hilbert Hotel paradox is closely related to the B-T paradox, it only adds a few more complications to make a stronger statement.
The key step in the B-T paradox is setting up a two-dimenional free group in just three-dimensional rotations. This is an infinite discrete set, a sort of analogue of the integers (which is the one-dimensional free group). In the integers it is no surprise (or maybe it is) that you can take the positive integers and shift them all to the right by one unit and then suddenly you have made a "hole" from apparently nothing (the Hilbert's hotel paradox). One worrisome part of this is that the "rooms" must get infinitely far away, but B-T exploits the fact that by using irrational rotations you an wrap the whole construction into the space taken up by a single sphere.
Add just a little more mathematical mumbo jumbo and you have a two-dimensional free group, which is a hotel such that every room is as big as all the rest of the rooms and you can set up the hotel so that each room is as big as all the rest of the rooms, and then you can do magic such that a move to the left actually leaves as much space as the whole hotel. And since it's all wrapped up in a ball it doesn't even run off to infinity to do this.
You do need the rooms to be points with zero volume, though; otherwise even with one-atom rooms you still need to shove the entire infinite hotel into a finite space and you will run out of room.
We use sets to model the real world. They do remarkably well, and the math we've developed to work on them includes calculus to make bridges that stay up, fluid dynamics that make aeroplanes fly, and discrete math that helps us understand routing, scheduling, and all sorts of stuff.
We use the real line to model distances. In the real world we are limited as to the accuracy we can use, but modelling those limitations is nasty. It's easier to assume that things are continuous. In its turn, we make choices that make working with these models easier, and they turn out to be amazingly useful.
But then we start poking the dusty corners. The choices we make in the development of the theory have consequences, and math is about exploring both the choices, and their consequences.
So we can choose that between every two numbers there's another number. We can choose that there's a number whose square is 2. We can choose that the sum of the reciprocals of squares : 1+1/4+1/9+1/16+1/25+1/36+ ... : is a real number.
And we can choose that there is no smallest positive number. That has a consequence. If you believe that there is no smallest positive number, then 0.99999... has to equal 1. You can't have one without the other.
So we can talk about "the length of a line." Then we can talk about the "length" of a set of points on a line. Then we start to find that these simplistic models, these obvious and natural choices, even though they are amazingly useful have some unexpected consequences.
Does that help you to understand the context?
I'd be really interested in developing an agreed dialog about this. Will you send me an email?