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The Point of the Banach-Tarski Theorem

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Re: The Point of the Banach-Tarski Theorem

#61

Post by harry8 »

mikewarot wrote: post #55

Give every point in the Sphere B a room in the infinity hotel. If the room number is odd, it belongs to sub-sphere B1, otherwise sub-sphere B2. Odd things happen when you divide by infinity.

This has always been my struggle with all of this "infinity" maths.

Infinity/Infinity is anything you like and many things you don't and these things always seem to boil down to dividing by infinity.

Visit each room in the hotel in turn, each one for half the total time spent going to and visiting the last. Without explanation of why we can divide infinity by infinity when we want to and not get total garbage.

Banach tarski, infinity/infinity = 2 because the numerator is exactly equaly to (infinity 2) then you cancel the infinity from top and bottom. Also (infinity 3) so clearly 2=3 and this is useful. infinity also equals infinity * infinity so 1 = infinity. As you say odd things happen when you divide by infinity and mostly they don't seem to be helpful things. They seem to be wholly invalid things. But not here?

I'm sure that's not it and I'm totally missing the point, but that point is being skipped over and handwaved away with a sneer and a muttering of "mathematical maturity" rather often.

Re: The Point of the Banach-Tarski Theorem

#62

Post by kgwgk »

theemathas wrote: post #48
thaumasiotes wrote: post #45

Earlier quoted context omitted.

> Actually, an ever better example is: "The set of reals that are not the solution to any equation that can be written with a finite number of symbols." -- an infinite set that has no nameable members! That's not a good example; the problem you're creating is due to sloppy use of language, not any cleverness in the definition. All real numbers, and all numbers of any other variety, can be written with a finite number…

> All real numbers, and all numbers of any other variety, can be written with a finite number of symbols. This is false. In some sense, there exist numbers that can't be referred to. We can refer to the set of real numbers as a whole, but not some of the elements. https://en.wikipedia.org/wiki/Definable_real_number But then there are issues with defining "definable numbers", which complicates things by a lot. https:/…

Apparently by “finite number of symbols” he means, for example, {0 1 2 3 4 5 6 7 8 9 .} but he allows the representation of a number to contain infinitely many of them.

Re: The Point of the Banach-Tarski Theorem

#63

Post by ThrustVectoring »

mikewarot wrote: post #55

Give every point in the Sphere B a room in the infinity hotel. If the room number is odd, it belongs to sub-sphere B1, otherwise sub-sphere B2. Odd things happen when you divide by infinity.

The points in the sphere are uncountably many - you can't give each one a room number.

Re: The Point of the Banach-Tarski Theorem

#64

Post by rocqua »

The article states that all known versions of these 'paradoxes' arise from the axiom of choice. But I believe it isn't known yet whether denying the axiom of choice categorically prevents these paradoxes and allows 'perfect' measures to exist.

Anyone care to clarify?

Re: The Point of the Banach-Tarski Theorem

#65

Post by pfortuny »

rocqua wrote: post #64

The article states that all known versions of these 'paradoxes' arise from the axiom of choice. But I believe it isn't known yet whether denying the axiom of choice categorically prevents these paradoxes and allows 'perfect' measures to exist. Anyone care to clarify?

Yes, see the math overflow question quoted above. Not choice gives you measures where all sets are measurable.

Re: The Point of the Banach-Tarski Theorem

#66

Post by plopilop »

thaumasiotes wrote: post #56
jiggawatts wrote: post #49

Earlier quoted context omitted.

> All real numbers, and all numbers of any other variety, can be written with a finite number of symbols. That's what it means to give something a name. Counter-intuitively, this is not true. The vast, vast majority of real numbers cannot be named, not even in principle. Their definitions would have to be infinitely long. Or to put it another way, no matter how close two named numbers are, there is an infinite number…

> The vast, vast majority of real numbers cannot be named, not even in principle. Their definitions would have to be infinitely long. So what? That doesn't stop them from being named.

Usually definitions are considered to be finite. Otherwise yes you can just name a number by enumerating its decimals but what's the point.

Edit : actually does not even work, courtesy of the Chaitin's numbers

Re: The Point of the Banach-Tarski Theorem

#68

Post by schoen »

harry8 wrote: post #61
mikewarot wrote: post #55

Give every point in the Sphere B a room in the infinity hotel. If the room number is odd, it belongs to sub-sphere B1, otherwise sub-sphere B2. Odd things happen when you divide by infinity.

This has always been my struggle with all of this "infinity" maths. Infinity/Infinity is anything you like and many things you don't and these things always seem to boil down to dividing by infinity. Visit each room in the hotel in turn, each one for half the total time spent going to and visiting the last. Without explanation of why we can divide infinity by infinity when we want to and not get total garbage. Banach…

What I got from this post is that the weirdness here comes in, not so much from the Hilbert's Hotel phenomenon about cardinalities and sets that can be put into correspondence with their own proper subsets, but from looking more deeply at something relatively familiar and something that we ordinarily use to tame infinities: volume.

Even though there are infinitely many real numbers in [0,1], we have the idea that the unit cube [0,1]³ should have a finite volume of exactly 1, or the unit sphere { (x,y,z) | √(x²+y²+z²)≤1 } should have a finite volume of exactly 4π/3. Or indeed the unit line segment [0,1] should have a finite length of exactly 1, even though it contains infinitely many points.

This stuff has felt totally normal and appropriate in mathematics ever since Euclid: Euclid would probably agree that you can't count how many points are in a line segment, but still endorses talking about lengths of line segments (or at least ratios of lengths of different line segments).

While it feels like we know how to work with volumes, and that they're comfortably finite and well-behaved, things like Banach-Tarski suggest that if we want to have every set of points have a well-defined volume in any given number of dimensions, we're actually going to run into bad trouble. But the article suggests that there are several ways to avoid this trouble, including just saying that some weird geometric objects don't have a defined n-dimensional volume. Instead, maybe only some "nice" sets should have one?

Re: The Point of the Banach-Tarski Theorem

#69

Post by Smaug123 »

thaumasiotes wrote: post #56
jiggawatts wrote: post #49

Earlier quoted context omitted.

> All real numbers, and all numbers of any other variety, can be written with a finite number of symbols. That's what it means to give something a name. Counter-intuitively, this is not true. The vast, vast majority of real numbers cannot be named, not even in principle. Their definitions would have to be infinitely long. Or to put it another way, no matter how close two named numbers are, there is an infinite number…

> The vast, vast majority of real numbers cannot be named, not even in principle. Their definitions would have to be infinitely long. So what? That doesn't stop them from being named.

You're using "being named" in a very unusual sense. Most people would consider you not to have named something if you're literally never allowed to stop speaking (or writing) the name. Most reals "have infinitely long names" under your definition, and there is no finite time at which you've distinguished between reals with the same initial segment of "name", so really what's the use of the naming scheme at all?

(For a more compelling argument, the equality of real numbers is uncomputable, so it must be impossible to name them. If we could name them usefully, then we could determine equality by asking which ones had different names.)

Re: The Point of the Banach-Tarski Theorem

#70

Post by Smaug123 »

harry8 wrote: post #61
mikewarot wrote: post #55

Give every point in the Sphere B a room in the infinity hotel. If the room number is odd, it belongs to sub-sphere B1, otherwise sub-sphere B2. Odd things happen when you divide by infinity.

This has always been my struggle with all of this "infinity" maths. Infinity/Infinity is anything you like and many things you don't and these things always seem to boil down to dividing by infinity. Visit each room in the hotel in turn, each one for half the total time spent going to and visiting the last. Without explanation of why we can divide infinity by infinity when we want to and not get total garbage. Banach…

If I asked you to divide an apple by a stone, what would your answer be? Infinity is simply not a thing that your carefully-honed real-world intuitions about division apply to. Hilbert's hotel is precisely a thought experiment to prove this: divide an infinitely large set among infinitely many people, and you can get many different results. Any true statement about infinities you construct with division is true only by coincidence, not for the usual maths reason that things are true (namely that they are the result of following some broadly-applicable rules).

There is no number called "infinity", so there's no reason to expect the rules of the arithmetic of numbers to apply to it; you have to carefully go and prove that the rules you want to apply have some meaning when you extend the language in this way, and moreover that these rules are true. For example, there are multiple meaningful ways to add infinite quantities, e.g. depending on whether you are considering the cardinals or the ordinals. Those concepts are the same in finite-land, but are different in infinite-land. Cardinals do not admit the notion of "division"; ordinals at least have the division algorithm.

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