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The Point of the Banach-Tarski Theorem

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Re: The Point of the Banach-Tarski Theorem

#71

Post by Smaug123 »

ur-whale wrote: post #41

So, Banach-Tarski says you can split a sphere of volume S into a finite number of pieces and reassemble the pieces into two spheres of volume S. Is the proof constructive? As in: does the proof actually show how to build the pieces? If yes, is the boundary of the pieces of measurable surface? Can the pieces be rendered in 3D? Or is it just another one of those "proofs" where if the set of pieces doesn't exist we land…

Moreover, there cannot be a constructive proof using this construction, because the Banach-Tarski construction will fail if all sets are measurable (which can happen in the absence of Choice). A constructive proof would continue to succeed in such settings.

Re: The Point of the Banach-Tarski Theorem

#72

Post by msla »

soVeryTired wrote: post #26

Mathoverflow has some good discussion of Banach-Tarski [0]. The top-rated posts argue that - The 'problem' with B-T may be related to our notion of "space" (i.e. point-set topology) rather than any issues with the axiom of choice - Most practical uses of the axiom of choice could get by with the axiom of countable choice, under which B-T doesn't hold - Mathematics need not directly model the 'real' world or match our…

> Mathematics need not directly model the 'real' world or match our physical intuition

There's so many other ways to demonstrate this. For example, it's possible to map the integers to the rational numbers one-to-one, which is a good definition for "as many as" when dealing with infinite sets, even though there are infinitely many rational numbers between any two rational numbers. Moreover, there are as many integer multiples of a trillion as there are integers total.

This is to do with set cardinality:

https://en.wikipedia.org/wiki/Cardinality

... which again comes back to the axiom of choice:

> If the axiom of choice holds, the law of trichotomy holds for cardinality. Thus we can make the following definitions:

> * Any set X with cardinality less than that of the natural numbers [...] is said to be a finite set.

> * Any set X that has the same cardinality as the set of the natural numbers [...] is said to be a countably infinite set.[10]

> * Any set X with cardinality greater than that of the natural numbers [...] is said to be uncountable.

(... with some mathematical notation that doesn't work here elided)

Re: The Point of the Banach-Tarski Theorem

#73

Post by harry8 »

schoen wrote: post #68
harry8 wrote: post #61

Earlier quoted context omitted.

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…

>...we have the idea that the unit cube [0,1]³ should have a finite volume of exactly 1...

>...even though it contains infinitely many points.

There is the division by infinity.

Cut the volume in half. Still has infinite points. infinity/2 = infinity.

1 = 1/2, or 1=2 if you can cancel those.

Any set of points is some number of points 1,2, ..infinity. Points are infinitely small (division by infinity) so you'd have to infinitely many of them to have a volume other than zero. And you're back to infinity points * (1/infinity) vol of a point.

So yeah, no way I can see to have any set of points have a well defined volume because the volume of a point is a division by infinity, unless the set is finite in which case it is zero (or whatever you define a finite number divided infinitely to be - define it to be gerald if that helps? - Mathematical immaturity on display right there).

Re: The Point of the Banach-Tarski Theorem

#74

Post by schoen »

harry8 wrote: post #73
schoen wrote: post #68

Earlier quoted context omitted.

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…

>...we have the idea that the unit cube [0,1]³ should have a finite volume of exactly 1... >...even though it contains infinitely many points. There is the division by infinity. Cut the volume in half. Still has infinite points. infinity/2 = infinity. 1 = 1/2, or 1=2 if you can cancel those. Any set of points is some number of points 1,2, ..infinity. Points are infinitely small (division by infinity) so you'd have to…

Yes, I see your point (no pun intended), but I would still maintain that this issue didn't necessarily bother mathematicians in the days before formalized set theory. The issue you mention is a reason that any definition of volume should not be based only on set cardinality, since there are those one-to-one correspondences with proper subsets (for infinite sets of points), yet volume should not be preserved by cutting something in half, or doubling something, in an ordinary way.

That is, if X is the unit line segment and Y is a pair of unit line segments joined end-to-end, we want the measure function µ to obey µ(Y) = 2µ(X), even though |X|=|Y| in set theory. And even though that's weird, formalized mathematics didn't get existentially ambitious enough to make anything truly bizarre out of it, I suppose, from Euclid all the way up until Vitali!

Re: The Point of the Banach-Tarski Theorem

#75

Post by quchen »

yellowapple wrote: post #34

"So for those of you who don't know the result, here it is in simple, non-technical terms:" proceeds to immediately use a character that can't even be copy/pasted due to needing MathJax to render it, refuses to elaborate further Thankfully, it ain't hard to find the Banach-Tarski theorem in actual simple, non-technical terms, so for those wondering what in tarnation that character is: it just means three-dimensional…

Had a laugh about that too. It’s sometimes hard to tell what’s »non-technical« when you’ve spent a lot of time on something, and even then what it means to be »technical« depends a lot on the audience. Non-technical here means knows basic math but is not familiar with measure theory; it does not mean _anyone_.

I like that they added my favorite math joke in their (linked at the very top) limited audience jokes section [1] (bottom left, starts with »Two mathematicians are in a bar«), which is also a about what basic math knowledge is.

[1]: https://www.solipsys.co.uk/new/LimitedAudienceJokes.html

Re: The Point of the Banach-Tarski Theorem

#76

Post by housecarpenter »

thaumasiotes wrote: post #45
jiggawatts wrote: post #33

Earlier quoted context omitted.

The Axiom of choice has never felt completely self-evident to me. E.g.: what if you have sets where the elements are non-computable? How do you "choose" objects that cannot even be named ? Something like: "the set of all programs that cannot be proven to halt" and the like can be used to create pathological sets where the set itself obviously exists, but you cannot name any of the members. Actually, an ever better ex…

> 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…

It's true that any real number can be written with a finite number of symbols. In fact, I can write any real number x with exactly one symbol, namely x, if I define x to mean that particular real number.

Now if you fix a particular formal language for defining real numbers, with a finite alphabet, then the language only has countably many words, hence there are reals not definable in the language. But the notion of "definability" here is not independent of the choice of formal language.

So "choosing an undefinable real number" amounts to "choosing a real number not definable in L", where L is some fixed formal language---and this isn't particularly hard to imagine; given a specific L, you can probably quite concretely construct a real number not definable in L.

Re: The Point of the Banach-Tarski Theorem

#77

Post by thaumasiotes »

Smaug123 wrote: post #69
thaumasiotes wrote: post #56

Earlier quoted context omitted.

> 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…

> You're using "being named" in a very unusual sense.

No, I'm not.

> Most people would consider you not to have named something if you're literally never allowed to stop speaking (or writing) the name.

What's the connection? Names do not completely describe their referent. They're names. I know someone named Ko. Given that name, what can you tell me about Ko?

Re: The Point of the Banach-Tarski Theorem

#78

Post by jiggawatts »

thaumasiotes wrote: post #77
Smaug123 wrote: post #69

Earlier quoted context omitted.

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…

> You're using "being named" in a very unusual sense. No, I'm not. > Most people would consider you not to have named something if you're literally never allowed to stop speaking (or writing) the name. What's the connection? Names do not completely describe their referent. They're names. I know someone named Ko. Given that name, what can you tell me about Ko?

Naming a number in mathematics has different rules to assigning names to objects in the physical world.

You can "name" a number such as: "a positive number, that when multiplied by itself, the result is 2".

This is a finite statement that requires only a handful of 'bits' to represent, but it exactly and uniquely identifies the square root of two. If written with decimal digits, then the square root of two would require an infinite number of digits, but it can be defined ('named') with a finite number of bits.

It turns out that this is a rare property for real numbers. The vast majority do not have a shorthand name like this, only the infinitely long description exists.

Re: The Point of the Banach-Tarski Theorem

#79

Post by andrewaylett »

camjw wrote: post #25
moloch-hai wrote: post #20

Earlier quoted context omitted.

That is a symbolic manipulation. Wherever it comes down to actual numbers, you use adequate approximations to infinite summations for x and iy. Even nominally exact rational values are often idealizations of measurements: your house has no actual right angles, but eh, close enough.

It feels to me like you're redefining what a number is to be very different to what anyone with a maths background would say a number is. Essentially you're saying that neither e nor pi are numbers?

The only numbers that may exist in nature are the non-negative rationals. Negative numbers are widely useful, but they don't exist in any meaningful way: they're an abstraction over reality. Irrational numbers definitely don't exist, and don't get me started on so-called "imaginary" numbers -- they're no less present in nature than the negative numbers are.

But the concepts are useful, and the symbols have meaning. The article mentions normal words having different meanings in mathematical language, and that impacts even every-day use of maths.

Re: The Point of the Banach-Tarski Theorem

#80

Post by smcl »

yellowapple wrote: post #34

"So for those of you who don't know the result, here it is in simple, non-technical terms:" proceeds to immediately use a character that can't even be copy/pasted due to needing MathJax to render it, refuses to elaborate further Thankfully, it ain't hard to find the Banach-Tarski theorem in actual simple, non-technical terms, so for those wondering what in tarnation that character is: it just means three-dimensional…

I did get a chuckle at that, like the running joke about how a monad is “just a monoid in the category of endofunctors”. To be fair, you could delete the “R^3” part and the explanation can still be understood.
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