41–50 of 116 posts
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 on a contraction?
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…
https://www.youtube.com/watch?v=uFvokQUHh08
Instead of saying "Barbara Streisand", they say "Banach-Tarski" and a number of oranges double:
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…
* The boundary of the pieces has to be unmeasurable, in that sense it's similar to Vitali's set[0];
* No, the pieces are more-or-less a "fog" of points.
* I don't know why you have "proofs" in quotation marks, and I don't know what you mean by the second half of the sentence (even if I replace "contraction" with "contradiction").
I do feel like a careful reading, or perhaps re-reading, would let you ask more specific questions that we can help you with.
As the author points out that Banach-Tarski theorem is an example of hard-to-accept result that comes out of the easy-to-accept axiom of choice. There is a popular quote that related to this: > The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma? From https://en.wikipedia.org/wiki/Axiom_of_choice Axiom of choice, the well-ordering principle and Zorn'…
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…
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 of symbols. That's what it means to give something a name.
I asked a mathematician about what a wacky conclusion it is. He said that whenever you allow infinity, you get results like that. It relies on uncountably-infinite division of an object, which corresponds to no real-world experience anywhere in the universe. Real objects have, you know, atoms. We use real numbers a lot, but we are careful never to rely on their more extreme properties anywhere it would matter. In pra…
As the author points out that Banach-Tarski theorem is an example of hard-to-accept result that comes out of the easy-to-accept axiom of choice. There is a popular quote that related to this: > The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma? From https://en.wikipedia.org/wiki/Axiom_of_choice Axiom of choice, the well-ordering principle and Zorn'…
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…
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…
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://mathoverflow.net/questions/44102/is-the-analysis-as-...
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…
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 of reals in between them. If you say, okay, sure, but some of those might be named, then pick the two closest and then there is still an infinite number of other reals in between those two!
Another way to look at it is that the amount of information (measured in bits) between any two real numbers is literally infinite. If the reals were represented with binary digits, then a sequential subset of them would have a common finite prefix, and then all possible infinite bit strings would be the suffixes!
I asked a mathematician about what a wacky conclusion it is. He said that whenever you allow infinity, you get results like that. It relies on uncountably-infinite division of an object, which corresponds to no real-world experience anywhere in the universe. Real objects have, you know, atoms. We use real numbers a lot, but we are careful never to rely on their more extreme properties anywhere it would matter. In pra…
The main reason that we use real numbers is because we need real numbers to do calculus, at least in the normal way. And to do physics, we need to do calculus. You can keep approximating integrals as sums of very large number of terms, but it gets unwieldy to handle symbolically.