The Point of the Banach-Tarski Theorem
solipsys.co.uk
The Point of the Banach-Tarski Theorem
1–10 of 116 posts
Re: The Point of the Banach-Tarski Theorem
#2I always feel part of the confusion with Banach-Tarski is that lots of words don't use their "natural definitions", which makes the proof more surprising. People (not this article) often talking about "cutting" a sphere, which is really misleading.
This result is, in many ways, quite similar to the idea I can "cut" the integers into the odd integers and even integers (but with many more fine details).
This is still a nice article, which explains the actual result well.
Re: The Point of the Banach-Tarski Theorem
#3As 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's lemma are equivalent statements (any one proves the other two). But each has a very different "believability" feel to it.
Re: The Point of the Banach-Tarski Theorem
#4Here is a potentially daft question that I nonetheless would appreciate if someone could answer. Is it possible to deny the axiom of choice for the purposes of measures while accepting it for vector spaces? I am wondering if you could say, "there are two kinds of sets, ones equipped with a choice function and ones without it, and measurable sets are of the latter kind."
Re: The Point of the Banach-Tarski Theorem
#5I 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 practice, in fact, we use floating-point numbers, not reals, when doing actual calculations, and use numerical analysis to stay well clear of nonsensical results. If you tried to rely on BT in a real calculation, you would find a lot of NaNs and Infs.
Re: The Point of the Banach-Tarski Theorem
#6Here is a potentially daft question that I nonetheless would appreciate if someone could answer. Is it possible to deny the axiom of choice for the purposes of measures while accepting it for vector spaces? I am wondering if you could say, "there are two kinds of sets, ones equipped with a choice function and ones without it, and measurable sets are of the latter kind."
If you accept that _all_ vector spaces have a (Hamel) basis, you can then prove the Axiom of Choice: http://www.math.lsa.umich.edu/~ablass/bases-AC.pdf
This means if you want to deny the Axiom in some cases, you will also have to allow for the existence of vector spaces without a basis.
Re: The Point of the Banach-Tarski Theorem
#7Here is a potentially daft question that I nonetheless would appreciate if someone could answer. Is it possible to deny the axiom of choice for the purposes of measures while accepting it for vector spaces? I am wondering if you could say, "there are two kinds of sets, ones equipped with a choice function and ones without it, and measurable sets are of the latter kind."
Not in a meaningful way I think. I mean you could weaken it to 'all finite vector spaces have a basis', but I think regular induction is enough to prove that, you don't need the axiom of choice.
Re: The Point of the Banach-Tarski Theorem
#8Here is a potentially daft question that I nonetheless would appreciate if someone could answer. Is it possible to deny the axiom of choice for the purposes of measures while accepting it for vector spaces? I am wondering if you could say, "there are two kinds of sets, ones equipped with a choice function and ones without it, and measurable sets are of the latter kind."
If you accept that _all_ vector spaces have a (Hamel) basis, you can then prove the Axiom of Choice: http://www.math.lsa.umich.edu/~ablass/bases-AC.pdf This means if you want to deny the Axiom in some cases, you will also have to allow for the existence of vector spaces without a basis.
[deleted]
Re: The Point of the Banach-Tarski Theorem
#9I 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…
In the real world we have atoms, but atoms are disturbances in the wave function of the universe, which may not be discrete.
We know atoms pop in and out of existence in matter anti-matter pairs. Maybe the universe is performing Banach Tarski under the hood, to make something from nothing?
> we use floating-point numbers, not reals, when doing actual calculations
I don't believe only calculations made by computers to be "actual calculations". Humans were calculating thousands of years before computers were invented, using integer or real numbers (e.g pi).
Re: The Point of the Banach-Tarski Theorem
#10I 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…
Atoms move according to path integrals which would be at least countably infinite in number even if time and space were discrete because they are allowed to loop back on themselves.