I agree that there are a ton of strange things you can do with infinity. There's the Cantor set, the cone which can hold a finite amount of paint but would take an infinite amount of paint to cover, etc, etc. But these are in a completely different set than the Banach-Tarski paradox.
Are you so sure?
From http://www.kuro5hin.org/story/2003/5/23/134430/275
The thing is, the Banach-Tarski paradox doesn't involve infinity and lives in a completely different realm of mathematics. Plus, it's a paradox meant to prove something false, unlike the others.
The ones you mention are indeed interesting, but they are fundamentally different than the B-T.
They might be fundamentally different, I do not know. But here is another quote from the article that I think helps my point:
I really think this article is a good read
Don't you mean "arc" instead of "circle"? Anyway, what you say is not entirely true. I think you are talking about transcending different coordinate systems and realities. The B-T paradox does neither. An axiom is an axiom and is always true. If you are talking about coordinate system A, then the axiom must say as much. An axiom can't be true in some situations and false in others. The B-T paradox is the same. It can't be true sometimes and false other times. It either is, or isn't.
"Great Circle" is the technical term for a straight line on a sphere. Considering we're not really talking about the Sphere in free space, but instead as THE space, a circle is an okay word to use.
An axion is not always true. An axiom is only true if you assume it to be true. Here's an axiom:
"All men are pigs"
This axiom could be true in general, I don't know. But we only know it to be true if we assume it to be so.
Now. once you've assumed your axiom, it can only be false if you're mathematicaly system is inconsitent (you've made two contrary assumptions, something that happens a lot in proofs by contradiction).
You can develop different mathematical systems (groups, fields, etc) based on what axioms you assume to be true. The common field axioms doesn't apply to the set of all 3X3 matricies (no multiplicative commutitivity) for example.
Matt Fahrenbacher
Are you so sure?
From http://www.kuro5hin.org/story/2003/5/23/134430/275
Quote:
| In fact, if we assume that spheres are not infinitely divisible, then the Banach-Tarski paradox doesn't apply, because each of the "pieces" in the paradox is so infinitely complex that they are not "measurable" (in human language, they do not have a well-defined volume; it is impossible to measure their volume). Immeasurable pieces can only exist if the sphere can be cut into infinitely-detailed pieces; this obviously isn't true for real spheres, since you cannot cut atoms into arbitrary shapes, especially not into infinitely complex shapes. |
The thing is, the Banach-Tarski paradox doesn't involve infinity and lives in a completely different realm of mathematics. Plus, it's a paradox meant to prove something false, unlike the others.
The ones you mention are indeed interesting, but they are fundamentally different than the B-T.
They might be fundamentally different, I do not know. But here is another quote from the article that I think helps my point:
Quote:
| (Before you dismiss this notion outright, let me state that mathematically infinite objects do not always behave intuitively. As a comparison, we use a more intuitive example of duplicating the set of integers: given N, the (infinitely large) set containing all the integers, we can split them up into two sets, E containing all the even integers, and F containing all the odd integers. Are E and F each smaller than N, the set of all integers? Intuitively, it appears to be so; however, I will convince you that they are, in fact, the same size. First, we take E, and rename each member of E so that a number x is renamed to x divided by two. What do we get? We now find that E=N. Similarly, we take each member y from F, and rename y to (y-1)/2. Whoopie, we also find that F=N. We have just duplicated the set of integers using nothing more than just the original integers. We didn't even need to use infinitely-divisible freak objects to achieve this.) |
I really think this article is a good read
Don't you mean "arc" instead of "circle"? Anyway, what you say is not entirely true. I think you are talking about transcending different coordinate systems and realities. The B-T paradox does neither. An axiom is an axiom and is always true. If you are talking about coordinate system A, then the axiom must say as much. An axiom can't be true in some situations and false in others. The B-T paradox is the same. It can't be true sometimes and false other times. It either is, or isn't.
"Great Circle" is the technical term for a straight line on a sphere. Considering we're not really talking about the Sphere in free space, but instead as THE space, a circle is an okay word to use.
An axion is not always true. An axiom is only true if you assume it to be true. Here's an axiom:
"All men are pigs"
This axiom could be true in general, I don't know. But we only know it to be true if we assume it to be so.
Now. once you've assumed your axiom, it can only be false if you're mathematicaly system is inconsitent (you've made two contrary assumptions, something that happens a lot in proofs by contradiction).
You can develop different mathematical systems (groups, fields, etc) based on what axioms you assume to be true. The common field axioms doesn't apply to the set of all 3X3 matricies (no multiplicative commutitivity) for example.
Matt Fahrenbacher