Quote:
|
Originally posted by Ghoser777: Sure, but you get all kinds of wacky things that don't make any sense in the real world with infinite divisibility - the only reason they seem wacky is because in the real world you don't have infinite divisibility for matter (although you do for time and such, or so we think). Case in point was the set with length 1 but measure 0 - the Cantor middle third set. What this paradox shows us is that with infinite divisibility, area and volume and the like are not necessarily as nicely defined and easy to think about as in the "real world." It only seems paradoxical from an intuitive stand point, because it does seem to imply that 2 equals 1, but the cantor set seems to imply that 1 equals 0 and that's not really paradoxical - it just assumes infinite divisibility. |
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.
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.
Quote:
| In any given situation, you can assume whatever axioms you want to be true. Like you can assume that the shortest distance between two points is a circle - then instead of talking about Euclidean geometry, you're talking about Spherical geometry. Essentially, based on what axioms you assume to be true, you get different results. So, all I was saying is an axiom is only true if we assume it to be so. There's plenty of different mathematical structures that are created where certain field, group, etc axioms are assumed to be true or false. |
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.
I think we've done a pretty good job of derailing this thread from "truth" to mathematics. It's all your fault, Stradlater!
