Gah, I had a response, and then iTunes and safari started freaking out together and hogging as much processor as they could 
Of course it is impossible in terms of real world matter. That's not the point. The point is that given our system of mathematics and the axioms we hold "true", it is possible.
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.
They did assume the Axiom of Choice to be true. They were then (basically) attempting to disprove it by contradiction.
Sounds good to me.
I'm not really sure what you mean by "given situation". An axiom, by definition, is always true. If it isn't, it isn't an axiom. If the Axiom of Choice turns out to not really be an axiom, then we are going to have to go back and re-prove a lot of things...
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.
Matt Fahrenbacher
Of course it is impossible in terms of real world matter. That's not the point. The point is that given our system of mathematics and the axioms we hold "true", it is possible.
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.
They did assume the Axiom of Choice to be true. They were then (basically) attempting to disprove it by contradiction.
Sounds good to me.
I'm not really sure what you mean by "given situation". An axiom, by definition, is always true. If it isn't, it isn't an axiom. If the Axiom of Choice turns out to not really be an axiom, then we are going to have to go back and re-prove a lot of things...
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.
Matt Fahrenbacher