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Originally posted by Ghoser777: Actually it was infinite divisibility. Sure, you were breaking up the circle into 5 (or four) sections, but the edges weren't exactly your everyday cuts. They were ragged, and probably impossible in terms of real world matter. |
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.
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| I'm just not sure they assumed the axiom of choice to be true when presenting the problem. |
They did assume the Axiom of Choice to be true. They were then (basically) attempting to disprove it by contradiction.
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| If the axiom of choice turns out to not be true in the given situation, then the paradox is invalid in any system where the axiom of choice is valid. |
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...