The Axiom of Choice (AC) is a controversial topic in mathematics primarily due to its independence from other axioms of set theory and its implications, such as the Banach-Tarski Paradox. The paradox demonstrates that a solid ball can be decomposed into a finite number of non-measurable sets and reassembled into two solid balls of the same size, which challenges intuitive notions of volume and measure. While AC is deemed useful for proving the existence of certain mathematical objects, its acceptance varies among mathematicians, particularly in practical applications where the Axiom of Countable Choice (ACC) suffices.
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I don't know you could call it controversial. The issue is that it is an independent axiom.RandomAllTime said:Hi guys. So I've been wondering, what's so controversial about the axiom of choice? I heard it allows the Banach-Tarski Paradox to work. A little insight would be much appreciated, thanks.
I see. I guess it's because I heard that it sort of let's the Banach Tarski Paradox hold true. Thanks for the link.mathman said:I don't know you could call it controversial. The issue is that it is an independent axiom.
https://en.wikipedia.org/wiki/Axiom_of_choice
I see. Thanksgill1109 said:If you like Banach Tarski then you like Axiom of Choice. If you don't like Banach Tarski then you are free to trash the axiom of choice, and now no Banach Tarksi.
Axiom of Choice is usually thought to be a useful thing in mathematics since (a) it seems intuitive (b) it makes it easier to prove theorems claiming that certain things exists. Well that is fine as long as those are things you kind of like to exist, but at some point it also starts allowing things to exist which see counter-intuitive, and maybe you don't like that.
For most of practical mathematics, the axiom of countable choice is quite enough to do everything you want to do. https://en.wikipedia.org/wiki/Axiom_of_countable_choice
And you even need it to make sure that the characterisation of epsilon-delta defined convergence in terms of sequences is indeed a true theorem.
Banach Tarski requires the existence of uncountably-many atoms, which does not hold " in this universe" . And, AFAIK, it requires infinitely-many operations.RandomAllTime said:Hi guys. So I've been wondering, what's so controversial about the axiom of choice? I heard it allows the Banach-Tarski Paradox to work. A little insight would be much appreciated, thanks.
Isnt this equivalent to the existence of infinitely-many ( at least countably -) atoms? And isn't the cardinality of the operations resulting in the partition infinite?mathman said:Banach-Tarski uses a mathematical fact that the number of points in a sphere is uncountable, and with the axiom of choice it can be divided into a finite number of unmeasurable sets.
No, atoms do not form a continuum. In any case you cannot "prove" anything about the real world using maths.WWGD said:Isnt this equivalent to the existence of infinitely-many ( at least countably -) atoms?
I mean one can argue reasonably -well that a ball containing uncountably-many points will contain infinitely-many atoms. But , yes, this would have to be laid out carefully.MrAnchovy said:No, atoms do not form a continuum. In any case you cannot "prove" anything about the real world using maths.
WWGD said:Isnt this equivalent to the existence of infinitely-many ( at least countably -) atoms?
And isn't the cardinality of the operations resulting in the partition infinite?
I mean the number of steps needed to do the partition.micromass said:Yes, the Banach-Tarski paradox assumes the existence of uncountably many atoms. Although the word atom is confusing, since it has nothing to do with the real world atoms. Here, atom is just an indivisible point with zero volume.
I don't really know what you mean with this.
Because we are assumming it is a physical , "real world" ball. How else would we go from a standard ball into the collection of non-measurable pieces?micromass said:Why would you need to transform the ball to partition it??
If it were possible, I would be rich by now, buying .1 oz of gold and doubling its volume many times. I don't know if there are physical models of non-measurable sets.micromass said:I don't know, but Banach-Tarski isn't about how you would do it in practice. It involves the axiom of choice and thus an infinite amount of choices which is impossible in the real world anyway.
WWGD said:If it were possible, I would be rich by now, buying .1 oz of gold and doubling its volume many times.
Well, maybe contrived, but if I can come up with a way and convince someone of it, pretty sure I can borrow enough to have it done. But this may be far OT. And this practically feasible aspect has to see with the fact that this cannot be done in a finite number of steps, if at all. EDIT: maybe tautological, but if it could be done in a number of steps, it would be feasible.micromass said:Not necessarily. Just because it is possible doesn't mean it's practically feasible. Physics still doesn't know whether there are nonmeasurable sets out there. So they might still exist.
glaucousNoise said:so, if we are an ultrafinitist, we needn't bother with the axiom of choice?