# Axiom of Choice

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1. Aug 23, 2015

### RandomAllTime

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.

2. Aug 23, 2015

### mathman

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

3. Aug 23, 2015

### RandomAllTime

I see. I guess it's because I heard that it sort of lets the Banach Tarski Paradox hold true. Thanks for the link.

4. Sep 2, 2015

### gill1109

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.

5. Sep 23, 2015

### RandomAllTime

I see. Thanks

6. Sep 24, 2015

### WWGD

Banach Tarski requires the existence of uncountably-many atoms, which does not hold " in this universe" . And, AFAIK, it requires infinitely-many operations.

7. Sep 24, 2015

### mathman

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.

8. Sep 24, 2015

### WWGD

Isnt this equivalent to the existence of infinitely-many ( at least countably -) atoms? And isnt the cardinality of the operations resulting in the partition infinite?

9. Sep 25, 2015

### MrAnchovy

No, atoms do not form a continuum. In any case you cannot "prove" anything about the real world using maths.

10. Sep 25, 2015

### WWGD

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.

11. Sep 25, 2015

### micromass

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.

12. Sep 25, 2015

### WWGD

I mean the number of steps needed to do the partition.

13. Sep 25, 2015

### micromass

What would be a step?

14. Sep 25, 2015

### WWGD

A transformation on the Ball , to decompose it into the nonmeasurable parts. Let me see how to define it more clearly.

15. Sep 25, 2015

### micromass

Why would you need to transform the ball to partition it??

16. Sep 25, 2015

### WWGD

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?

17. Sep 25, 2015

### micromass

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.

18. Sep 25, 2015

### WWGD

If it were possible, I would be rich by now, buying .1 oz of gold and doubling its volume many times. I dont know if there are physical models of non-measurable sets.

19. Sep 25, 2015

### micromass

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.

20. Sep 25, 2015

### WWGD

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.