3 ms·
As I understand it, the Banach-Tarski construction depends on the axiom of choice. And Banach-Tarski violates my intuitions so thoroughly that I think I might
by surrealize 14y ago
As I understand it, the Banach-Tarski construction depends on the axiom of choice. And Banach-Tarski violates my intuitions so thoroughly that I think I might prefer to dispense with the axiom of choice than accept B-T.
If you do give up the axiom of choice, is there a lot of cool/useful math that you'd also have to give up?
- flebron 14y agoIt is equivalent to, for instance, "every vector space has a basis", and to Zorn's lemma, which I've often used.
- ColinWright 14y agoIt says that the product of non-empty sets is always non-empty. Do you really want to give that up?
- lmm 14y agoHilbert's basis theorem is the big one, and the reason the axiom was invented. But you can get very far without it - you just have to be very careful about it, because the operation of choosing an arbitrary representative of each of many sets is so natural it can be hard to realize you've used the axiom.
- ionfish 14y agoLots. For example, that every set has a cardinality (is bijective with some aleph). That's pretty intuitive, too, as (I think) are the following. * Let X and Y be sets. Then either they have the same cardinality, or one is smaller than the other. * Let X be an infinite set. Then there is a bijection between X and the cartesian product of X with itself, X × X. * Tychonoff's theorem: every product of compact topological spaces is compact.
- sesqu 14y agoYour third example is often cited as an unintuitive result, so I wouldn't mind getting rid of it. The first and second are easy enough to consider collateral damage, which we already have plenty of in basic math. But the fourth one is harder to give up. What would it look like without the axiom?
- ionfish 14y agoComing from a set-theoretic perspective, I suppose I've got so used to Tarski's theorem that I consider it intuitive. As far as Tychonoff's theorem goes, you might find this paper interesting: http://matwbn.icm.edu.pl/ksiazki/fm/fm113/fm11313.pdf http://matwbn.icm.edu.pl/ksiazki/fm/fm113/fm11313.pdf
- CJefferson 14y agoDoes it bother you that by dividing all the integers into odd and even numbers, I can end up with two infinite sets, each of the same size as the original set? While the Banach-Tarski violates your intuitions, it is probably because it is more about how there are "multiple copies" of the real numbers in the real numbers, than anything resembling a real cut.
- surrealize 14y agoYou might be right. The more I learn about the real numbers, the less real they seem :/
- Adrock 14y agoNot to nitpick or assume you we're saying this, but doesn't the cardinality of the evens and odds equalling the cardinality of the integers hold even without the axiom of choice?
- Adrock 14y agoCurse you, iPhone keyboard and Autocorrect. s/we're/were/
- CJefferson 14y agoYes, the argument is more complicated, and does require the axiom of choice. However, to me the "basic principle" is a similar argument. The important thing (for me) was to understand this is an argument about slicing up infinities in an "Infinite comb" type way, rather than anything resembling a "cut".
- mjn 14y ago> If you do give up the axiom of choice, is there a lot of cool/useful math that you'd also have to give up? Reverse mathematics (http://en.wikipedia.org/wiki/Reverse_mathematics http://en.wikipedia.org/wiki/Reverse_mathematics) is a reasonably recent branch of mathematics aiming to investigate those kinds of questions.
- andrewcooke 14y agomathematicians like it, but physicists and computer scientists seem to be able to get by without it (based on answers i've received asking the same q in the past).
- deleted 14y ago[deleted]
- SatvikBeri 14y agoThere is a decent amount, but perhaps more importantly, you can get really weird conclusions if you assume choice is false! Simplest example: you can have a product of non-empty sets be empty.
- btilly 14y agoThe axiom of choice makes everything simpler. Which helps a lot when we're talking about things like infinite dimensional vector spaces. That said, it is a theorem that absolutely no theorem of number theory can possibly depend on the axiom of choice. So absolutely no "real" consequence can come from accepting or not the axiom of choice.