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One of the troubling aspects of the Axiom of Choice is that both affirming and denying it results in some pretty unintuitive implications (but different ones).
by mjn 12y ago
One of the troubling aspects of the Axiom of Choice is that both affirming and denying it results in some pretty unintuitive implications (but different ones).
This MathOverflow discussion has some interesting examples of unintuitive implications of the axiom, along with a bonus answer giving examples of unintuitive implications of not having the axiom: http://mathoverflow.net/questions/20882/most-unintuitive-application-of-the-axiom-of-choice http://mathoverflow.net/questions/20882/most-unintuitive-app...
The history of it is full of people wrestling with both sides of that, including many participants in the debate having trouble keeping their own positions consistent—the axiom of choice is equivalent to so many other things (many only discovered fairly recently) that consistently affirming or denying it has tripped up many world-class mathematicians! I am currently about 1/3 of the way through this really interesting history that tries to trace that development, Zermelo's Axiom of Choice by G.H. Moore (2013): http://www.amazon.com/gp/product/0486488411/ref=as_li_tl?ie=UTF8&camp=1789&creative=390957&creativeASIN=0486488411&linkCode=as2&tag=kmjn-20&linkId=A73WQE7M7G3ZH5BD http://www.amazon.com/gp/product/0486488411/ref=as_li_tl?ie=... (ToC: http://www.apronus.com/math/zermelos.htm http://www.apronus.com/math/zermelos.htm)
- j2kun 12y agoI particularly like the one about some vector spaces having no basis, or bases of different cardinalities (should we deny the Axiom of Choice).
- andrewflnr 12y agoThat one really struck me, too. Personally, I'll take the creepy prisoner paradox if I need it to keep my vector space bases.
- emillon 12y ago"The axiom of choice is obviously true, the well-ordering principle obviously false, and who knows about Zorn's Lemma?" -- Jerry Bona
- hypersoar 12y agoThat's a great list! Being a grad student in mathematics, I find several of those way more upsetting than anything like Banach-Tarski or the hat problem in the linked post.
- dnautics 12y agoBanach-Tarski is too easily dismissed as a problem with real numbers being an intuitive model of experience, versus a problem with AC itself.
- hypersoar 12y agoOh, I agree. I'm totally comfortable with it, by now. I think the von-Neumann quote might apply here: "In mathematics, you don't understand things. You just get used to them."
- judk 12y agoIndeed, this is what intuition is. We often fail to notice where our childhood intuitions come from (experience), and so our adult desire to understand interferes with our ability to form new intuitions.
- judk 12y agoWhat's the difference? Uncountable sets does not respect out intuition about countable sets. AC is independent from ZF, meaning it doesn't matter for any application to anything in the Universe, including physics or computing. Theory of uncountable objects is a game to play with symbols, the only thing "real" about them is that they are the firm boundary between the potentially reachable and the infinite void. The name itself conveys an important truth.
- portlander12345 12y agoSo clearly, it's the Axiom of Infinity where things really go wrong.