5 ms·
Best quote so far: > If we have infinitely many pairs of shoes we don’t need the axiom of choice, since we can just take the left shoe from each pair; but if w
by nixpulvis 5y ago
Best quote so far:
> If we have infinitely many pairs of shoes we don’t need the axiom of choice, since we can just take the left shoe from each pair; but if we have infinitely many pairs of socks, we do need the axiom of choice.
- Bertrand Russell
So randomness seems like a good fit for "choice" here, no? Or maybe call it ambivalence, or indifference. If the objects are truly unidentifiable, can it possibly matter which one you choose?
As the argument surely goes...
- lmm 5y agoThe original statement of the axiom I heard was "it's possible to make an infinite number of arbitrary choices" - I think that's a good word for what you're getting at. Of course it doesn't matter which you choose - the axiom doesn't tell you anything about which you can choose - but whether you can choose at all is important. E.g. Hilbert's basis theorem relies on the idea that given an infinite-dimensional space you can always "start somewhere" - it doesn't matter specifically where, but it matters that you can pick a specific point in the space to start from. (The first journal he submitted it to rejected the paper with the comment "this is not mathematics, this is theology")
- nixpulvis 5y agoThe axiom of arbitrarity has an interesting ring to it... slightly avoiding the theological connotation, while being a bit bleak. Anyway, I hope they gave him a Ph.D for that comment, lolol. Oh and yikes, did Hilbert just bring roots of polynomials into the mix?
- siraben 5y ago> So randomness seems like a good fit for "choice" here, no? The axiom of choice is exactly the thing that tells us that we can make "random" choices. The way set theory works is that you have axioms to talk about intuitions of sets. So you might say that such a reasonable intuition is that we can make "make unlimited random choices", but this is exactly the axiom of choice! It's not about a choice from an infinity but an infinity of choices.
- Ostrogodsky 5y agoI have always thought that Mr Russell is wrong with that cheeky quote. With the socks you also dont need the AOC. Why? because if you have a pair of identical socks (I think that was what Russell had in mind) then effectively you have a set of cardinality 1.If the socks are different, choose the lightest, if there is a tie between them choose the longest, and so on,if for all the tests the socks are tied declare them the same and back to point 1. This is my preferred quote about the AOC: > The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma? Jerry Bona. https://en.wikipedia.org/wiki/Jerry_L._Bona https://en.wikipedia.org/wiki/Jerry_L._Bona
- naveen99 5y agoAtoms are able to select which indistinguishable electrons will fill the lowest energy orbitals.
- nixpulvis 5y agoOr might this imply that the electrons are actually distinguishable in some way? Or maybe I should re-phrase; could each electron actually have some (possibly hidden) identity?
- naveen99 5y agoAlso if the universe is infinite, is it the one with or without axiom of choice ? Can you always pick the center of each universe in a universe of infinite universes ? Any point can be the center, you just need to pick one.
- nixpulvis 5y agoI'm sorry if this is a stupid question... but what on earth is the center of the universe? Must there even be such a concept? If the universe is truly infinite, I would think that implies it can be both expanding and centerless at the same time.
- dsr_ 5y agoA circle has a center which is a point in that circle such that the distance(s) from the center to every point on the boundary of the circle is the same. The center is within the space of the circle but not on the boundary. A sphere has a center which is in that sphere but not on the surface (boundary) of the sphere. A 4-sphere has a center which is in that hypersphere but not in the volume boundary of the hypersphere. ... It is entirely possible that the universe is a 3s1t shape which is part of the boundary of an NsNt shape, and there is a center to it which is not usefully accessible -- or even discoverable -- from anywhere in the universe.
- 5y ago
- shadowgovt 5y agoIt can definitely matter---if I mutate one of the socks and the mutation doesn't show up on the other, I've got a problem if I'm assuming equivalence is equality. I suspect that's key to why the axiom of choice has to be an axiom (and how this may pop up from time-to-time in the realm of computer science problems).
- nixpulvis 5y agoHow can you mutate it if you can't select it?
- shadowgovt 5y agoSomething I'm unclear on is whether the axiom of choice implies determinism. if it does, then "I select one at random." That could be accomplishable without a guarantee that I can tell you how to select (in finite time) the same element I selected.
- patrickthebold 5y agoMy favorite quote: > The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma? [1]https://en.wikipedia.org/wiki/Jerry_L._Bona#Quotes https://en.wikipedia.org/wiki/Jerry_L._Bona#Quotes
- jodrellblank 5y agoWhy can't we take the left sock from each pair?
- bmm6o 5y agoBecause there isn't a left/ right distinction with socks.
- jodrellblank 5y agoI have never considered that there might not be a distinction, it always seems to me that the heels are different at least. Feet are different, shoes are different, so why would socks not also be shaped for the feet and mirrored left/right in the same way? Turns out, it depends which socks you buy - there are makes which are different, and ones which aren't. examples at https://www.quora.com/Is-there-a-left-and-right-sock-in-a-pair https://www.quora.com/Is-there-a-left-and-right-sock-in-a-pa... including toe-socks but not limited to them.
- cynix 5y agoSays who? My socks are distinctively left or right.