I don't get the Axiom of Choice. All it seems to say is that if you have a bunch of piles of objects, it is possible to pick exactly one item from each pile. Well big fucking whoop, what's the big deal? Why is this so controversial?
It's not controversial. Famous would be a more accurate word. It's one of those things, like Euclid's parallel axiom, that was just sort of assumed for a good amount of time because it was so "obvious" that no one bothered to inspect it. Then someone did, and guess what? They couldn't prove it. Furthermore, a number of well known results in set theory, analysis, and algebra require the axiom of choice (or a weaker variant of it).
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Anonymous2007-05-26 2:56 ID:IMW3qAfx
It faced some resistance when it was introduced. Banach and Tarski intended to show that it was an unsuitable axiom with the Banach-Tarski Paradox, but it is still almost always accepted.
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Anonymous2007-05-26 5:17 ID:4eYyvrxh
To add to the previous poster, while the axiom is almost always accepted the few people who reject it are often very adamant about rejecting it.
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Anonymous2007-05-26 7:52 ID:H8fB0MsL
The axiom of choice is a very strong axiom. It implies boolean logic. In situations where you have weaker logics, like intuitionistic logic, you will find that the axiom of choice is not true.
Why would you want to do this? Well, fields like synthetic differential geometry have true infinitesimals in them, but if we use classical logic everything breaks down.
Look up topos theory for stuff about this, but I'm afraid the wikipedia article is a bit difficult.
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Anonymous2007-05-26 14:48 ID:jK1lWmXe
Remember, you might have an infinite amount of sets. Then you are claiming you are able to pick one item from each of an infinite amount of sets. This amounts to an infinite process and a small number of people (often computer scientists) aren't too happy with it. I've always thought of it as tantamount to allowing an algorithm with infinite steps.
No matter if you are for or against, weird "paradoxes" occur. But really, there is little controversy anymore, and it is widely accepted.
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Anonymous2007-05-26 16:57 ID:nP+VYt4d
Hmm... thanks, I kinda sorta get it now. (OP here) Basically, it's something that's so painfully obvious that you can't really prove it's true, and just have to assume it is (hence an axiom)- but in some cases you can get weird things to happen by assuming it's false. Right?
The closest thing you can get to a 'truth' value for axioms is considering whether they agree with your intuitions. However, our intuitions are shoddy whenever infinity gets involved. So people will always disagree, and the best we can do is investigate the consequences both of accepting it and of not accepting it.
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Anonymous2007-05-26 23:51 ID:IMW3qAfx
"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"
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Anonymous2007-06-03 6:12 ID:+swJ0LDU
Trying to bump out my post. Please ignore this reply.
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Anonymous2007-06-03 8:35 ID:tK6seCbA
Divide by zero. Oh shi-
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Anonymous2007-06-03 9:13 ID:nA9PnZef
>>2
How exactly is it unprovable? Because it is based on empirical evidence? Why couldn't they define the "axiom of choice" in a little more abstract manner. Something more like this...
You can only make a rational judgements after making initial judgements based on experience with which to reason with.
If this is case, then people who run around screaming "LOL I AM TEH PHILOSOPHY LOL AXIOM OF CHOICE BITCHES!!!!!!111" need to gb2 18th century because Immanuel Kant has already been through this. I think Descartes and probably some Chinese and Greeks already have aswell.