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[1180144626] Axiom of Choice

Anonymous ID:nP+VYt4d Sat 26 May 2007 10:32:00 No.18362 View View Reply Original Report

Quoted By:

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?

Anonymous

2

Anonymous ID:Heaven Sat 26 May 2007 10:49:00 No.18363 Report

Quoted By:

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).

Anonymous

3

Anonymous ID:MW3qAfx Sat 26 May 2007 10:56:00 No.18364 Report

Quoted By:

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.

Anonymous

4

Anonymous ID:4eYyvrxh Sat 26 May 2007 13:17:00 No.18365 Report

Quoted By:

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.

Anonymous

5

Anonymous ID:H8fB0MsL Sat 26 May 2007 15:52:00 No.18366 Report

Quoted By:

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.

Anonymous

6

Anonymous ID:jK1lWmXe Sat 26 May 2007 22:48:00 No.18372 Report

Quoted By:

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.

Anonymous

7

Anonymous ID:nP+VYt4d Sun 27 May 2007 00:57:00 No.18374 Report

Quoted By:

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?

Anonymous

8

Anonymous ID:Heaven Sun 27 May 2007 02:04:00 No.18379 Report

Quoted By:

>>7
Weird things either way.

Anonymous

9

Anonymous ID:Heaven Sun 27 May 2007 06:30:00 No.18389 Report

Quoted By:

It doesn't matter whether you accept it or not.

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.

Anonymous

10

Anonymous ID:MW3qAfx Sun 27 May 2007 07:51:00 No.18391 Report

Quoted By:

"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"

Anonymous

11

Anonymous ID:+swJ0LDU Sun 03 Jun 2007 14:12:00 No.18665 Report

Quoted By:

Trying to bump out my post. Please ignore this reply.

Anonymous

12

Anonymous ID:tK6seCbA Sun 03 Jun 2007 16:35:00 No.18713 Report

Quoted By:

Divide by zero. Oh shi-

Anonymous

13

Anonymous ID:nA9PnZef Sun 03 Jun 2007 17:13:00 No.18721 Report

Quoted By:

>>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.

Anonymous

14

Anonymous ID:Heaven Sun 03 Jun 2007 18:42:00 No.18722 Report

Quoted By:

>>13
Come back after you've learned what mathematical axioms and formal systems are.

Anonymous

15

Anonymous ID:T1p9aFEE Sun 03 Jun 2007 19:15:00 No.18723 Report

Quoted By:

>>10
erm the axioms of choice, well ordering priciples and the zorn's lemma are equivalent statements.

Anonymous

16

Anonymous ID:Heaven Sun 03 Jun 2007 21:29:00 No.18727 Report

Quoted By:

>>13
You have no idea what you're talking about, and the "more abstract definition" you give has nothing at all to do with the axiom of choice.

>>15
Yes, which is what makes that joke funny.

Anonymous

17

Anonymous Wed 18 Mar 2009 10:43:00 No.42317 Report

Quoted By:

Don't call me gay, but I need some mary jay!

Marijuana MUST be legalized.

Anonymous

18

Anonymous Wed 30 Oct 2013 21:25:00 No.56540 Report

Quoted By:

I don't get it =/

Anonymous

19

Anonymous Thu 31 Oct 2013 07:36:00 No.56541 Report

Quoted By:

>>13
It's not unprovable. It's independent. Big difference.

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[1180144626] Axiom of Choice