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/sci/ - Science & Math

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6182150 No.6182150 [Reply] [Original]

Hi /sci/ . I cannot understand the meaning / usefulness of the axiom of choice . the cartesian product of non empty sets is non empty . So what ?

>> No.6182179

>>6182150
http://en.wikipedia.org/wiki/Axiom_of_choice

>> No.6182198

>>6182150
I like my vectorial spaces to have basis, thank you.

>> No.6182414

>>6182150
Soooo... you're okay with the idea of a product of nonempty sets being empty?

>> No.6184664

>>6182414
How does that work?

>> No.6184679

>>6182414
has nothing to do with it but matrix multiplication cough

>> No.6184685

If we don't take the axiom of choice, it breaks Banach-Tarski, so that you can't cut up any object and reassemble it into any larger or smaller object.

Since that clearly violates our experience of the real world, if we want a relevant and meaningful geometry, we need the axiom of choice.

>> No.6184701
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6184701

Are you implying zorn's lemma wasn't absolutely obvious to you upon introduction?

>> No.6184712

>>6182414
You're okay with drawing firm conclusions on important questions by making assumptions about infinite sets with indistinguishable members?

The alternative to the axiom choice isn't to accept an absurdity, but to avoid relying on absurd constructs. By using these, even with the axiom of choice, you still find it impossible to avoid reaching obvious absurdities.