The Person Above Me II

[Magrathean]

6:

I know. That's like the third time i've seen this demonstration. Thanks, though, for resurfacing this subject in my mind. It's a beautiful subject; too bad there aren't many people interested in it.

Can you demonstrate the existence of aleph two, though? Hint: Aleph two is the cardinality of the set of all possible functions of RxR.

 

[Zack]

^^Is damn straight!

^Is currently discussing this "aleph" business, when he should be debating my theory of square roots (hint, see sig.)

 

[Magrathean]

The square root of 30 is roughly 5.47722557505.

Now, back to infinite set theory...

 

[6 Stringed Fingers]

>_< my book stopped at Aleph 1. But as for RXR, wouldn't that be points in a square? If it is, then it should have the same cardinality as Aleph 1, since points are infinitismal anyways.

 

[Magrathean]

The number of points in RxR has indeed an aleph one cardinality. But the number of all possible functions of RxR has the same cardinality as the number of all possible subsets of R, and the continuum hypothesis makes the following two statements (which, i think, are also appliable to finite sets, but i have to double-check that):

1. For any given set A with cardinality |A|, there is another set B which contains all the possible subsets of A (as long as |A|>0). The cardinality of B is |B|=2^|A|.

2. For infinite sets, aleph N+1 is equal to 2 ^ aleph N.

Thus, the cardinality of the set of all possible subsets of R is aleph 2. The continuum hypothesis has been proved to be non-provable, though, so it's up to you to believe all of this.

 

[6 Stringed Fingers]

Was it because you can't chose in a set of infinite elements?

 

[Magrathean]

What?

 

[La Rocque]

^ still here

 

[6 Stringed Fingers]

As to why the continuum hypothesis can't be proven, is it because the axiom of choice doesn't apply to it?

The axiom of choice - for every set we can provide a mechanism for choosing one member of any non-empty subset of the set.

 

[Magrathean]

I think it is because it makes sense both if you suppose the basic axioms of set theory to be false and also if you suppose them to be true. Let me recheck that.

 

[La Rocque]

^^ was created -

 

[6 Stringed Fingers]

^ is the second one to state the obvious.

^ was created.
By whom, i wonder...

 

[Zack]

^Has 77.84 posts a day

 

[La Rocque]

^ gazes at the stars

 

[Zack]

^Just watched Lord of War (I think)

 

[RampageSword]

^Is ready to blow 4000$ on a 1978 Les Paul....

 

[Zack]

^Is talking to the wrong person

La Rocque, Rampage... La Rocque

 

[hyena]

^opens interesting threads

 

[Zack]

^Likes harry potter?

Thanks a bunch, hyena

 

[marduk1507]

^is about to become a devoted fan of Necrophagist