What is [math]\aleph_2[/math]? How is it different from [math]\aleph_1[/math] ?
What would [math]\aleph_{\aleph_{\aleph...}}[/math] be?
As much as it pains sci to hear this, one or the more recent vsauce videos does a pretty decent amateur treatment of the topic of cardinality
>in b4 REEEEEEEEEEEEEEE
>>8002372
It's what I saw that made me want to ask this. He gets to [math]\aleph_1[/math] as the set of all reals but leaves off there. I'm trying to figure out how the later cardinalties are different.
>>8002381
I'm pretty sure the way he explains it in the video is using the powerset of N1. I don't think this powerset is N2 but it proves that there's a set that's larger than N1 out there...then he says some complicated shit that basically means "we can define N2 because math is all about definitions and doing whatever the fuck we want". Good luck getting a better explanation than this without taking a set theory course.
>>>>8002381
Oh your question is what sets are of these cardinalities? AFAIK he touches in this briefly that you can take the set of cardinality aleph 1 and take the power set P( reals) to get a set of cardinality aleph 2. Repeating this process gives your result I think. This is off memory since 1) I'm on mobile atm 2) I saw the video a day or two ago 3) my set theory knowledge is a few years old
>>8002325
ℵ2 is the cardinality of the set of ordinals with cardinality ≤ ℵ1
>>8002381
For the record I was wrong about iterated power sets. From wiki. Here's a construction of a set of arbitrary aleph cardinality. You might have to review ordinal numbers
>>8002432
reals are just as big as complex numbers
>>8002432
There exists a bijection between the reals and the complex (if you permit the complex to be isomorphic to R^2) in fact, for any finite n there's a bijection between the reals and R^n. So a set of cardinality aleph 2 must be "greater" (in some sense) than R^n. The typical construction is the power set (set of all subsets) since there is probably no bijection between a set and it's power set (maybe I forget some shitty extreme cases)
>>8002442
Probably = provably
