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So uh, just to clarify, this isnt by me. Some fella was talking about VSBW's dimensional tiering being wrong and what not, so I asked him to compile the reasons and such, and paste it here to see whats right, wrong or not. Just interested. (If its fine)
Ok, here: ////"My main issue with dimensional tiering is
Infinite 4d logically shouldn't be above infinite 3d since R^4(or any finite number) = R^3 = R^2 = R
The explanation for dimensional tiering I most commonly see is that a cube has a more than infinite(specifically an inaccessible amount of them)amount of squares in it because infinity×0 = 0, but infinity*0 is actually undefined, and there are an infinite amount of points on a line, but there is also the same amount of points on a square or any other finite dimensional object
And for the argument of infinity^3(or however many dimensions you are using not neccesarily 3) > infinity, that is objectively incorrect since infinity*infinity isnt even a higher infinity(which can be shown by making a bijection between the two with Cantor's proof for the rational numbers having the same cardinality as the natural numbers, which seems unintuitive at first since there would be infinite rational numbers per natural number)
And, even if dimensions did infinitely transcend eachother, there is the same amount of points on a line as on a finite dimensional object, which is aleph one assuming CH, and since by definition the next aleph number is the smallest infinity bigger than the previous one, i.e. there is no infinity between aleph one and aleph two
So if dimensions infinitely transcended eachother than, for instance, an infinite line would be aleph null size, an infinite square aleph one sized, and an infinite cube would be aleph two, which would mean it somehow has more units of size than its amount of points which logically wouldn't make sense
This is only really "proof" for infinite 3d = infinite 4d or any other finite dimension, not necessarily dimensional tiering as a whole"\\\\
k. Done.
(There is no dimensional tiering tag. So I went with "tiering system" instead. If its ok? Or well, idk.)
Ok, here: ////"My main issue with dimensional tiering is
Infinite 4d logically shouldn't be above infinite 3d since R^4(or any finite number) = R^3 = R^2 = R
The explanation for dimensional tiering I most commonly see is that a cube has a more than infinite(specifically an inaccessible amount of them)amount of squares in it because infinity×0 = 0, but infinity*0 is actually undefined, and there are an infinite amount of points on a line, but there is also the same amount of points on a square or any other finite dimensional object
And for the argument of infinity^3(or however many dimensions you are using not neccesarily 3) > infinity, that is objectively incorrect since infinity*infinity isnt even a higher infinity(which can be shown by making a bijection between the two with Cantor's proof for the rational numbers having the same cardinality as the natural numbers, which seems unintuitive at first since there would be infinite rational numbers per natural number)
And, even if dimensions did infinitely transcend eachother, there is the same amount of points on a line as on a finite dimensional object, which is aleph one assuming CH, and since by definition the next aleph number is the smallest infinity bigger than the previous one, i.e. there is no infinity between aleph one and aleph two
So if dimensions infinitely transcended eachother than, for instance, an infinite line would be aleph null size, an infinite square aleph one sized, and an infinite cube would be aleph two, which would mean it somehow has more units of size than its amount of points which logically wouldn't make sense
This is only really "proof" for infinite 3d = infinite 4d or any other finite dimension, not necessarily dimensional tiering as a whole"\\\\
k. Done.
(There is no dimensional tiering tag. So I went with "tiering system" instead. If its ok? Or well, idk.)
