Why is Aleph0 > Aleph1 considered a 1d gap?

[Vietthai96]

Strange. By this logic, a 3-A universe with a Time axis that isn't infinite (like for example if it's X years of time only) should also be Low 2C, yet we only consider it as such if Time is infinite like a line, and not finite like a Line segment, as in the latter case we only consider it as High 3A.

I also remember reading an article that said otherwise about a line and line segment. Will try to find it and link it when I do find it, cuz I'm pretty sure I used it in one of my sandboxes on some other wiki.

it is the problem of Low 2-C, just similar to how tiny section of space-time or small length of time could result in Low 2-C due to no matter how small the space is, time still multiplying those space snapshot to uncountable infinite amount, thus Low 2-C, but if we doing this, fiction verse rating could get inflated to an absurd level, that why we arbitrary put a requirement for space-time to be universal in spatial size first, and timeline need to be continuous and is infinite in length, at leat infinite to the future since most fiction wrote timeline have starting point

 

[IDK3465]

it is the problem of Low 2-C, just similar to how tiny section of space-time or small length of time could result in Low 2-C due to no matter how small the space is, time still multiplying those space snapshot to uncountable infinite amount, thus Low 2-C, but if we doing this, fiction verse rating could get inflated to an absurd level, that why we arbitrary put a requirement for space-time to be universal in spatial size first, and timeline need to be continuous and is infinite in length, at leat infinite to the future since most fiction wrote timeline have starting point

Yeah, this is pretty much what I thought the reason was for finite amounts of time not qualifying for Low 2-C, not because segments of time aren’t actually uncountably infinite, but simply due to dumb arbitrary bullshit rules to prevent inflated values.

(Imo, finite sections of time really SHOULD more often than not qualify for Low 2-C, as you wouldn’t call something High 3-A simply because it only destroyed 13.8 billion years as apposed to infinite years, but I digress)

 

[Ezaru]

Because you'll need an extra dimension to fit the extra stuff that can't fit otherwise due to size.

Why though? Like people keep saying you CAN have an uncountable set with the same cardnality

 

[Agnaa]

Why though? Like people keep saying you CAN have an uncountable set with the same cardnality

"uncountable set" is just "a set with cardinality aleph-one or greater". You cannot have such a set within a set of cardinality aleph-zero or lower.

(Unless you use a very weird definition of "within", measuring cardinality differently between those two cases. I could say that the set {ℝ} contains one element, it's just a letter, but then say it contains an uncountable set due to what ℝ represents. But this would be exceedingly silly)

 

[Eclipso]

Because you need a "space" with a larger size that is comparable to that of a significant Low 1-C space to contain uncountable infinite amount of low 2-c. Kinda like an equalizer. It is also somewhat based on set theory iirc, where uncountable infinite amount of point make a 1D line, uncountable infinite amount of 1D line make a 2D plane, uncountable infinite amount of 2D plane make a 3D cube, and well, continuing up to whatever D want


This is kinda verse specific thing, which is an exception, not general rule. But again i have yet too see such a thing

The simplest example of an uncountable collection of points, that is, simply taking a point and adding another one uncountably infinitely many times(that is, the uncountable coproduct of Aleph 1 many points, in Top) is definitely not 1D. It’s 0D.

The fact that R, a specific uncountable collection of points, with a specific topology, is 1 dimensional, is not a “general rule”, that’s the exception. R^n for any n>=1 has uncountably many points, but only dim 1 if n=1. Most ways of taking uncountably many copies of a point won’t result in it having non-zero dimension, either. You have to give it a very specific topology. Similar thing applies to, say, taking uncountably many copies of R^3. It will not be 4-dimensional unless you give it some very specific topologies, say, if you’re taking the product of R^3 with R, which is not nearly necessary to having uncountably many copies of R^3.

 

[Eclipso]

To put it simply, you can completely fill an infinite X-dimensional plane with a countably infinite amount of "things" that have finite non-zero extensions in those X-dimensions.

I think a simple example of dots and lines can explain this.

I think you already know the basics, a point is 0-Dimensional with zero extensions in any direction/dimension. A line segment is a set of countably infinitely many points, while a line is an a countably infinite amount of points.

But a point has zero extensions in any dimension, so let's drop that and move to another unit. A line segment.

It has a finite extension in a 1-dimensional plane. Now, if you take a countably infinite amount of line segments, it'll make a line. This is equivalent to how you can fill an infinite 4D space (2-A) with a countably infinite amount of universes.

Now, let's move on. What if there's an uncountably infinite amount of line segments? Well, we know that a set of cardinality Aleph1 cannot be put on a one-to-one correspondence with a set of cardinality Aleph0.

To explain a one-to-one correspondence in case you don't know, it's basically when you can assign one unique element of Set A to one unique element of set B, insofar as that each and every unique element of Set A is in with each and every unique element of set B, such that there's no "extra" elements from either set that don't make a pair with an element of the other set, and neither are there any overlapping elements. (Like for example, a set of 12 apples is in one-to-one correspondence with a set of 12 oranges, but a set of 11 apples is not in a one-to-one correspondence with a set of 12 oranges).

Now that that's explained, we can already see that you can't fit an uncountably infinite amount of line segments such that they occupy space equivalent to only a single line. So where do all the extra elements go? They move to another dimensional direction!

Simply put, you can just add them not in "length" but I'm "height" then, and that'll give you a higher dimension: 2-D.

In the same sense, you can't put an uncountably infinite(Aleph0) amount of universes in a countably infinite 4D plane. So you add a new direction for those universes to take place.

You can apply the same to any arbitrarily higher dimension and Infinity, upto any arbitrary layer of high 1-B+. ¯⁠\⁠⁠(⁠ツ⁠)⁠⁠/⁠¯

Just wanted to mention two things, first, line segments(and lines) contain uncountably many points, and, secondly, you can have uncountably many line segments “glued together” that is 1-dimensional(in the usual sense), without any form of width or anything else. Meet the long line.

 

[Ezaru]

"uncountable set" is just "a set with cardinality aleph-one or greater". You cannot have such a set within a set of cardinality aleph-zero or lower.

(Unless you use a very weird definition of "within", measuring cardinality differently between those two cases. I could say that the set {ℝ} contains one element, it's just a letter, but then say it contains an uncountable set due to what ℝ represents. But this would be exceedingly silly)

I am refering to the fact they said this:

The Cantor set is an example of a set with continuum-many elements, but is "nowhere dense": in practical terms, it takes up 0 space.

Is this an example of what you are saying in the parenthesis?

They are saying uncountable 3d isn't 4d because uncountable x can fit in 0d or whatever ( idk how Cantor set works )

 

[Ezaru]

To put it simply, you can completely fill an infinite X-dimensional plane with a countably infinite amount of "things" that have finite non-zero extensions in those X-dimensions.

I think a simple example of dots and lines can explain this.

I think you already know the basics, a point is 0-Dimensional with zero extensions in any direction/dimension. A line segment is a set of countably infinitely many points, while a line is an a countably infinite amount of points.

But a point has zero extensions in any dimension, so let's drop that and move to another unit. A line segment.

It has a finite extension in a 1-dimensional plane. Now, if you take a countably infinite amount of line segments, it'll make a line. This is equivalent to how you can fill an infinite 4D space (2-A) with a countably infinite amount of universes.

Now, let's move on. What if there's an uncountably infinite amount of line segments? Well, we know that a set of cardinality Aleph1 cannot be put on a one-to-one correspondence with a set of cardinality Aleph0.

To explain a one-to-one correspondence in case you don't know, it's basically when you can assign one unique element of Set A to one unique element of set B, insofar as that each and every unique element of Set A is in with each and every unique element of set B, such that there's no "extra" elements from either set that don't make a pair with an element of the other set, and neither are there any overlapping elements. (Like for example, a set of 12 apples is in one-to-one correspondence with a set of 12 oranges, but a set of 11 apples is not in a one-to-one correspondence with a set of 12 oranges).

Now that that's explained, we can already see that you can't fit an uncountably infinite amount of line segments such that they occupy space equivalent to only a single line. So where do all the extra elements go? They move to another dimensional direction!

Simply put, you can just add them not in "length" but I'm "height" then, and that'll give you a higher dimension: 2-D.

In the same sense, you can't put an uncountably infinite(Aleph0) amount of universes in a countably infinite 4D plane. So you add a new direction for those universes to take place.

You can apply the same to any arbitrarily higher dimension and Infinity, upto any arbitrary layer of high 1-B+. ¯⁠\⁠⁠(⁠ツ⁠)⁠⁠/⁠¯

I was under the impression that a square was made of uncountably infinite lines already, not just infinite

Because like take a line, it is made of uncountably infinite points, and each of those points needs to be "matched" to make a square

That's practically the same as asking why a block of 100 meter cube cannot exist inside a space that, in all extensions, is only 10 meter cube.

If you mean an infinite line x infinite line, yes, I get your PoV up till this point;

Thing is, we by default assume that a dimension (R) is made up of uncountably infinitely many points, so compared to a line segment (a non-zero finite length), it's only countably infinitely bigger. Like we assume a High 3-A Space is equivalent to a countably infinitely sized cube or something equivalent.

Same applies to squares; a Square that is countably infinitely bigger than a finite square is equivalent to the 2-Dimensional plane (uncountably infinitely bigger compared to a single line still).

So a square that's uncountably infinitely bigger than a finite non-zero area square won't be able to fit in a regular 2-dimensional plane. So it's a default assumption that anything bigger than the infinite plane of any X-dimension (insofar as the difference between Aleph0 and Aleph1) is 1 dimension higher than X.

Or at least this is what I understand of it.

Because you'll need an extra dimension to fit the extra stuff that can't fit otherwise due to size.

Idk, I don't see why it can't just be uncountably infinite big.
Another dimension isn't a size, it's a direction

Strange. By this logic, a 3-A universe with a Time axis that isn't infinite (like for example if it's X years of time only) should also be Low 2C, yet we only consider it as such if Time is infinite like a line, and not finite like a Line segment, as in the latter case we only consider it as High 3A.

I also remember reading an article that said otherwise about a line and line segment. Will try to find it and link it when I do find it, cuz I'm pretty sure I used it in one of my sandboxes on some other wiki.

Every line segment has uncountably infinite points for the same reason there are uncountably infinite numbers between 1 and 2 or .1 and .2 or .01 and .02 etc

And yes, by the logic we use even erasing a second of time would be a low 2-C feat, but we don't treat it like that, for like, the same reason we don't count destroying a 1x1x1x1x1 hypercube as 2-C, even though it is

 

[Agnaa]

I am refering to the fact they said this:

The Cantor set is an example of a set with continuum-many elements, but is "nowhere dense": in practical terms, it takes up 0 space.

Is this an example of what you are saying in the parenthesis?

It is not an example of what I'm saying in parentheses.

It has uncountably infinitely many elements; it has more elements than a set with countably infinitely many elements, and it has more elements than a finite set.

However, it has a measure of zero in one dimension. But if we use an extension of the notion of dimensionality, that works with non-integers and fractals, then the Cantor set has a dimensionality of log3(2), which is around 0.631

Our system does not work well with fractal dimensions, and we're aware of this, but they don't come up too often. Practically, we treat them as weaker than characters at higher integer dimensions, and stronger than characters at lower integer dimensions.

They are saying uncountable 3d isn't 4d because uncountable x can fit in 0d or whatever ( idk how Cantor set works )

This isn't really true, as I hope this explanation can show. It is too large for 0-D, exceeding countably infinitely many points within that space, and being at home in a fraction between them. I think it's fine to generally assume that pieces of fiction make a whole integer jump, since they rarely have the sort of fractal gaps in cosmology that would necessitate using Hausdorff dimensions explicitly.

 

[Ezaru]

It is not an example of what I'm saying in parentheses.

It has uncountably infinitely many elements; it has more elements than a set with countably infinitely many elements, and it has more elements than a finite set.

However, it has a measure of zero in one dimension. But if we use an extension of the notion of dimensionality, that works with non-integers and fractals, then the Cantor set has a dimensionality of log3(2), which is around 0.631

Our system does not work well with fractal dimensions, and we're aware of this, but they don't come up too often. Practically, we treat them as weaker than characters at higher integer dimensions, and stronger than characters at lower integer dimensions.

This isn't really true, as I hope this explanation can show. It is too large for 0-D, exceeding countably infinitely many points within that space, and being at home in a fraction between them. I think it's fine to generally assume that pieces of fiction make a whole integer jump, since they rarely have the sort of fractal gaps in cosmology that would necessitate using Hausdorff dimensions explicitly.

I guess I don't really understand fractal or non-integer dimensions

How does that even work?

And how does that relate to the universe thing

Could an uncountably infinite amount of universes have a dimensionality of like 3.631 then? (theoretically, like I said idk how Cantor set works or how infinite size changes this )

Why does uncountably infinite universes = 4d? ( 5 with time )

 

[Vietthai96]

I guess I don't really understand fractal or non-integer dimensions

How does that even work?

Fractal or non-integer dimensions based on non-integer number in mathematic. Simply talking, non-integer numbers are number that aren't whole, whole number is 0, 1, 2, 3, etc.....then non-integer number are number such as 0.73, 3.56, 2/3, etc....any number that have decimals or fractions

Applies this to dimensionality, you have "whole" dimension is 1D, 2D, 3D, 363D then fractal or non-integer dimension is 3.5D, 7.3538D, 10.34D, etc.....

Like how a non-integer number is bigger than lower integer number and smaller than higher integer number, fractal/non-integer dimension will be bigger than lower integer dimension and smaller than higher integer dimension. For example, we have non-integer number 3.5, it will be bigger than 3 and smaller than 4 with 3 and 4 is whole number, then applies this to dimensionality, thus 3.5D is bigger than 3D and smaller than 4D with 3.5D is fractal/non-integer dimension

 

[Agnaa]

I guess I don't really understand fractal or non-integer dimensions

How does that even work?

Dimensionality can be thought of as meaning "If we increase the scale of an object, how much does its measure increase?"

If you make a line twice as long, it gets twice as large. 2^1

If you make a square twice as long, it gets four times the area. 2^2

If you make a cube twice as long, it gets eight times the volume. 2^3

Fractal objects, like the Cantor set, increase in a non-integer way. When you make the Sierpinski triangle twice as big, you have three copies of the original, which corresponds to about 2^1.58

And how does that relate to the universe thing

It more relates to the counterexample provided of the Cantor set. It has a measure of 0 in 1 dimension, despite having uncountably infinitely many 0-D points, because its dimensionality is ~0.631

Could an uncountably infinite amount of universes have a dimensionality of like 3.631 then? (theoretically, like I said idk how Cantor set works or how infinite size changes this )

Theoretically, but if we assume that those universes are lined up on a new axis, then for it to be fractional dimension like this, rather than an integer dimension, it would have to strictly occupy the larger space it's in as a fractal.

Like, the Cantor set occupies the space of a line, but because of the specific way it's constructed, it never occupies certain points in it. The Sierpinski triangle occupies a 2-D space, but does so as a fractal slice of it, leaving massive gaps.

This doesn't seem particularly likely for fiction.

Why does uncountably infinite universes = 4d? ( 5 with time )

I already explained that:

But ultimately, it's hard to find particularly good alternatives. Alephs themselves are unwieldy due to how each new one is the power set of the previous, resulting in the gaps between them increasing, rather than staying constant, which isn't a good thing to equalise verses to. Ultimately, we'd kind of have to create our own pseudo-math to have proper structures to equalise series to, and would that really be better than moderately butchering math and physics as we are now? I think that's an open question.

Yes, it is not mathematically correct. I can spin you some reasons why it's better than you might be interpreting it as, but ultimately, it's just because there aren't good alternatives.

 

[Ezaru]

Dimensionality can be thought of as meaning "If we increase the scale of an object, how much does its measure increase?"

If you make a line twice as long, it gets twice as large. 2^1

If you make a square twice as long, it gets four times the area. 2^2

If you make a cube twice as long, it gets eight times the volume. 2^3

Fractal objects, like the Cantor set, increase in a non-integer way. When you make the Sierpinski triangle twice as big, you have three copies of the original, which corresponds to about 2^1.58

Hmmm
That makes sense but I am a little confused

Ok so if you make a square twice as long you get 4x the area

If you make a Sierpinski Triangle twice as big (longer? increase the area I assume?) you get 2^1.58 x more what? Area is to length as as what is to what here

You get my question?

Not sure what you mean by 3 copies

It more relates to the counterexample provided of the Cantor set. It has a measure of 0 in 1 dimension, despite having uncountably infinitely many 0-D points, because its dimensionality is ~0.631

Oh ok I see like you said here:

This isn't really true, as I hope this explanation can show. It is too large for 0-D, exceeding countably infinitely many points within that space, and being at home in a fraction between them. I think it's fine to generally assume that pieces of fiction make a whole integer jump, since they rarely have the sort of fractal gaps in cosmology that would necessitate using Hausdorff dimensions explicitly.

Theoretically, but if we assume that those universes are lined up on a new axis, then for it to be fractional dimension like this, rather than an integer dimension, it would have to strictly occupy the larger space it's in as a fractal.

Like, the Cantor set occupies the space of a line, but because of the specific way it's constructed, it never occupies certain points in it. The Sierpinski triangle occupies a 2-D space, but does so as a fractal slice of it, leaving massive gaps.

This doesn't seem particularly likely for fiction.

Yeah, I missed your earlier example quoted above, that makes sense

I already explained that:

Yes, it is not mathematically correct. I can spin you some reasons why it's better than you might be interpreting it as, but ultimately, it's just because there aren't good alternatives.

Yeah, sorry, I didn't really get that part


With the power set, wouldn't the gaps in amounts of universes required to make the 1d jump also be increasing?

Yeah, I mean I'd like to hear any reasoning if you feel otherwise it's all good.


Thanks

 

[Agnaa]

Hmmm
That makes sense but I am a little confused

Ok so if you make a square twice as long you get 4x the area

If you make a Sierpinski Triangle twice as big (longer? increase the area I assume?) you get 2^1.58 x more what? Area is to length as as what is to what here

You get my question?

Not sure what you mean by 3 copies

If you look at the image of the Sierpinski triangle, you'll see that each third of it is a (smaller) copy of the whole, achievable due to this recursively going downwards infinitely. And these are arranged such that each side of the "big triangle" is twice the length of the side of the "small triangles".

So, making it twice as large gives you 3x as much stuff, which corresponds to ~2^1.58

With the power set, wouldn't the gaps in amounts of universes required to make the 1d jump also be increasing?

Not sure what you mean by this.

Yeah, I mean I'd like to hear any reasoning if you feel otherwise it's all good.

Apologies but I've got a bit on my plate rn so I'd rather not.

 

[Eclipso]

Just pointing out, I don’t think the cantor set example is too important, as there are uncountable subsets of R with zero dimension, even zero Hausdorff dimension. https://en.m.wikipedia.org/wiki/Liouville_number, for example.

 

[Agnaa]

Just pointing out, I don’t think the cantor set example is too important, as there are uncountable subsets of R with zero dimension, even zero Hausdorff dimension. https://en.m.wikipedia.org/wiki/Liouville_number, for example.

Damn alright.

Yeah, at that point the only saving grace would just be the idea that, due to them having the same cardinality, they could be considered sufficiently equivalent, since they could be rearranged to form each other.

 

[Eclipso]

Damn alright.

Yeah, at that point the only saving grace would just be the idea that, due to them having the same cardinality, they could be considered sufficiently equivalent, since they could be rearranged to form each other.

I don’t think that’s a particularly good argument either, since, as I’m sure you already know, |R^1|=|R^2| etc, so you could say the same about that set and, say, a square in R^2. Same cardinality, so can be rearranged. Unless I’m misunderstanding you

 

[Agnaa]

Yeah, but we're not just going off of that. We don't give 8-D aliens big tiers, we require indications that it does necessarily encompass a higher size (by covering full axes), or that each infinitesimal slice of higher-dimensional objects has non-zero mass, or that each additional dimension corresponds to a significant increase in strength.

 

[Eclipso]

Yeah, but we're not just going off of that. We don't give 8-D aliens big tiers, we require indications that it does necessarily encompass a higher size (by covering full axes), or that each infinitesimal slice of higher-dimensional objects has non-zero mass, or that each additional dimension corresponds to a significant increase in strength.

I mean, in this example, the aforementioned uncountable subset of the real numbers with 0 Hausdorff dimension, we are just going off of that, no? It doesn't cover any full axis, or have any mass(or any relevant equivalent to them, since it's just points), or necessarily correspond to significant increases in strength, etc, it's just an uncountable subset of the real numbers which is 0-dimensional in every notion of dimension that could possibly be relevant, as far as I can tell. I simply don't see how it having the same cardinality as, say, a 1D line, or that it could be rearranged into one, means it's sufficiently equivalent. I am assuming by sufficiently equivalent you meant that it could still be considered 1D for tiering purposes, right? If not, apologies for misunderstanding.

 

[Agnaa]

We're talking about universes here; each of those "points" (universes) would have mass.

 

[Eclipso]

We're talking about universes here; each of those "points" (universes) would have mass.

I see.

So, 4D is stronger than 3D *in the specific case that, say, the 4D space has uncountably infinitely many 3D universes or things with mass etc, so to speak? In which case the answer to the OP's question is essentially that it's by definition, since we define 4D ap to be that, not literally just being a 4D thing in a vacuum?

 

[Agnaa]

Yeah, the primary definition is not dimensionality, that's just a useful shorthand.