Tiering System

Sources:

https://vsbattles.com/threads/analyzing-the-tiering-system.63840/#post-2148447

Original Video Transcription

Understanding Dimensional Tiering: A Simplified Guide

Dimensional tiering is one of the most controversial, misunderstood, and difficult concepts in power scaling. It is commonly used to measure the scale and true power of fictional characters, particularly those who can destroy entire space-time continuums or exist in realms beyond human comprehension. Although it can seem complex at first, dimensional tiering is actually simpler than many think. This essay aims to clarify what dimensional tiering is, why it matters, and how it works in the context of fiction.

What is Dimensional Tiering?

Dimensional tiering is a scaling method that considers the number of dimensions an object or character occupies. In other words, to tier something dimensionally means to take into account how many dimensions it operates within. For example, if a character destroys a tesseract (also known as a hypercube), this is considered a fourth-dimensional feat because a tesseract is a four-dimensional object.

This method differs from traditional power scaling, which might focus on raw destructive force without considering the dimensionality of the target. Understanding dimensionality is crucial because the higher the dimension, the greater the complexity and the scale involved.

Why Does Dimensionality Matter?

To understand the importance of dimensionality, imagine comparing a two-dimensional square with a three-dimensional cube. The square has length and width but no height. No matter how many squares you stack, they will never form a cube because the square lacks height. This is analogous to stacking an infinite number of zeros—no matter how many zeros you add together, you will never reach one.

This dimensional difference is important when it comes to understanding how powerful a character or object is. A character who can destroy a cube is vastly more powerful than one who can only destroy a square. To destroy something of higher dimensionality requires significantly more power.

Joules and Dimensionality

When we talk about power, we often measure it in joules, which is a unit of energy. A joule is defined as the energy transferred when a force of one newton moves an object one meter in the direction of the force. However, joules are tied to three-dimensional measurements—mass, distance, and time. Applying joules to objects of higher or lower dimensionality can create absurd results, such as "beyond infinite" joules, which aren't practically useful.

For instance, a square, being two-dimensional, has no mass in the third dimension, and thus, applying joules (a 3D measurement) to it becomes nonsensical. The same principle applies to characters or objects that operate beyond three dimensions.

Higher Dimensions: What is a Fourth-Dimensional Object?

To understand higher dimensions, we can start by extending familiar concepts like squares and cubes. A square is two-dimensional, and a cube is three-dimensional. A fourth-dimensional object, such as a tesseract, is an extension of these concepts into a fourth spatial axis, one that we cannot perceive directly.

Importantly, the fourth dimension in this context does not refer to time, as many people mistakenly believe. Time is a separate concept from spatial dimensions. A tesseract occupies four spatial dimensions, and the difference between it and a cube is just as significant as the difference between a cube and a square. Just as a square cannot stack into a cube, a cube cannot stack into a tesseract.

Infinite Dimensions and Beyond

Dimensional tiering can be extended beyond just four dimensions. In theory, there can be any number of dimensions—five, ten, even infinite. An object or space with an infinite number of dimensions is referred to as "high hyperversal," while an object that exists beyond all dimensions is called "outerversal." These terms are used to describe beings or spaces that are beyond conventional dimensional scaling.

The difference between low outerversal and high hyperversal is greater than the difference between a three-dimensional object and a high hyperversal one. While you could theoretically stack an infinite number of dimensions on top of each other, an outerversal entity is beyond dimensionality itself, meaning it exists in a realm where dimensions do not apply.

Beyond Dimensionality: Outerversal Entities

An outerversal entity is one that exists beyond the very concept of spatial dimensions. To understand this, we need to consider the nature of concepts. A concept, in this context, refers to an abstract rule or object that governs a particular idea. For example, the concept of an apple is the abstract idea that makes all apples possible. Similarly, the concept of space is what makes spatial dimensions possible in the first place.

An outerversal being is one that exists beyond the concept of space and, by extension, beyond all dimensionality. This is much more impressive than simply being beyond three-dimensional space, as it encompasses all possible dimensions.

High Outerversal and Beyond

Just as there are tiers of dimensional beings, there are tiers beyond outerversal. High outerversal beings exist beyond other outerversal entities in the same way that a normal universal being is beyond an average human. Although it is possible to imagine tiers even higher than high outerversal, this is typically where most discussions on dimensional tiering conclude.

Transfinite Dimensionality

Lastly, it’s important to note that simply having a transfinite (or infinite) number of dimensions does not automatically make an entity outerversal. Being bound by any number of dimensions, even an infinite number, still places that entity within the realm of dimensionality. To be considered outerversal, a being must transcend dimensions entirely.

Conclusion

Dimensional tiering offers a way to scale fictional characters in a manner that acknowledges the complexity of dimensions. Characters or objects that operate in higher dimensions are far more powerful than those limited to lower ones. However, true power is achieved when a character exists beyond dimensionality altogether, in the realm of outerversal or even high outerversal beings. Understanding these concepts is essential for accurately gauging the power and scale of characters in fiction.

Debunking

To understand the importance of dimensionality, imagine comparing a two-dimensional square with a three-dimensional cube. The square has length and width but no height. No matter how many squares you stack, they will never form a cube because the square lacks height. This is analogous to stacking an infinite number of zeros—no matter how many zeros you add together, you will never reach one.

This dimensional difference is important when it comes to understanding how powerful a character or object is. A character who can destroy a cube is vastly more powerful than one who can only destroy a square. To destroy something of higher dimensionality requires significantly more power.

No.

First, let’s examine how higher dimensions function in reality. Quantities like mass, density, and energy are unaffected by the existence of higher geometric dimensions—they can flow through them equally because these are scalars, not vectors. A higher geometric dimension is not inherently larger or more "powerful" than a lower one, nor does it contain more or a superior form of energy.

Additionally, higher geometric dimensions are contingent upon lower dimensions for their existence. The fourth dimension relies on the third as its foundation, just as the third depends on the second, and so forth. If the length and width from lower dimensions were removed, the third dimension could not exist, as it is built upon those fundamental properties. Just as apples cannot grow without trees, higher dimensions cannot exist without the lower ones they are composed of. Therefore, it is illogical to view higher geometric dimensions as vastly superior to lower ones when they are, by nature, dependent on them.

This principle also suggests that a three-dimensional being could harm a four-dimensional being by targeting its structure on the three-dimensional plane, as the body of the higher-dimensional being would still be comprised of lower-dimensional components.

Geometric dimensions are fundamentally about adding more directions—not about enhancing mass or energy. Mass is not dependent on the number of dimensions an object exists in. Even if one were to argue that density could be relevant, this only further emphasizes the point: density is mass per volume, but the mass itself is independent of the object’s volume. For instance, if you compress an object and reduce its volume, its mass remains the same. Similarly, if you stretch the object and increase its volume, the mass remains unchanged. This demonstrates that the mass of an object is not tied to its size or dimensionality.

A singularity provides the most extreme example of this, with stellar masses compressed into an infinitely small space. Despite the unique interplay between dimensions, scalars like mass and gravity are not confined by vectors or higher-dimensional constraints. Gravity, for instance, is believed to interact freely with extra dimensions. This is one reason why gravitons, the hypothesized carriers of gravity, might vanish into these dimensions, potentially leaving behind imbalances in momentum and energy—a possible avenue for evidence in support of string theory.

Or, as the OP of the VSBATTLE thread says:

I will simply start by saying that this notion is completely wrong even when going by our basic understanding of physics: This is due to the fact that Energy is a Scalar Quantity, which means that it is described by only one value, its magnitude, and thus does not depend on direction or displacement of any kind.

A Vector Quantity, on the other hand, is a quantity that is dependent not only on its Magnitude, but on its Direction and Displacement as well. A good example of the distinction between Scalar and Vectors would be the concepts of Speed and Velocity - Speed, as it is commonly defined, is the rating which is used to describe how fast an object is moving, or in more formal terms, the rate in which an object covers distance: this definition is a constant regardless of motion and direction, as it is only defined by Magnitude and nothing else, a car moving at 70 km/h will have the same Speed regardless if it is heading towards North, West or East. Thus, Speed is a Scalar Quantity, one that never changes regardless of directions.

Velocity, on the other hand, is the rating in which an object changes position, that is, it defines not only the rate in which an object covers distance, but also the direction in which it is doing so. Indeed, to shamelessly quote Wikipedia on this:

If a car is said to travel at 60 km/h, its speed has been specified. However, if the car is said to move at 60 km/h to the north, its velocity has now been specified.

The big difference can be noticed when we consider movement around a circle. When something moves in a circular path (at a constant speed, see above) and returns to its starting point, its average velocity is zero but its average speed is found by dividing the circumference of the circle by the time taken to move around the circle. This is because the average velocity is calculated by only considering the displacement between the starting and the end points while the average speed considers only the total distance traveled

Now that the difference between Scalar and Vector Quantities is established, it is fairly easy to see where I am going with this, as I mentioned earlier: energy is a scalar quantity, and so are mass and speed, the quantities which are used to measure energy, and by extension, power, in the first place. Hence, energy is a dimensionless concept, and is fully independent from direction or displacement. This obviously doesn't mean that any higher-dimensional being is about as strong as lower-dimensional ones period, rather, it just means energy is a constant that is independent of dimensionality, and would exist regardless even in a universe of more than three dimensions

Of course, higher-dimensional beings would probably be a lot bigger and stronger than us, but that would be more due to external factors, rather than strictly about their dimensionality per se.

Now, it is true that we cannot calculate the "Area" of a 3-dimensional object, or the "Volume" of 4-dimensional constructs. Trying to do such is indeed a fruitless endeavour that won't get you anywhere, but this is not at all because they are Infinite in size compared to 2 and 3-dimensional objects respectively, far from it.

This is because they have an additional parameter which describes their size and displacement in an area of space, another axis of movement to dislocate themselves through. Thus, another variable has to be taken into account in the context of whatever equation you are using to calculate the size of a given shape. In fact, in most cases you just have to use another equation altogether. It is basically a case of overall incompatibility, not difference in magnitude.

There is also the argument that, since, a higher-dimensional object possess an infinite number of slices of lower-dimensional objects within itself, then they are necessarily infinitely larger than them, and as a consequence of that, you can't really stack up lower-dimensional objects to get yourself a higher-dimensional one.

The former is actually geometrically accurate, there is, say, an infinite number of squares within a cube, but the issue is that this is mostly a very basic geometrical notion that doesn't really conform to how this stuff would play out within the physical world we live in, similar to how any object can be mathematically visualized as an uncountably infinite set of points, no matter how large or small. Higher and lower-dimensional objects are just... arbitrarily large, and can really be of literally any size.

When we talk about power, we often measure it in joules, which is a unit of energy. A joule is defined as the energy transferred when a force of one newton moves an object one meter in the direction of the force. However, joules are tied to three-dimensional measurements—mass, distance, and time. Applying joules to objects of higher or lower dimensionality can create absurd results, such as "beyond infinite" joules, which aren't practically useful.

For instance, a square, being two-dimensional, has no mass in the third dimension, and thus, applying joules (a 3D measurement) to it becomes nonsensical. The same principle applies to characters or objects that operate beyond three dimensions.

No.

However, I would like to note that this has little relation to Size per se; rather, it is due to the fact that the 3-dimensional object extends into another direction of space entirely, which the set of 2-dimensional objects can't reach because they lack the parameters needed to do so.

Remember, Dimensions are merely axis of space-time movement, and under Coordinate Spaces the above argument falls apart even much more clearly - Suppose that we have a Coordinate System with Three Axes (System A):

X = 999

Y = 999

Z = 999

W = 0

And then we have another one, comprised of Four Axes (System B):

X = 1

Y = 1

Z = 1

W = 1

Obviously, System B can never be reached by the System A, but this is simply because it has a greater value on the W axis, which the latter System lacks entirely. It doesn't translate to "lol W axis is infinitely more bigger than (999,999,999)", rather, it translates to "W axis is to the left and even if you go infinitely up and up and up it'll never reach it".

Even then, you can just increase the value of the W axis to 1 in System A. You don't need to accelerate the parameters X, Y and Z to infinity to try and make W increase to 1.

In layman's terms, what the argument I detailed above (and Dimensional Tiering as a whole, really) proposes is analogous to saying that North is infinitely far away from West, since you can walk infinitely in the direction of the former and never reach the latter, when this only happens because West is positioned in another direction relative to North, and you can simply take a step in its direction and reach it with no problem whatsoever.

Additionally, we are actually capable of calculating Hypervolume. This is even more evidence that Dimensions don't represent orders of infinity even in a geometrical sense.

In fact, it is indeed possible to calculate the radius of higher-dimensional spheres without the values accelerating to infinity, so that should tell you something. Of course, there are also examples for actual physics, such as Strings, which are 1-dimensional objects, yet are planck-length, not infinitely smaller than we are.

The latter is also intuitively accurate, but not so much diving deeper into it: Yes, you can't really stack up a lower-dimensional thing to get you a higher-dimensional thing, but this isn't really because the latter is infinitely bigger, rather it is because it has another direction through which its body can displace itself into.

The lower-dimensional object can't actually displace itself in said direction because it doesn't exist to them. It's a matter of position, rather than size: 1 meter in an additional axis isn't really bigger than 2 meters in the other axis, the former is just inaccessible to the latter because it is displaced in another direction, much like how west isn't infinitely far away from north because you can accelerate as far as you want in the direction of one and never reach the other, or how one second of time isn't really bigger than the entire Universe. The distance between them can be better described as "undefined", rather than infinite.

The thing about them is that they exist in different "sets". Dimensions are, for the most part, defined through real coordinate spaces, which are basically coordinate systems that denote said coordinates using the real numbers.

They are denoted as R^n (where R = the line of real numbers, equivalent to an axis of the space, pretty much a line. And n = the number of dimensions, which you get by taking the cartesian product/multiplication of R), with R^3 (R x R x R) being 3-dimensional space, R^4 (R x R x R x R) being 4-dimensional space and so on.

This relates to what I said in the sense that, for example, 4-dimensional objects are subsets of R^4, while 3-dimensional ones are subsets of R^3, and so on. The spaces don't "intermingle", and the difference between subsets in R^3 and subsets in R^4 is undetermined, since, again, they are in different sets altogether with a different number of axes. When you get down to it, you can only rely on the "lol uncountably infinitely larger" thing for the whole set, not just some random subsets of it.

(On a side note, I’ll disagree with this part. All R^n spaces are the same cardinality, each being bijectable to each other.

https://math.stackexchange.com/questions/183361/examples-of-bijective-map-from-mathbbr3-rightarrow-mathbbr

In terms of cardinality (or "size" of the set), all geometric dimensions of ℝ^n spaces are the same size. Here's why:

1. Bijection Between ℝ and ℝ^n:

2. Implications for Dimensions:

Conclusion:

All geometric dimensions (1D, 2D, 3D, etc.) have the same cardinality, c. So, in terms of "size" as measured by cardinality, they are all the same, even though they differ geometrically and conceptually.)

Dimensional tiering can be extended beyond just four dimensions. In theory, there can be any number of dimensions—five, ten, even infinite. An object or space with an infinite number of dimensions is referred to as "high hyperversal," while an object that exists beyond all dimensions is called "outerversal." These terms are used to describe beings or spaces that are beyond conventional dimensional scaling.

The difference between low outerversal and high hyperversal is greater than the difference between a three-dimensional object and a high hyperversal one. While you could theoretically stack an infinite number of dimensions on top of each other, an outerversal entity is beyond dimensionality itself, meaning it exists in a realm where dimensions do not apply.

Sure, but it doesn’t matter lol.

An outerversal entity is one that exists beyond the very concept of spatial dimensions. To understand this, we need to consider the nature of concepts. A concept, in this context, refers to an abstract rule or object that governs a particular idea. For example, the concept of an apple is the abstract idea that makes all apples possible. Similarly, the concept of space is what makes spatial dimensions possible in the first place.

An outerversal being is one that exists beyond the concept of space and, by extension, beyond all dimensionality. This is much more impressive than simply being beyond three-dimensional space, as it encompasses all possible dimensions.

To be beyond space, one would have to not have any physical size or shape. To be beyond time, one would have to be changeless. Most fictional characters claimed to be outerversal aren’t. Like 99% of them aren’t.

Also this wouldn’t be an attack potemcy. Your energy transcends time…so you can’t punch…and it transcends space…meaning it can’t enter physical space to harm someone.

The only thing that would be outerversal are concepts by default.

Just as there are tiers of dimensional beings, there are tiers beyond outerversal. High outerversal beings exist beyond other outerversal entities in the same way that a normal universal being is beyond an average human. Although it is possible to imagine tiers even higher than high outerversal, this is typically where most discussions on dimensional tiering conclude.

This is the old system of VSBW, and it doesn’t fucking work. From the horse’s mouth itself.

https://vsbattles.com/threads/on-the-many-many-incoherences-of-the-tiering-system.159803/#post-6111278

Lastly, it’s important to note that simply having a transfinite (or infinite) number of dimensions does not automatically make an entity outerversal. Being bound by any number of dimensions, even an infinite number, still places that entity within the realm of dimensionality. To be considered outerversal, a being must transcend dimensions entirely.

Outerversal beings couldn’t fight lol. And an infinite dimensional being would die horribly to anyone Aleph 1 and up…that is, if you could use transfinites to measure physical size, mass, and energy…but you cannot.

Debunking Hausdorff Dimensions

One of the main justifications for Higher Dimensions equaling greater orders of Infinities is the use of the concept of a Hausdorff dimension, which, according to our own pages elaborating on this aspect of the Tiering System, proposes that each Higher Dimension is uncountably infinitely greater than a lower one.

However, Felix Hausdorff's formalisms are extremely specific, and assuming all Higher-Dimensions in fiction need to abide to them is frankly an absurd idea that borders on the pseudo-intellectual.

To start with, Hausdorff Dimensions are simply a way of measuring Fractal Sets and Topologies, which usually have non-integer Dimensions and thus can't be measured using regular Euclidean Geometry and all that junk. They have absolutely no relation to Spatio-Temporal Dimensions that are used in Cosmological Models or Manifolds, which is generally what we use when dealing with the rating of fictional verses.

Yes, theoretically you can obviously model a Formalism based on Hausdorff Fractal Topologies, but, as I stated above, it would be an insanely specific framework that is unseen in any notable cosmology, both in Fiction and in Theoretical Physics, and once again, acting like every single Higher-Dimension needs to act like a Hausdorff Dimension is a fine example of Cherry-Picking.

Furthermore, even stating that Hausdorff's Formalisms establish that Higher Dimensions are infinitely superior to lower ones is erroneous anyway. It effectively argues that due to the existence of Non-Integer Dimensions under Fractal Geometry, and the fact you can infinitely extend any subsets and decimals between any two Integer, the distance between 1 and 2 is Uncountably Infinite, something which, as I keep repeating, is not used in any Manifold or Model describing our Universe / Multiverse, and is generally inapplicable to them unless you model a very specific framework.

So, no, Hausdorff Fractal Geometry doesn't state that Higher Dimensions are infinitely greater than lower ones, it just uses Dimensions as a way of measuring the complexity of shapes and nothing more.

On the Many, Many Incoherences of the Tiering SystemVS Battles Wiki Forum