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Q: When are higher dimensions not viable to use as evidence for Tier 2 and above?

A: Whether higher-dimensional entities qualify for such high tiers or not depends on several different factors, which may take root both in and out-of-verse. To explain this situation, we must first clarify what exactly being higher-dimensional entails.

Are higher-dimensional beings infinitely larger than lower-dimensional equivalents?

In a way, yes, though not how most would think when using this word. Basically, an arbitrary object of dimension n is essentially comprised by the total sum of uncountably infinite objects of one dimension less, which may be described as lower-dimensional "slices", each corresponding to one of the infinite points of a line. For instance, a square is made of infinitely many line segments (Lined up on the y-axis), a cube of infinitely many squares (Lined up on the z-axis), and so on.

One may think of it as a multiplication between sets: For instance, the unit square [0,1]² may be expressed as the product of two unit intervals [0,1] x [0,1], which itself can be visualized as taking "copies" of the first interval and lining them up along each point of the second interval, of which there are uncountably infinitely-many, thus forming a square out of infinite line segments.

Are higher-dimensional beings infinitely stronger than lower-dimensional equivalents?

Unintuitive as that may be: Not necessarily, as a number of characteristics through which we quantify the strength or power of a character can remain unchanged when transitioning between higher and lower dimensions. For example: Mass is a quantity that is detached from the dimension of the object which it is inherent to, and unlike volume is not divided in units corresponding to each particular dimension (1-volume [length], 2-volume [area], 3-volume, 4-volume...). It is singular in nature and its units equally apply to all dimensions; whether it is distributed over an area or a volume only tells us about the span of space in which it is spread, not about the quantity itself.

As a consequence of that, much of the calculation methods which are used to measure strength apply equally to both higher and lower dimensions, as they do not care about the extra variables and often work with a single one of them. Examples of this are kinetic energy (Ek=0.5*M*V^2), force (F=M*A), work (W=F*d), and etc.

An intuitive example of that is found in the general definition of Work as defined in physics: In essence, as work itself denotes the energy applied to an object as it is displaced along a given path, the basic formula for calculating it only takes into account a single variable, and the path itself is treated as an one-dimensional object, regardless of the dimension of the space in which the action itself takes place.

Hence, a higher-dimensional entity can be both stronger or weaker than a lower-dimensional one, and thus, they are usually quantified based on their own feats, instead of dimensionality alone. If a character is merely stated to be higher-dimensional and simultaneously has no other feats to derive anything noteworthy from, then they are put at Unknown, and the same applies to lower dimensions as well.

Do note, however, that them not qualifying for Tier 2 and above doesn't mean they are "fake" higher-dimensional beings or anything of the sort. It is simply that being higher-dimensional does not inherently mean they have infinite power in the first place, as explained above.

Q: When are higher dimensions valid, then?

A: One of the more straightforward ways to qualify for Tier 2 and up through higher dimensions is by affecting whole higher-dimensional universes which can embed the whole of lower-dimensional ones within themselves. For example: A cosmology where the entirety of our 3-dimensional universe is in fact a subset of a much greater 4-dimensional space, or generalizations of this same scenario to higher numbers of dimensions; i.e A cosmology where the four-dimensional spacetime continuum is just the infinitesimal surface of a 5-dimensional object, and etc.

However, vaguer cases where a universe is merely stated to be higher-dimensional while existing in a scaling vacuum with no previously established relationship of superiority towards lower-dimensional ones (or no evidence to infer such a relationship from) should be analysed more carefully. In such cases where information as to their exact nature and scale is scarce, it is preferable that the higher dimensions in question be fully-sized in order to qualify.

Furthermore, higher-dimensional entities can also qualify for higher tiers when the verse which they are from explicitly defines them as being infinitely above lower-dimensional ones in power and/or existential status. An example of this being verses such as Umineko no Naku Koro ni. However, lower-dimensional beings being stated to be "flat" in comparision to higher-dimensional aliens is not necessarily grounds for assuming the latter has infinitely more power (For reasons outlined in the answer above), and thus, such scenarios must also be analyzed case-by-case.

Q: How do temporal dimensions impact on tiering?

A: The relationship between the spatial dimensions of a universe and the additional temporal dimension(s) may be visualized as something akin to the frames of a movie placed side-by-side. Basically, the time-like direction may be thought of as a line comprised of uncountably infinite points, each of which is a static "snapshot" of the whole universe at any given moment, with the set of all such events comprising the totality of spacetime.

This structure can then be generalized to any amounts of dimensions, and is also the reason destroying a spacetime continuum is a greater feat than destroying only the contents of the physical universe (Low 2-C, rather than 3-A or High 3-A). So, for example, a spacetime continuum comprising two temporal dimensions (Instead of just one) would have an additional time direction whose "snapshots" correspond to the whole of a 4-dimensional spacetime, and so on and so forth.

Q: Do higher-dimensional entities automatically get Immeasurable speed?

A: No. To put it simply: Although the presence of the additional axis results in a higher-dimensional space being infinitely larger in comparision to a lower-dimensional one, the numerical values themselves remain unchanged, as a "dimension" is nothing more than a continuum of numbers representing a direction of space. Consequently, the euclidean distance between a Point A and a Point B is always represented by an one-dimensional path (Regardless of the dimension of the space in which they are embedded), whose length is always measurable and given by a generalization of the Pythagorean Theorem to n dimensions. That is:

Pythagoras.png

Or, in plain english: Subtracting each of the coordinates of the starting and ending points (Distance is always given by the absolute value, so whether the result is negative or not is irrelevant), squaring the results, summing them up, and then taking the square root of the resulting value.

So, for example, in two-dimensional space, the distance between the points (4,4) and (8,8) is calculated through the following formula:

d(4,8) = |√(4 – 8)² + (4 – 8)²|

d(4,8) = √4² + 4²

d(4,8) = √16 + 16

d(4,8) = √16 = 4

Then, to generalize this to higher dimensions, we only have to take into account the additional variables. For example, in 6-dimensional space, the distance between the coordinates (2,2,2,2,2,2) and (8,8,8,8,8,8) is:

d(2,8) = |√(2 – 8)² + (2 – 8)² + (2 – 8)² + (2 – 8) + (2 – 8)² + (2 – 8)²|

d(2,8) = √6² + 6² + 6² + 6² + 6² + 6²

d(2,8) = √36 + 36 + 36 + 36 + 36 + 36

d(2,8) = √216 = 14.6969384567 ≈ 15

Taking the "15" to be some arbitrary unit of distance, it is then perfectly possible to gauge a defined speed for a character who crosses it in a given length of time. For example, if 15 in this case is 15 meters, then a character who crosses this in a second would naturally be moving at 15 m/s

Similarly, moving in a higher-dimensional space also doesn't qualify as Immeasurable speed, and would be more appropriately rated as Interdimensional range

Of course, this formula doesn't always illustrate how the distance between two points works in real life, as the Earth has curvature and is obviously not a perfectly flat plane like Euclidean Space (At least non-locally), and the same applies to the universe at large. This is no issue, however, as there are plenty of metrics that can be applied to non-euclidean spaces: For example, the distance between two points on the surface of a sphere is given by a geodesic, as opposed to a straight line passing directly through the sphere's interior, whose length is itself given by its spherical distance.

They can qualify for Immeasurable Speed, however, if the regular dimension of time appears like a spatial dimension from their higher dimensional perspective. That is to say, that it can freely be traversed both forward and backwards, allowing them to access any point in it and move unbound by the notions of time inherent to the lower space. An example of this are the Bulk Beings from Interstellar.

Q: Is destroying multiple infinite multiverses a better feat than destroying a single one?

A: In spite of what our intuitions may tell us, destroying or otherwise fully affecting multiple infinite-sized multiverses is in fact not a better feat than doing the same to a single infinite multiverse, and thus, not above the "baseline" for 2-A

The reason behind this is that the total amount of universes contained in a collection of multiple infinitely-sized multiverses (Even one consisting of infinitely many of them) is in fact equal to the amount of universes contained in a single one of the multiverses that form this ensemble: It is countably infinite, as the union of countably-many countable sets is itself countable, and thus does not differ in size from its components. Thus, only an uncountably infinite number of universes actually makes any difference in terms of Attack Potency, at this scale.

This illustrates some of the more unintuitive properties of sets with infinite elements: Namely, given a set X, it being a subset of another set Y does not imply that Y > X in terms of size. An example of this is how the set of all natural numbers contains both the odd numbers and even numbers, yet all of these sets in fact have the same number of elements.

However, such a feat may indeed qualify as stronger if the verse itself treats it as such.

Q: How do cardinal numbers relate to tiering?

A: Firstly, it should be highlighted that asking about the tier of a cardinal number is effectively a meaningless question when the quantity which it is denoting is not specified in the question as well, and makes as much sense as asking "What tier is the number 8?"

Let's take the smallest infinite cardinal (aleph-0, or ℵ0, the cardinality of countably infinite sets) as an example in this case: A set comprised of a countably infinite number of 0-dimensional points is itself a 0-dimensional space under the usual notions of dimensionality, being thus still infinitely small. Meanwhile, a countably infinite number of planets is High 3-A, a countably infinite number of universes 2-A, and countably infinite dimensions High 1-B.

We then move on to the power set of ℵ0, P(ℵ0), which is an uncountably infinite quantity and represents the set of all the ways in which you can arrange the elements of a set whose cardinality is the former, and is also equal to the size of the set of all real numbers. In terms of points, one can say that everything from 1-dimensional space to (countably) infinite-dimensional space falls under it, as all of these spaces have the same number of elements (coordinates, in this case), in spite of each being infinitely larger than the preceding one by the intuitive notions of size that we regularly utilize (Area, Volume, etc)

On the other hand, an P(ℵ0) number of universes is Low 1-C, and a similar number of spatial dimensions/layers of reality is Low 1-A

However, the same does not necessarily apply when approaching sets of higher cardinalities than this (Such as P(P(ℵ0)), the power set of the power set of aleph-0), as they would be strictly bigger than all of the spaces mentioned above, by all rigorous notions of size, regardless of what their elements are. From this point and onwards, all such sets are Low 1-A at minimum.

Do note, however, that these infinities must specifically refer to elements that physically exist within a verse's cosmology. Them existing as in-universe mathematical concepts is not sufficient for anything to scale to them, unless there is a direct comparision that allows scaling to be made.

Q: How do I determine if something is "transcendent"?

A: "Transcendence" is a vague term which can be used in several contexts, many of which do not at all align with how it is normally used in our forums, as it simply means "to go beyond the ordinary", first and foremost. For example, statements of "transcending space and time" can refer to things like time travel, dimensional travel, or even agelessness in some cases. Hence, it is very preferable to ascertain the meaning of statements involving this term from background context (If there is any), being especially careful around flowery language or purple prose.

Now, one of the most common scenarios where this question might arise is when dealing with cosmologies involving "higher planes of existence" or similar structures. In such cases, it's very important to note what exactly being a "higher plane" entails in the context of the setting: For instance, it's very common for Heaven and Hell to be defined as higher and lower planes of existence respectively in relation to the normal universe, in which case, "higher" and "lower" tends to simply indicate their position in a cosmology, as opposed to any kind of existential status, which is obviously not enough for anything remotely Tier 1.

They can qualify, however, if said "higher plane" is defined as having a relationship of qualitative superiority over lower realms in one way or another, such as by perceiving them as literal fiction/unreality (or being comparatively more "real" in nature), encompassing them in an infinitesimal portion of itself, residing in a higher state of being altogether, and etc.

Q: What tier is transcending space and time?

A: As said above, "transcending space and time" is a very vague statement by itself and can mean multiple things depending on the context in which it is made, as well as how this characteristic is portrayed in the first place. It is perfectly possible for such a statement to mean that a character is simply "untied" from the universe's spacetime, and is thus unaffected by alterations in the timeline and similar meddlings. Likewise, it's not exactly uncommon for time travel (Or any action / process that affects something through different points in time) to be described as "transcending time and space."

However, if it is specified that they "transcend space and time" in the sense that they exist on some higher level of reality that is outright superior to a spacetime continuum in nature, then they should be put at Low 1-C, assuming the continuum in question is one comprised of four dimensions. The answer may vary depending on this factor.

It should also be noted that simply existing in some alternate state of existence that lacks time and/or space is not really grounds for any tier in particular, as lacking such things does not translate to being superior to them, and would most often overlap with abilities like Acausality or Nonexistent Physiology. A good example of a case like this is Dormammu (Marvel Cinematic Universe), who is stated to exist in a realm "far beyond time," yet never actually displays any superiority over it, and is in fact vulnerable to time-based abilities due to his timeless nature.

Q: How do I differentiate between 1-A, High 1-A and 0 in practice?

A: To put it simply, these three tiers are all defined by the property of being "inaccessible" in relation to a certain starting point: For instance, Low 1-A is defined as a size that is unreachable in relation to any countable number of dimensions and/or higher realms of existence, while 1-A trivializes such sizes in a similar manner. High 1-A is an extension of this idea, and marks states that are undefinable in relation to 1-A realms and beyond any extensions thereof, while 0 transcends them in a similar manner.

Do note, however, that these definitions only involve transcending all forms of size in contrast to a specific scale. Though it's indeed vastly easier to get these levels when an infinite-dimensional space (or similar structure) is present, such things are not necessary to qualify for them, in principle.

Q: How can a character be 1-A and above without an infinite-dimensional/infinitely-layered cosmology, then?

A: A good way to accomplish this would be to show that whatever state of being in which they exist is completely independent of the number of layers/dimensions present on the setting. For example, if they are unaffected by dimensions being arbitrarily added or removed from physical space by virtue of transcending it entirely, or if they exist as a "background" or canvas of sorts in which any amount of them can be inserted. This argument generalizes to tiers higher than 1-A as well.

Q: Is transcending an 1-A character to the same degree they transcend normal humans High 1-A?

A: No. Due to making use of a much larger measuring stick in comparision to lower tiers (Power sets of infinite sets, as opposed to adding individual dimensions), the gap between any two levels in 1-A actually exceeds the entire system below them, and is equivalent to repeating the whole process which led to the previous level on a much higher scale. Thus, most statements that make use of such comparisions would only amount to one further level into the tier, unless some additional context showing it to be higher is present.

Q: Is predating the concepts of space and time an 1-A feat?

A: No. As said above, predating a certain concept does not necessarily imply any form of superiority over it, especially not to the degree where it warrants an 1-A rating.

Q: What tier is transcending dimensions?

A: As specified above, a "dimension" is nothing more than a set of values representing a given direction within a system, and a multi-dimensional space can itself be thought of as a multiplication of several "copies" of these sets. For instance, the 3-dimensional space in which we live is often visualized as the set of all 3-tuples of real numbers (Thus, taking its values from the real number line, R), and is thus the result of the iterated multiplication: R x R x R = R³, likewise, 4-dimensional space is the set of all 4-tuples of real numbers, and is thus equal to R x R x R x R = R⁴, and so on and so forth.

Practically speaking, this means that there is no limit for the number of dimensions which a space can have whatsoever, and one can construct spaces whose dimension corresponds to any cardinal number, including the infinite ones mentioned above. It is not even necessary for us to restrict ourselves to values taken from the real numbers, either: It is also possible to define the space of all n-tuples of cardinal numbers (Which takes its values from V, the class of all sets)

As a result, it is not at all feasible to take any statements involving a character existing "beyond dimensions" at face value, as this would lead to extremely inflated ratings largely dependent on No-Limits Fallacies. Therefore, such descriptors are to be evaluated while taking into account the number of dimensions which the verse has been shown to entertain; for example, a character stated to exist above physical dimensions in relation to a 4-dimensional cosmology would be Low 1-C with no further context.

See also