Dimensional Tiering
Dimensional Tiering Standards
The dimensional theory that this tiering system primarily adheres to is that of the Euclidean Vector Space.
Unit Interval
Understanding what a unit interval is in mathematics is essential. A unit interval is an open interval between 0 and 1, which is a subset of real numbers. Real numbers can represent points along an infinitely long line. The line segment of length 1, which represents the unit interval in this case, is a part of the infinitely long line that real numbers represent points along and has the same number of points as the whole line. This means the unit interval or open interval between 0 and 1 is equivalent to the entire set of real numbers. This is why it's considered an uncountable set.
How is this significant for dimensions? Because dimensions are n-tuples of real numbers. I defined N-dimensional spaces, which are what dimensions are generalized to be.
The first dimension is equivalent to the unit interval since it's part of the infinitely long line, a subset of it with length 1. Even for the 1st dimension, it would be equivalent to the entire set of real numbers. The next dimension, the second dimension, is practically a multiplication of two sets of real numbers. This means the 2nd dimension is uncountably greater than the first dimension. When I explained set theory in absolute infinity, I discussed the cardinal of the continuum and power setting.
Essentially, a set of real numbers is equivalent to natural numbers to the power of natural numbers. In this case, having real numbers to the power of real numbers to represent the second dimension implies it's a greater set in cardinality than the first dimension, uncountably infinitely greater. We know that a set of real numbers to the power of real numbers, or all functions from real numbers to real numbers, is a strictly greater set in cardinality than real numbers.
This concept is generalized to every other dimension and higher dimensions since N-dimensional space uses the same arithmetic to obtain higher dimensions.
Any higher dimension is uncountably infinitely superior to lower dimensions. If you explore the continuum hypothesis, you'll find an explanation as to how the first uncountably infinite cardinal is equivalent to a set of real numbers. You can visualize dimensions as being akin to ℝ (real number line), where 1 dimension in the line has infinitely many subsets.
The demonstration is as follows:
ℝ^1 contains infinitely many smaller subsets of 1 (like 0.111111111, 0.11111112, etc.), while ℝ^2 represents the next dimension, equivalent to multiplying the previous subsets as ℝ^2 / ℝ×ℝ, which also contains uncountably infinite subsets.
It's important to note that higher dimensions do not automatically imply greater attack power (AP). In fiction, we consider higher dimensions superior only if they demonstrate qualitative advantages over others. This is crucial for understanding, as this wiki does not adhere strictly to dimensional tiering. Such an approach often leads to inconsistencies since many fictional works feature higher dimensions that do not align with what one might call "reality fiction" or the actual mathematical theorems explaining the differences in magnitude between dimensions.
Unless it is specifically proven in context that higher-dimensional beings possess certain characteristics that warrant a tier based on their dimensional existence, they are not automatically assigned higher tiers.
A higher-dimensional construct in a given universe must be shown to be embedded within all preceding dimensions, following the principles of real coordinate space.
R = real number
N = tuples
Real coordinate space utilizes tuples to define dimensional layers. The set of all points with n coordinates is denoted as RN, which is known as n-dimensional Euclidean space or simply n-dimensional space.
In R3, the three coordinates are typically referred to as x, y, and z. For four or more dimensions, it’s more convenient to label the coordinates as x1,x2, and so on.
In real coordinate space, dimensions are represented by real numbers (R).
R×R×R=3 Dimensions (R3=3 Dimensions).
This example illustrates how a higher dimension operates. You can simplify it by viewing dimensions as products of earlier subsets, with the addition of the 4th dimension embedding it within the entire 3rd dimension. Thus, R4 can be seen as a multiplication of the previous subset, adding the next dimension perpendicularly, which makes the entire 3rd dimension a subset of the embedded 4th dimension. As a result, the 4th dimension perceives the 3rd, and this reasoning applies to all higher dimensions, analogous to continuous Cartesian products of (R∞∞∞∞/5th dimensional/R5).
Claims that higher-dimensional characters can perceive lower-dimensional characters within a fictional context provide justification for why higher dimensions should confer greater attack power (AP), particularly for low complex multiverses and beyond.
Summary
Real coordinate space relates to Euclidean space and is often referred to as n-dimensional spaces in vector calculus. Each dimension corresponds to a tuple on the real number line.
Example:
The first dimension is R1, equivalent to the number 1 on the real line, followed by the second dimension R2, and so on.
Unit intervals in calculus form an uncountable set for various reasons, supported by mathematical proofs. The unit interval [0,1] represents the first dimension and extends similarly to other dimensions. For the second dimension, the unit interval can also be [0,1].
In set theory, any subset of the real number line shares the same cardinality as the entire line, as no subset can be matched one-to-one with natural numbers, indicating a greater cardinality. Therefore, R1 as a subset of the real number line is also uncountable, demonstrating this point further.
Next, let's discuss the continuum hypothesis, which illustrates that each dimension is not only uncountably infinitely larger than the previous one but also inaccessibly greater. The continuum hypothesis asserts that there is no cardinality between aleph null and aleph 1. Therefore, any set with a larger cardinality than aleph null is classified as aleph 1. In this context, consider the first dimension as aleph null, while the next dimension represents a greater cardinality, thus aligning with the next aleph.
This demonstrates that each dimension possesses inherent properties of inaccessibility and greater magnitude. Aleph null is a strong limit cardinal, meaning it cannot be derived from repeated successor operations.
To illustrate: the set of natural numbers, represented by aleph null, exceeds any finite number and encompasses all natural numbers. In basic arithmetic within first-order logic, for any natural number x, there is always a successor, indicating that the set of natural numbers is infinite. Continuously applying the successor function S(0),S(S(0)), and so on will never yield the complete set of natural numbers.
Since each dimension has this property due to being an uncountable set, which includes countably infinite sets (regular limit cardinals), every dimension in vector calculus inherently possesses attributes of inaccessibility and is uncountably infinitely greater.
Higher Dimensions
For n ≥ 4, the space is considered higher-dimensional. While all dimensions are of the same order of infinity, higher-dimensional spaces differ in terms of topology and structure rather than cardinality.
For instance, in ℜ², fixing z = 0 defines the x-y plane, meaning all points on this plane have a z-coordinate of zero. Similarly, an n-dimensional hypercube consists of points {x₁, x₂, ..., xₙ}, where each coordinate xᵢ ranges from 0 to 1. Each coordinate can be seen as a unit interval, and fixing one coordinate creates a subspace of lower dimension. For example, fixing x₁ = 0 in ℜ⁴ results in a 3D subspace within the 4D hypercube.
This principle extends to any dimension, demonstrating how higher-dimensional spaces embed lower-dimensional ones. Each additional dimension allows for greater complexity and degrees of freedom, enabling configurations that are impossible in lower dimensions. Despite sharing the same order of infinity, higher-dimensional spaces contain richer structures and more intricate relationships between points.
Temporal Dimensions
Time can be treated as either discrete or continuous. If time t is discrete, it means that it consists of distinct intervals indexed by integers. A discrete time sequence is represented as {t₀, t₁, t₂, ...}, where t₀ is the starting point, and t₁, t₂, etc., are successive moments in time. The time difference between consecutive indices remains constant, ensuring uniform progression.
Mathematically, the next time step follows:
tₙ₊₁ = tₙ + Δt
tₙ = t₀ + nΔt
For continuous time, t belongs to the real numbers (t ∈ ℜ), meaning it can take any value within an interval. This allows for infinitely many moments between any two time points. Unlike discrete time, where functions are only defined at specific intervals, continuous time allows functions to be evaluated at every possible moment.
Metric Expansion
In a hypothetical expanding universe within a given fictional setting, expansion does not necessarily mean an increase in the literal size of space. Instead, it can be visualized as an infinite grid where the distance between points increases over time, rather than the grid itself growing.
A flat universe follows Euclidean geometry and is infinite, meaning it has no boundaries and extends infinitely. The Friedmann-Lemaître-Robertson-Walker (FLRW) metric describes cosmic expansion:
ds² = c²dt² + a(t)²(dx² + dy² + dz²)
where:
- ds represents the space-time interval
- c is the speed of light
- t is the time coordinate
- a(t) is the scale factor describing expansion
- x, y, and z are spatial coordinates
Space can remain infinite while expanding, as the scale factor a(t) increases over time. The proper distance d(t) between two points changes as:
d(t) = a(t) ⋅ d₀
where d₀ is the initial proper distance when a(t) = 1.
In comoving coordinates, points remain fixed while their proper distance increases over time. For example, if two galaxies start 2 billion light-years apart and the scale factor doubles, their proper distance increases to 4 billion light-years, but their comoving distance remains constant.
Closed and Open Universes
A closed universe is finite and bounded, meaning it has finite size but no edges. If one were to travel in a straight line, they would eventually return to their starting point. Expansion in such a universe means it will eventually contract back into a singularity. The metric describing a closed universe includes:
ds² = ℜ²[dψ² - dχ² - sin²χ dΩ²]
where:
- sin²χ indicates closed geometry
- χ ranges from 0 to π, describing a curved space looping back on itself
- ψ is a temporal coordinate
- χ is a spatial coordinate
- R is the universe’s radius
This bounded nature means that while the universe has no edges, it is finite in extent, similar to the surface of a sphere.
An open universe has hyperbolic geometry, curving outward with no boundaries, extending infinitely. It is also described by the FLRW metric but with a different spatial component. In such a universe, expansion continues indefinitely without leading to eventual contraction.
Conclusion
Higher-dimensional spaces provide a framework for understanding complex spatial structures. Whether considering discrete or continuous time, metric expansion, or the nature of the universe’s geometry, these concepts help describe and analyze the vastness of mathematical and physical spaces. The interplay between lower and higher dimensions illustrates how space and time can be extended beyond intuitive three-dimensional understanding, offering deeper insight into the nature of the universe.
