Dimensional Tiering

## **Dimensional Tiering Standards**

**The dimensional theory that this tiering system primarily conforms to is that of the Euclidean Vector Space.**

## **Unit interval**

It is important to understand what a unit interval is in mathematics. A unit interval is an open interval between 0 and 1, it is a subset of real numbers. Because of the nature of real numbers, specifically how they can be used to represent points along an infinitely long line. The line segment of length 1, which is practically 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 a set of real numbers entirely. This is why it's regarded as an uncountable set.

Now how is this significant for dimensions? Because dimensions are n-tuples of real numbers. I defined N-dimensional spaces, which is practically what dimensions are generalized to be.

This means the first dimension is an equivalent of the unit interval since it's part of the infinitely long line a subset of it with length 1. Again, it's important to remember that even for the 1st dimension it would be equivalent to the entire set of real numbers. The next dimension would be the second dimension, which is practically a multiplication of 2 sets of real numbers. It entails that the 2nd dimension is uncountably greater than the first dimension. When I explained a little bit of set theory in absolute infinity, I explained the cardinal of continuum and power setting.

Essentially a set of real numbers is an equivalent of natural numbers to the power of natural numbers, likewise in this case having the real numbers to the power of real numbers to represent the second dimension, implies it's a greater set-in cardinality to the first dimension uncountably infinitely greater at that. We even know that a set of real numbers to the power of real numbers of all functions from real numbers to real numbers is a strictly greater set in cardinality to real numbers.

This is generalized to every other dimension and higher dimension since N-dimensional space uses the same arithmetic in that sense to obtain higher dimensions.

Any higher dimension is uncountably infinitely superior to lower dimensions now if you went over the continuum hypothesis, you'd get an explanation as to how the first uncountably infinite cardinal is equivalent to a set of real numbers, you can virtualize dimensions as being something akin to ℝ (real number line) in which 1 dimension in the line has infinitely many subsets.

The demonstration of this goes as follows.

ℝ^1 which within 1 would contain infinitely many smaller subsets of 1 (0.111111111,0.11111112......, R^2), the next dimension would be an equivalent to making multiplications of the previous subset demonstrated here as ℝ^2 / ℝ×ℝ, which would also have yet even more uncountably infinitely many subsets.

Now it's important to note higher dimensions don't inherently grant you greater AP within fiction we only treat them as such if it's demonstrated that they have qualitative superiority to each other as further extrapolated below, This is important to learn because in this wiki we don't strictly operate under dimensional tiering as such tendencies cause a lot of inconsistencies as a result of a lot of fictional works having higher dimensions but don't necessarily conform to what one would call "reality fiction" or the actual mathematic theorems that describe the gap in magnitude between dimensions.

Unless it's proven within the specific context that higher dimensional beings are following these certain aspects that constitute for them gaining a tier within the tiering that relates to their dimensional existence.

A higher dimensional construct within set verse has to be shown to be embedded into the whole of the previous dimensions as per real coordinate space.

R= real number

N=tuples

Real coordinate space uses a set of tuples to constitute dimensional layers.

The set of all points with n coordinates is denoted R^n and is referred to as n-dimensional Euclidean space or simply n-dimensional space.

In R^3 3the three coordinates of a point are usually called x, y, and z. When working with four or more dimensions, though, it is too cumbersome to use a different letter for each coordinate. Instead, we refer to the n coordinates in Rn as x1, x2, and so forth.

In real coordinate space dimensions are represented by real numbers /ℝ

R×R×R=3 Dimensions/ℝ^3= 3 Dimensions

The following example above is what a higher dimension should look like you can view it in a simplified way as dimensions being multiplications of previous subsets, with adding the 4th dimension embedding it into the whole of the 3rd dimensions which would appear like this.

ℝ^4 because of it being a multiplication of the previous subset adding the next dimension in a perpendicular position essentially making the whole of the 3rd dimension to be a previous subset of the embedded 4th dimension because of this the 4th effectively views the 3rd as for what goons would call "fiction" this is generalized for all higher dimensions akin to continuous Cartesian products of (ℝ ∞^∞^∞^∞^∞/ 5 dimensional/ℝ^5).

Statements such as higher dimensional characters viewing lower dimensional characters within a fictional setting suffice as well as a justification for why higher dimensions should grant greater AP (Low complex multiversal and up).

## **Summary**

Real coordinate space has a relation to Euclidean space in general sometimes denoted as N-dimensional spaces in vector calculus. For every number of space, it is a tuple in a real number line.

### **Example**

*The first dimension would be R^1 which is an equivalent of the number 1 in a real number line and the second dimension R^2 and so forth.*

When it comes to unit intervals in calculus, they're an uncountable set, for several reasons. There are even demonstrable mathematical proofs for this. So essentially the unit interval [0, 1] is an uncountable set and you can view [0, 1] as the first dimension this can get carried on to every other dimension. The unit interval for the second dimension can just be applicable a ls [0, 1]. There's another reason for every dimension being an uncountable set which we'll get to again which pertains to unit intervals.

Essentially if you look at it here, they explain how in set theory (which is true). Any subset of the real number line has the same cardinality as the entire real number line. Because any subset of the real number can't have a 1-1 correspondence with a set of natural numbers therefore its inherently a greater cardinality which I'll get to in a minute. So, R^1 as a subset of the entire real number is an uncountable set this is another way to show it.

Moving on to the continuum hypothesis, this is where you can demonstrate how each dimension is not only uncountably infinitely greater than the last but inaccessibly greater too, now the continuum hypothesis states that there's no cardinality between aleph null and aleph 1. Hence anything with an arbitrarily larger cardinality than aleph null is aleph 1, in this case, take the first dimension to be an aleph because the next dimension is a greater cardinality which entails that it'll be an equivalent of the next aleph.

Now we got this, this shows how any dimension inherently has properties of inaccessibility and greater. Aleph null is a regular strong limit cardinal; meaning it cannot be obtained from repetitive successor operations.

Think of it like this a set of natural numbers which is aleph null, is beyond any finite number, and is a collection of all of them. The language of basic arithmetic in first-order logic infers that for any natural number x there's always going to be the successor to the natural number meaning there can never be an end to them hence a collection of them is infinite if you keep using the successor function S(0), S(0) .... And so forth over and over again you'll never reach the entire set of natural numbers.

Because each dimension has this property as a result of being an uncountable set hence encompassing countable infinite sets which are regular limit cardinals each dimension intrinsically on vector calculus has properties of inaccessibility and is uncountably infinitely greater.

If you require the explanation in a video format you can watch this