Tiering System Explanation

## **Introduction**

Welcome to the explanation page for the wiki! In this section we will be talking about some commonly asked questions. Please feel free to direct yourself to the Q&A section!

## **Explanation**

### **Cardinality:**

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The transfinite cardinal numbers, often denoted using the Hebrew symbol (aleph) followed by a subscript, describe the sizes of infinite sets.

Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets. In the case of finite sets, this agrees with the intuitive notion of size. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers. It is also possible for a proper subset of an infinite set to have the same cardinality as the original set—something that cannot happen with proper subsets of finite sets.

Large Cardinals are, speaking very roughly, certain cardinal numbers with the property of being very "large". That is, they are cardinals numbers whose existence is not provable from the standard axioms of most set theories, and thus have to be explicitly defined on them by the addition of another axiom postulating their existence, much like how an infinite set (ω) cannot be obtained from the standard tools and operations of arithmetic, and thus its existence needs to be added separately.

The Hierarchy of Large Cardinals is fairly extensive and contains many exotic-named numbers, but the most conventional and easily-understood entry point to it are the Inaccessible Cardinals: Infinite cardinal numbers that are uncountable, regular (cannot be defined as the union of quantities smaller than themselves) and strong limit (cannot be attained through repeated power set operations), which in layman's terms, means that they cannot be reached from lesser cardinals in any way, shape or form, standing beyond their scope entirely. The following is a good illustration of an Inaccessible Cardinal's relation to lesser numbers.

### **Platonic and Abstract Forms:**

The logic that follows our tiers known as "Outer verse Level" is based off pure and abstract forms of purely metaphysical ideologies. Things such as Platonism and other pure abstract forms cannot be attributed to things such as numbers or cardinal systems. We gave the "Layering" system as a way to buffer and weed out the difference between characters scaling's which falls within the margin of error in terms fictional scaling. We need to allow for some of the fictional inconsistencies and drawbacks when referring to scaling. Keeping the numerical system of mathematics out of purely abstract concepts will allow for proper gravity and less inflated scaling within characters and the tiers. We do not believe that allowing for mathematical transcendences would make logical sense considering the nature of abstract and platonic forms.

What is Platonism? Platonism is the view that there exist abstract (that is, non-spatial, non-temporal) objects (see the entry on abstract objects). Because abstract objects are wholly non-spatiotemporal, it follows that they are also entirely non-physical (they do not exist in the physical world and are not made of physical stuff) and non-mental (they are not minds or ideas in minds; they are not disembodied souls, or Gods, or anything else along these lines). In addition, they are unchanging and entirely causally inert — that is, they cannot be involved in cause-and-effect relationships with other objects. All of this might be somewhat perplexing; for with all of these statements about what abstract objects are not, it might be unclear what they are. Platonists of this sort say the same thing about other properties as well: in addition to all the beautiful things, there is also beauty; and in addition to all the tigers, there is also the property of being a tiger. Indeed, even when there are no instances of a property in reality, Platonists will typically maintain that the property itself exists. This isn't to say that Platonist's are committed to the thesis that there is a property corresponding to every predicate in the English language. The point is simply that in typical cases, there will be a property. For instance, according to this sort of Platonism, there exists a property of being a four-hundred-story building, even though there are no such things as four-hundred-story buildings. This property exists outside of space and time along with redness. The only difference is that in our physical world, the one property happens to be instantiated whereas the other does not. Basically Platonism is outside of space and time as a whole and we ourselves can only start to see the very shadow of what Platonism will overtake. We have no words no descriptions no numbers that can truly be attributed to people on this tier.

### **Higher Dimensional Existence:**

Speaking in more rigorous terms, a "dimension" in a system can be most effectively equated to the real number line, which is itself the most basic example of a 1-dimensional coordinate space (commonly denoted as simply as ℝ), and contains bounded subsets (smaller, finite lines if you will) whose exact measurements can be taken by "pointing" at a real number represented as a specific point in the line (Example: A line with length 3 is a subset of the real number line, whose length is infinite)

In this context, adding another dimension to a given space is effectively adding another real number line in a different, perpendicular position, which is mathematically represented by continuous cartesian products of ℝ. For example, 2-dimensional space is represented by ℝ^2 (ℝ x ℝ), 3-dimensional space by ℝ^3, and so on and so forth.

In this context, measurements of positions in space are once again obtained by directing oneself at arbitrary numbers contained in the axes of the given expanse, thus obtaining one's own coordinates in it. Simplifying this case, one could also informally state that the "Dimension" of a space is the exact number of coordinates needed to specify any position within it.

Likewise, any n-dimensional object can be thought of as being a subset of a real coordinate space of corresponding dimension.