Cardinal Tiering System

**Massive Credit to Versus Battles Wiki's tiering system**

We notify infinity as *omega (ω)*, which is the set of all natural numbers ℕ, which are numbers like 1,2,3,...

As we will go on later in the system, the use of tetrations will be seen. A tetration is basically an exponent of exponents. The notation for tetration that we use is double caret notation, or *a^^n*. This becomes important in later tiers.

Tetrations of ω become important for the hierarchies of several verses. Given this, we should have a more convenient notation than writing their tier as **ω^^4 1-B** or however, which looks awkward. We have decided to use the Phoenecian letter equivalent to omega/omicron called *Ayin*, which we will notify with the georgian (For formatting convenience) letter *ჺ*. So for example:

* *Ayin-Naught (ჺ-0)* will = {0,1,2,3...ω,ω^2,ω^3...}
* *Ayin-One (ჺ-1)* will = ω^ω
* *Ayin-Omega (ჺ-ω)* will = {ω,ω^ω,ω^^2,...,ω^^ω). This is also known as *Epsilon-Naught (ε-0)*

All real numbers is signified by ℝ, which is equivalent to *Aleph-One (ℵ-1)*, though we will use *Omega-One (ω-1)* instead

### Tier 11: Lower Infinities

Existing below even our set of infinity, and is absolutely inconsequential to our existence

### Tier 10: Regular

#### **10-C:** Below Average Human level

This category is separated in the following manner:

* Sub-Microbial: (**Low 10-C**) This is for characters or things smaller and weaker than life. They can range from the size of molecules, atoms, quarks, or even planck units, and are as such extremely weak physically

* Microbial: (**10-C**) This is for characters or things the size or physical strength of viri, cells, organic macromolecules, or bacteria

* Below Average Human level: (**High 10-C**) Physically impaired humans. Small animals or insects

#### **10-A:** Athlete level

Athletes, most fighting characters from action movies.

### Tier 9: Superhuman

#### **9-C:** Street level

Peak Humans to Low Superhuman. Few physically very strong olympic level athletes and martial artists in real life. Most protagonists and final villains from action/martial arts movies. Large animals.

#### **9-B:** Wall level

Characters/Weapons who can destroy a wall, or those who can easily harm characters with wall level durability. Very large animals.

#### **9-A:** Room/Small Building level

Characters/Weapons who can destroy a room or a small building, or those who can easily harm characters with room level durability.

### Tier 8: Urban

#### **8-C:** Building level

Characters/Weapons who can destroy a building, or those who can easily harm characters with building level durability.

#### **8-B:** City Block level

Characters/Weapons who can destroy a city-block, or those who can easily harm characters with city-block level durability.

#### **8-A:** Multi-City Block level

Characters/Weapons who can destroy multiple city-blocks, or those who can easily harm characters with multi city-block level durability.

### Tier 7: Nuclear

#### **7-C:** Town level

Characters who can destroy a town, or those who can easily harm characters with town level durability.

#### **7-B:** City level

Characters/Weapons who can destroy a city, or those who can easily harm characters with city level durability.

#### **7-A:** Mountain level

Characters/Weapons who can destroy a large mountain, or those who can easily harm characters with mountain level durability.

### Tier 6: Tectonic

#### **6-C:** Island level

Characters/Weapons who can destroy a large island, or those who can easily harm characters with island level durability.

#### **6-B:** Country level

Characters who can destroy a country, or those who can easily harm characters with country level durability.

#### **6-A:** Continent level

Characters who can destroy a continent, or those who can easily harm characters with continent level durability.

### Tier 5: Planetary

#### **5-C:** Moon level

Characters who can destroy a moon, or an astrological object of similar proportion.

#### **5-B:** Planet level

Characters who can create/destroy a planet.

#### **5-A:** Large Planet level

Characters who can create/destroy large gas giants such as Jupiter and Saturn.

### Tier 4: Stellar

#### **4-C:** Star level

Characters who can create/destroy a star.

#### **4-B:** Solar System level

Characters who can create/destroy a solar system.

#### **4-A:** Multi-Solar System level

Characters who can create/destroy multiple solar systems.

### Tier 3: Cosmic

#### **3-C:** Galaxy level

Characters who can create/destroy a galaxy.

#### **3-B:** Multi-Galaxy level

Characters who can create/destroy multiple galaxies.

#### **3-A:** Universe level

Characters who can destroy all of the physical matter within an observable universe at full power. More specifically, usually via an explosion, omnidirectional energy blast, or a shockwave, that encompasses all of the stars and planets within a universe.

### **Tier 2: Multi-Universal**

#### **2-C:** Low Multiverse level

This category is separated in the following manner:

* Universe level+: ('*Low 2-C*) This is for characters who are exceedingly stronger than a standard universal character by a massive degree

* Low Multiverse level: (**2-C**) Characters who can destroy and/or create up to 1000 universes

#### **2-B:** Multiverse level

Characters who can destroy/create any number of finite universes greater than 1000.

#### **2-A:** Infinite Multiverse level

Characters who can destroy/create a number of universes equal to the set of all integers (Aleph Null), or can change/destroy/warp the laws of physics of infinite universes

* Composite Multiverse level: (**High 2-A**) Characters who can affect/create/destroy either: a 5-D to 10-D realities (ℝ^5 to ℝ^10) which are reasonably able to be assumed to be bigger than an infinite multiverse in existential status, or at least a multiplicative level of infinity above an infinite multiverse, and up to 6 multiplicative levels of infinity (So between ω^2 and ω^7).

### **Tier 1: Higher Infinities**

#### **1-C:** Hyperverse level

This category is separated in the following manner:

* Hyperverse level: (**1-C**) Characters who can affect 11-D to any higher finite number of dimensions’ worth of realities (ℝ^11 to ℝ^[finite number]) or characters who provably exist at/can affect/create/destroy at least 7 levels of infinity above an infinite multiverse, up to any finite numbered level of infinity (So any finite number above ω^8).

* High Hyperverse level: (**High 1-C**) Characters who can affect/create/destroy an infinite number of dimensions’ worth of realities (ℝ^ω), and thus exist on an *Ayin-One (ჺ-1)* scale. Also for characters who provably exist at/can affect/create/destroy an infinite number of levels of infinity above an infinite multiverse or more, up to an epsilon-naught (ω^ω^ω^ω^ω^ω...an infinite number of times) number of dimensions/higher infinities/planes of existence. It should be noted that if one achieves this rating through being infinite dimensional, they may have not necessarily transcended time or dimensions. If is this can be proven to be a problem for a character, weapon, or faction, it is reccomended that they are listed as a weakness. Also, to notate to what degree they are at, one will state their Ayin-Number, so for example equalling to ω^ω^ω would be written as *Ayin-Two (ჺ-2) High 1-C.*

#### **1-B:** Outerverse level

Characters who can create/destroy/affect typically abstract, metaphysical realms that exceed all real/complex variations of spatio-temporal dimensionality, and/or those that qualitatively exceed an uncountably infinite number of layers/dimensions/higher infinities to such a degree that said characters’ existential nature is impossible to encompass through simply stacking more dimensions/layers/powers of infinity. (Mathematical logic: We can assume that an arbitrary “meta-naught” to represent a “basic” “outerversal” level of existence outside a formalized multiverse and its dimensions/layers is a class of objects with the cardinality aleph-three, or specifically R ^ aleph-two, the power set of the real and complex number axis. We can say that since R ^ aleph-one (which really has the size cardinality of aleph-two) represents a truly uncountable number of dimensions, “Meta-naught” exceeds the ordering of uncountable real/complex space-time dimensions by an amount which dwarfs the rest of the tiering system in size, which must represent a qualitative degree of superiority/infinity over all real and complex arrangements of spatiotemporal dimensions.)

* Pleioverse level: (**Low 1-B**): *Epsilon-one* ordinal size to *omega-one (ω-1)* ordinal size (synonymous with aleph-one levels, or an uncountable number of dimensions/layers/higher infinities.) This tier is called "Pleioverse" for the Greek root word *pleion (πλεῖον)* which means "more," which is in reference to being more abstract classes beyond epsilon-naught which most hiearchies would reach.

* Outerverse level: (**1-B**): Transcending dimensional hierarchies and space-time as a real/complex concept or constant, existing outside the scope of an infinitely-layered system of dimensions/infinities, or having a provably aleph-three “meta-naught” cardinal size or greater.

* High Outerverse level: (**High 1-B**): Characters which exist at, or can affect, levels which are an infinite powerset of “baseline” 1-B levels, similar to the hierarchical superiority of High 1-C vs Low 1-C existentially.

#### **1-A:** Stratoverse level

Characters which can create/destroy/affect realms which utterly transcend the scope of even 1-B realms conceptually, but exist within composite abstract hierarchies of their own. Would basically be “omnipotent” to the standards of all outerversal characters if not for one weakness or due to the existence of an equal or superior character.

### **0: Boundless**

Characters which are all-encompassing to even the scope of 1-A characters. This is similar to the size of a strongly inaccessible cardinal (or Cantor’s absolute infinity) within mathematics, which is a conceptual size which surpasses in every manner all concepts of infinity or hierarchy before it, being a “set of all sets” which encompasses all ordinal and cardinal values of objects which precede it. Stronger axioms (concepts) and sets (collections of things) can always be formed with tier 0, and thus this tier has no end point, but battles at this level are largely arbitrary. Simply using “omnipotence” or other similar theological considerations to justify this tier is not allowed.