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Cardinality
Introduction
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = {2, 4, 6} contains 3 elements, and therefore A has a cardinality of 3. The cardinality of a set A is usually denoted |A| with a vertical bar on each side; this is the same notation as absolute value, and the meaning depends on context.
The cardinality of the natural numbers is denoted aleph-null (ℵ0) while the cardinality of the real numbers is denoted by "c" and is also referred to as the cardinality of the continuum. Cantor showed, using the diagonal argument, that c >ℵ0.
The continuum hypothesis says that ℵ1 =2^ℵ0 i.e. 2^ℵ0 is the smallest cardinal number bigger than ℵ0, i.e. there is no set whose cardinality is strictly between that of the integers and that of the real numbers.
An infinite set A is called countably infinite (or countable) if it has the same cardinality as N. In other words, there is a bijection A → N.
An infinite set A is called uncountably infinite (or uncountable) if it is not countable. In other words, there exists no bijection A → N.
These definitions suggest that even among the class of infinite sets, there are different "sizes of infinity." In the sense of cardinality, countably infinite sets are "smaller" than uncountably infinite sets. Of course, finite sets are "smaller" than any infinite sets, but the distinction between countable and uncountable gives a way of comparing sizes of infinite sets as well. Below are some examples of countable and uncountable sets.
A character's cardinality refers to either:
- The size of their energy source or energy capacity as a set.
- The size of the character's own existence as a set (e.g., a character whose true form spans a countably infinite number of dimensions or realities).
The size of their energy source or energy capacity (this can denote from themselves or from a Power Source that they use), considered as a set or their own existence as a set. In other words, if a character's energy can be divided into discrete units or states, the cardinality of that set of units/states is the character's cardinality.
For example:
- A character whose energy comes from a finite battery has finite cardinality.
- A character whose energy source is countably infinite (e.g., drawing power from a countable infinity of discrete energy units or states) has cardinality ℵ₀. In simplistic terms, someone who is naturally drawing from infinite energy or power.
- A character whose energy source is uncountably infinite (e.g., drawing power from a continuous spectrum of energy states, like the real number line) has cardinality 𝔠 (which equals ℵ₁ under the continuum hypothesis). In simplistic terms, someone who is naturally drawing from transfinite energy or power.
- A character who is a living multiverse.
This is separate from the tiering system, though a character's demonstrated feats may overlap with cardinality for their energy source/energy capacity.
Levels
The following levels list cardinalities from smallest to largest. These describe the size of a character's energy source or the cardinality of their existence as a set (this could also apply to characters who are living massive sizes of something). These levels are somewhat independent of the tiering system, though there can be some overlap. For example a living multiverse (Tier 2) would naturally have a cardinality above Finite and a living void (11-C) would naturally have a cardinality of None.
None: The character does not correspond to any set (e.g., proper classes, logical paradoxes, or entities outside set theory).
Finite: Any finite cardinality (0, 1, 2, 3, …).
ℵ₀: The smallest infinite cardinal. The cardinality of ℕ, ℤ, ℚ, and all countable sets.
ℵ₁: The first uncountable cardinal — the smallest cardinal strictly larger than ℵ₀. Under the continuum hypothesis, ℵ₁ = 𝔠 (the cardinality of ℝ).
ℵ₂ or Higher Accessible: Cardinals such as ℵ₂, ℵ₃, ℵ₄, and any other cardinals reachable from ℵ₀ via successor and limit operations, up to but not including inaccessible cardinals. These approach larger uncountable infinities or transfinite cardinals within an accessible cardinal.
ℵ_ω: The cardinality obtained by taking the union (or supremum) of all finite-index alephs: ℵ₀, ℵ₁, ℵ₂, ℵ₃, and so on, continuing through every natural number subscript. In other words, ℵ_ω is the smallest cardinal that is larger than ℵₙ for every finite n. The subscript ω (omega) represents the first infinite ordinal, indicating that this cardinal is a limit of the sequence ℵ₀, ℵ₁, ℵ₂, ….
ℵ_{ω+1} or Higher Accessible: Cardinals such as ℵ_{ω+1}, ℵ_{ω+2}, ℵ_{ω·2}, ℵ_{ω·3}, ℵ_{ω²}, ℵ_{ω^ω}, ℵ_{ε₀}, and so on. This covers all accessible cardinals from ℵ_{ω+1} upward, up to but not including inaccessible cardinals.
θ (Inaccessible Cardinal): An uncountable cardinal that is regular and a strong limit. Cannot be reached from smaller cardinals via powerset or replacement.
→Ω (Approaching Absolute Infinity): A limit concept larger than any definable cardinal, approaching but not reaching Cantor's Absolute Infinite.
Ω (Absolute Infinity): The mathematical equivalent to god. The totality of all sets, ordinals, or cardinals. In ZFC, this is a proper class, not a set. For characters that are in 1-Ω, though very rare cases exists where someone could have an energy capacity like this without being the tier.
Q & A
- Q: How does this apply to the Tiering System and Attack Potency?
- A: It technically doesn't, cardinality is merely an inherent property of sets (the number of individual objects something contains), so it has nothing to do with either of these. However, since energy is a scalar quantity whose capacity can be measured, cardinality can apply to a character's energy capacity in that sense.
- Q: Wasn't the tiering system based off levels of infinity?
- A: Initially it was but with the most recent revision back in March 5th, 2025, the tiering system is solely based off scalar quantities, with the only thing left of levels of infinity being Absolute Infinity remaining tier 1 due to it being something likened to God.
- Q: So what is the point of cardinality then?
- A: Cardinality is a way to further explain power source and show how much a character's capacity is. This is especially interesting for characters that can draw from infinite power sources.
- Q: Can a character with a ℵ₀ cardinality or higher be still below tier 2?
- A: Yes, as cardinality and tier are not directly correlated, all higher cardinalities means in terms for tier is if a character is something like an energy absorber, they wouldn't have a direct defined limit and can absorb infinitely, thus they can approach higher tiers given enough time and absorption.
- Q: What else does cardinality provide?
- A: It provides a way of further measuring characters for indexing purposes, it can even serve to explain random jumps in power in many stories.
- Q: Is there a reason Finite is not listed into different categories? All finites shouldn't be created equal.
- A: While this is true, Finite cardinals are all the same in meaning, which means a finite amount is contained within a cardinal, for the higher cardinals it goes into levels of infinity for how much one can contain.
- Q: Does this have anything to do with other stats?
- A: Unless the verse in question gives it a reason too, no. Cardinality is mostly a sole stat.
- Q: What about cases when characters are the personification or embodiment of a space-time continuum/multiverse?
- A: They would apply under the same thing of measuring their cardinality as this is directly their size and what they contain within that size.
