The Absolute Infinite (symbol: Ω) is an extension of the ultimate idea of infinity that is bigger than any other conceivable or inconceivable quantity, either finite or transfinite.

• The Absolute Infinite can be viewed generally as a maximal concept that transcends and encompasses all infinities. There is not only an absolute infinite multiplicity of sets of infinite size (like the set of natural numbers) but there is an absolute infinite multiplicity of sets (like the set of real numbers) whose infinite size is larger. One can generate an infinite sequence of infinite sets whose infinite sizes are larger and larger infinities ceaselessly.
• The Absolute Infinite contains an infinite amount of infinities, along with an infinite hierarchy of infinities (both for ordinals and cardinals).
• The Absolute Infinite contains the inconsistent actuality of the collection of all sets in a single "set of all sets."
• The Absolute Infinite contains the "inconsistent multiplicities" (called proper classes) in the universe of sets.
• The Absolute Infinite contains the world beyond sets.

The Absolute Infinite isn't defined by well-orderings and infinite axioms based on multi-sorted logic.

The concept of an ultimate idea of infinity that surpasses any other conceivable or inconceivable quantity is an abstract and philosophical notion that transcends our normal understanding of infinity. It represents the idea of an infinity that is beyond the scope of any finite or countable representation.

While traditional mathematical systems, such as set theory and the hierarchy of large cardinals, provide a framework for exploring different sizes of infinity, they are still bounded by logical and conceptual limitations. The concept of an ultimate idea of infinity goes beyond these frameworks and suggests a realm of magnitude that cannot be fully comprehended or captured by any formal system or human thought.

This notion of an absolute infinite multiplicity is the idea of an absolute or transcendent realm that is beyond our capacity to grasp fully. The Absolute Infinite is truly a quantity or magnitude that is greater than all formal and informal conceivable or inconceivable quantities.

• It is important to note that the concept of an ultimate infinity lies outside the realm of formal mathematics.

The Absolute Infinite is the notion of an ultimate idea of infinity that surpasses any conceivable or inconceivable quantity that represents a philosophical and metaphysical concept. It goes beyond the boundaries of formal mathematical systems and supersedes the existence of a magnitude that transcends human understanding and the limits of logical reasoning. It is the actuality of the ultimate existence of an unattainable and infinite realm.

The Absolute Infinite embodies the pinnacle a logical, illogical, possible, impossible abstract mathematical form can take. The Absolute Infinite embodies every higher infinity onto itself, as well as transcendence over every possible higher infinity. The Absolute Infinite is beyond mathematics completely - beyond any sort of measurement, which includes abstract mathematics.

• The Absolute Infinite maximally surpasses the notion to increase, i.e, the Absolute Infinite is such that it cannot be further increased. It is, as it were, the Greatest Maximal. In other words, the Absolute Infinite is absolute completion and encompasses everything conceivable and inconceivable. In itself, it is not characterized by any particular property and is thus ineffable. The Absolute Infinite can only be acknowledged and admitted, never known, not even approximately.

Limits, infinite cardinals, infinite ordinals, and transfinite numbers are all distinct mathematical formalizations of the concept of infinity. Each is useful in its corresponding field and with different properties. The Absolute Infinite, in contrast, is the name given by Professor Lionel L. Sierra to the original concept of infinity. It has no actual formal definitions because it is not definable. If you define something, you can, often, build something bigger than it. Sometimes by simply adding something to it, sometimes by some form of limiting process. Professor Lionel L. Sierra also links the Absolute Infinite to the End of Infinity, holding them together.

The Absolute Infinite is the set of everything. It is the biggest thing that can exist. It encompasses everything and anything. The Absolute Infinite includes all hierarchies in general. This also includes all hierarchies of power as well, as well as all transhierarchical expanses of power; meaning it is always at the top of scaling chains and cannot be lower than the top. The Absolute Infinite contains the entire hierarchy of absolute sets and all their analogs and subsets, as well as everything above them - that transcends and supersedes them.

• The Absolute Infinite is always greater than any quantification. The Absolute Infinite is "beyond" all conceivable quantification and "beyond" all inconceivable quantification.
• The Absolute Infinite is always "impossible to index, as the very notion of being indexed no longer applies to the scale."

The Absolute Infinite is not a number of a set, since we can prove an object containing all the numbers is not a number or an object containing all the sets is not a set. Every property it has, a smaller object also has. It's the ultimate terminal object in category theory. There is the smallest infinity (aleph-null, the size of natural numbers), but for every infinity, there's an infinity that is bigger. When we prove that there's a bigger infinity than |S| (the size of S) for any set S, we show that the set of all subsets of S (this is the set p(S) ) has a bigger size. The system Ω of all [ordinal] numbers is an inconsistent, absolutely infinite multiplicity.

The system Ω, also known as the class of all ordinal numbers, is a foundational concept in set theory that provides a well-ordered hierarchy of numbers. Ordinal numbers are a way to generalize and extend the concept of natural numbers to encompass the idea of order and progression beyond finite quantities.

The ordinal numbers in Ω are used to describe the order or position of objects in a well-ordered set. They serve as labels for the different stages or levels in the progression of the cumulative hierarchy of sets. Each ordinal number represents a distinct step in the hierarchy and signifies the size or magnitude of the sets reached at that level.

In the system Ω, ordinal numbers possess two fundamental properties: well-orderedness and transitivity.

1. Well-orderedness: The ordinals are totally ordered, meaning that any two ordinals can be compared and one will be greater than the other, or they will be equal. This property ensures that there is a clear linear order among the ordinals, establishing a sequence from the smallest to the largest ordinal.
2. Transitivity: The ordinals are transitive, which means that if one ordinal is a member of another ordinal, then the first ordinal is also a subset of the second ordinal. This property ensures that each ordinal number encapsulates all the ordinals that precede it.

The smallest ordinal number, denoted by 0, represents the empty set or the absence of any elements. The successor of an ordinal α is denoted as α+1 and represents the ordinal immediately following α. This process continues indefinitely, generating a sequence of ordinals: 0, 1, 2, 3, and so on.

Additionally, the system Ω includes limit ordinals. A limit ordinal is an ordinal that is not the successor of any other ordinal but can be seen as the "next stage" in the progression. For example, ω (omega) is the first limit ordinal and represents the smallest ordinal that is greater than all the finite ordinals.

As the progression of ordinals continues, it reaches higher and higher stages, culminating in what is known as the class of all ordinals or the class of all ordinal numbers, denoted as Ω. This class encompasses all the ordinals that can be conceived or defined within the framework of set theory.

The system Ω has a rich structure, with various properties and operations defined on the ordinals. For instance, ordinal arithmetic allows for the addition, multiplication, and exponentiation of ordinals, providing a way to navigate and manipulate the hierarchy of ordinals within Ω.

The class of all ordinal numbers plays a fundamental role in set theory as it provides a foundation for constructing and analyzing sets. It serves as a backbone for defining the hierarchy of sets and establishing a well-ordered structure within the universe of sets.

In summary, the system Ω, or the class of all ordinal numbers, is a well-ordered hierarchy that encompasses all the ordinal stages in the progression of sets. It provides a systematic way to assign labels to these stages, allowing for the comparison, ordering, and manipulation of the ordinals within the class.

The Absolute Infinite contains no end to the process of set formation, the totality of all sets, and the set hierarchy. The Absolute Infinite contains such totalities that they themselves have to be a set, thus lying somewhere within the hierarchy. The Absolute Infinite contains the unrestricted formation of sets from arbitrary properties. The Absolute Infinite contains the naive set theory. The Absolute Infinite contains the notion to allow a class to describe arbitrary ("large") entities. The Absolute Infinite contains the class of all sets that are a proper class.

• In set theory, a proper class is a collection of objects that is too large to be considered a set. Unlike sets, proper classes cannot be elements of other sets. Instead, proper classes are treated as a distinct category or class of objects in their own right.
• The class of all sets that are proper classes is referred to as the class of all proper classes. It represents the collection of all collections that are considered proper classes. In other words, it encompasses all the objects that are not sets but are instead too large to be sets.
• In the formal notation of set theory, the class of all proper classes is typically denoted by the symbol "V" with a bar on top, representing the "Vee" notation. So, we can denote the class of all proper classes as V̄.
• It is worth noting that the class of all proper classes is not a formal part of standard set theory, such as Zermelo-Fraenkel set theory (ZF) or Zermelo-Fraenkel set theory with the axiom of choice (ZFC). These formal systems typically deal with sets rather than proper classes. However, the concept of proper classes is still relevant within the framework of set theory as a means of distinguishing between sets and collections that are too large to be sets.
• The study of proper classes and their properties falls within the field of class theory, which extends set theory to encompass the notion of classes, including proper classes. Class theory provides a framework for exploring mathematical and logical structures that go beyond the traditional boundaries of sets.
• In summary, the class of all proper classes represents the collection of all collections that are too large to be sets. It cannot be a set itself and plays a significant role in distinguishing between sets and proper classes in set theory and class theory.

The Absolute Infinite contains all Cardinals that are indexed by ordinals. Every infinite cardinal is just Aleph of an ordinal. This is because cardinals are well-ordered, and ordinals are designed for the purpose of well-ordering.

• There are no ends to the number of ordinals, so there is no end to the cardinals either.
• The ordinals are too big to even form a set (in particular, they're not countable). So the cardinals can't form a set either.
• Aleph 0 is the only countable cardinal, everything else is uncountable, by definition.
• It's pretty much impossible to describe how the ordinals are ordered in explicit language. Essentially, any attempts to use operations to derive a big ordinal from a smaller one will always run into a stumbling block: an ordinal inaccessible from below it using those operations. This can be explained using the reflection principle, which intuitively said that a small amount of ordinal can "fake" being all the ordinals, so any notations that attempt to capture all the ordinals will only reach a small part of it.

For example, you can start from 0 and keep adding 1 to get all the finite ordinal. Then take the limit, you get omega. Then you can keep adding 1 to that, you get omega+1, omega+2, and so on. Then you can take the limit again, you get omega+omaga which is omega x2. Then you can keep going to get omega x3. Then you can do this an infinite number of times, getting omega x4, omega x5, and so on. Then you can take the limit of all these limits, you get omega x omega which is omega² . Then you can repeat this process, to get omega³ . Then you can do it again and again, taking limits of limits, to get omegaomega . And of course, that's still not done, as you can move up to tetration and so on. In fact, you're never done. All of these operations are computable, so there is a limit that they cannot breach, ever, the limit of computable ordinals. After that point, you're reaching ordinals that are so complicated there are no ways to compute them explicitly. And cardinals will follow exactly the same pattern. After all, every infinite cardinal is just Aleph of an ordinal.

The Absolute Infinite contains the Hierarchy of Large Cardinals

• "Small" cardinals: 0, 1, 2, ...,,..., , ... (so on...)
• worldly cardinals
• weakly and strongly inaccessible, α-inaccessible, and hyper inaccessible cardinals
• weakly and strongly Mahlo, α-Mahlo, and hyper Mahlo cardinals.
• reflecting cardinals
• weakly compact (= Π¹
  ₁-indescribable), Π^m
  _n-indescribable, totally indescribable cardinals
• λ-unfoldable, unfoldable cardinals, ν-indescribable cardinals, and λ-shrewd, shrewd cardinals
• ethereal cardinals, subtle cardinals
• almost ineffable, ineffable, n-ineffable, totally ineffable cardinals
• remarkable cardinals
• α-Erdős cardinals (for countable α), 0^# (zero sharp), γ-iterable, γ-Erdős cardinals (for uncountable γ)
• almost Ramsey, Jónsson, Rowbottom, Ramsey, ineffably Ramsey, completely Ramsey, strongly Ramsey, super Ramsey cardinals
• measurable cardinals, 0^†(zero dagger)
• λ-strong, strong cardinals, tall cardinals
• Woodin, weakly hyper-Woodin, Shelah, hyper-Woodin cardinals
• superstrong cardinals
• subcompact, strongly compact, supercompact, hypercompact cardinals
• η-extendible, extendible cardinals
• Vopěnka cardinals, Shelah for supercompactness, high jump cardinals
• n-superstrong (n≥2), n-almost huge, n-super almost huge, n-huge, n-superhuge cardinals
• Wholeness axiom, rank-into-rank (Axioms I3, I2, I1, and I0)
• HODHOD-Super Reinhardt cardinals, HODHOD-Reinhardt cardinals
• Reinhardt cardinal, Super Reinhardt cardinals
• Berkeley cardinal, hyper-Berkeley cardinal
• Suggs cardinal

The Hierarchy of Large Cardinals is a concept within set theory that describes a sequence of increasingly powerful and significant types of large cardinals. Large cardinals are certain types of infinite cardinal numbers that possess special properties and consistency strength beyond those of the standard axioms of set theory, such as Zermelo-Fraenkel set theory (ZF) or Zermelo-Fraenkel set theory with the axiom of choice (ZFC).

The hierarchy begins with the concept of an inaccessible cardinal and progresses through various levels, each representing a more profound level of "largeness" in terms of cardinality and consistency strength.

The hierarchy of large cardinals continues with even stronger and more complex types. Each level in the hierarchy brings additional consistency and strength and allows for the investigation of deeper set-theoretic conceptualizations.

• It is important to note that large cardinals have significance not only in set theory but also in various branches of mathematics and logic. They provide insights into the nature of infinite sets, the consistency of mathematical theories, and the complexity of mathematical structures.

See Inaccessible cardinals.

The Absolute Infinite (or Absolute Infinite Multiplicity) contains The Absolute Infinite (symbol: Ω), which is an extension of the idea of infinity proposed by mathematician Georg Cantor. It exists beyond all dichotomies and principles which exist within the realm of abstract mathematics.

"The Absolute Infinite" is not actually the largest infinity, because it transcends and defines the concept of infinity in the first place, and exists entirely apart from the whole hierarchy of infinities that he discovered.

The Absolute Infinite includes the hierarchy of cardinalities which includes the set of all real numbers (Which is represented by P(N), or the power set of the natural numbers), which is itself the cardinality of all spaces of countable dimension, from 1-dimensional space to infinite-dimensional space. The Absolute Infinite encompasses all infinite-dimensional space by a boundless amount of sets of boundless transcendences.

• Cantor's Absolute Infinity transcends the number that satisfies the full reflection principle: "All cardinality properties are satisfied in this number, in which held by a smaller cardinal."

• Cantor's Absolute Infinity actualizes that "no number can have a higher well-ordering in a set, and bigger than this."

Absolute Infinity, as introduced by Cantor, challenges conventional notions of infinity and expands our understanding of the infinite realm. Traditional conceptions of infinity often treat it as a single, undifferentiated concept that encompasses all possible infinities. However, Cantor's approach takes a more nuanced perspective, recognizing that there are different levels or sizes of infinity.

Cantor's investigations into infinity led him to develop the concept of cardinality, which measures the size or "number" of elements in a set. He established a hierarchy of infinities based on the notion of comparing the cardinalities of different sets. Cantor used aleph (ℵ) to denote the cardinality of an infinite set. The smallest infinity, known as aleph-null (ℵ₀), represents the cardinality of the set of natural numbers (1, 2, 3, …).

However, Cantor went beyond aleph-null and extended his exploration into the realm of larger infinities. He introduced a concept known as the power set, which is the set of all subsets of a given set. Cantor showed that for any set, the power set always has a strictly greater cardinality than the original set itself. This result implies that there is an infinite hierarchy of infinities beyond aleph-null.

To express these higher infinities, Cantor introduced the notion of transfinite numbers. He used aleph numbers, such as aleph-one (ℵ₁), aleph-two (ℵ₂), and so on, to represent larger and larger infinities. Each aleph number represents the cardinality of a set that is greater than all the aleph numbers preceding it. Cantor demonstrated that the aleph numbers form an ascending sequence, with each subsequent aleph being larger than the previous one.

However, Cantor's exploration of infinity did not stop there. He sought to reach the concept of Absolute Infinity, which represents the largest possible infinity that transcends all aleph numbers and the entire hierarchy of infinities. Cantor described Absolute Infinity as an unattainable concept that lies beyond the realm of any conceivable set or collection.

Cantor believed that Absolute Infinity could not be fully comprehended or captured by any formal or informal mathematical or logical framework. Instead, it exists as an idealized notion, forever beyond human understanding and grasp. In this sense, Absolute Infinity represents a philosophical and metaphysical concept rather than a strictly mathematical one.