Catafinity

### Class

෧ Dhaxarum [Final] (old), Λ Conkeptilum (Transcendental List), ᘮ Absoluilmies [XIV] (new)

### Examples of larger numbers

Stupid bandinal Jou-Shufinity, Colossus, Propertus

### Discoverer

TheGoogologist

**Catafinity** (♕) is a **Dhaxarum/Absoluilmies**-class number that serves as the turning point of both fictional googology and hypergoogology. Beyond this point, the very metaphysical notion of infinity that has governed **every** single other number on the wiki until this point completely breaks down, essentially rendering anything larger than this not even a fictional "infinity" anymore. Thus, a new category will be proposed known as **fictional transfinities** that exist beyond Catafinity to further distinguish between numbers bound and not bound by the concept of infinity.

Now that the preamble is out of the way, let us get into the proper definition.

## Transhierarchical Multiplicities

So far, everything within the Fictional Googology Wiki has followed *some* semblance of structure; with certain numbers being "larger" than others in some sense, therefore forming a pseudo-linear hierarchy. For example, using numbers from Thienem/Postfinitum, Outerconst is the B-mathematical equivalent of 0 while Postfinity is the limit of the entire []-mathematics function altogether. There is a visible distinction between numbers in terms of size and magnitude; binding everything within here to some form of hierarchy such as the main list of all numbers.

Let there be a new number known as **True Outerconst**, denoted as Ψ. While the original Outerconst rejected only A-mathematical hierarchical bounds, True Outerconst rejects absolutely **all** hierarchical bounds, regardless of their internal or external properties. This makes True Outerconst a **transhierarchical multiplicity**; in other words, a multiplicity that is detached from **all** hierarchies, regardless of their nature, simply because it has surpassed the very notion of such. Beyond this point, comparison becomes completely pointless, as the gaps between each number grow so vast that comparing them through **any** definition of such is completely wrong by nature. Because of this, any form, extension, and/or analogue of hierarchy attributed to Ψ and anything beyond it is completely arbitrary and is in no way reflective of their true nature, created only for the sake of wiki organization.

## The Absolute Infinite

The **Absolute Infinite** is usually associated with Absolute Infinity or the original philosophical notion of such detailed by Georg Cantor; however, for the sake of descriptiveness, this article will present a non-standard interpretation of the Absolute Infinite that encompasses most notions of such, both mathematical and philosophical.

In this text, the **Absolute Infinite** of a given, well-defined system of mathematics or transcendental logic, is the least boundless multiplicity not expressible within said system; with a boundless multiplicity being defined as a "collection" with a distinct least element yet no distinct last element. For example, the Absolute Infinite of the naturals is Aleph-0, while the Absolute Infinite of all α-sort irreducible collections is Absolute Everything (otherwise known as Absility). A list will be presented, detailing most known and relevant interpretations of the Absolute Infinite within this article:

* Finite numbers - Aleph-0
* Countable ordinals/cardinals - Aleph-1
* ZF without the Axiom Schema of Replacement - Aleph-ω
* Accessible cardinals - θ (inaccessible cardinal)
* Large cardinals/sets - Absolute Infinity (Ord)
* α-sort irreducible collections - Absolute Everything (Absility)
* Non-cyclon systems - Terminus
* Non-dimensional infinities - Totality
* A-mathematics - Outerconst
* []-mathematics - Postfinity
* First Existential Axiom - Conkept (Beyond this point, Absolute Infinites are no longer the "least" of anything due to Conkept's nature)
* Domains of Discourse - Perfect Conkept
* Formalization - K
* Degrees of Unknowability - Zenith
* Contradictions - Parapass & Paracard
* Hierarchical multiplicities - True Outerconst

Let us call any system one can construct an **Absolute Infinite** out of an **Ω-constructible theory**. Within the list, we have detailed only 16; however there can be any amount of Ω-constructible theories that have not been created or discovered yet. There will now exist a new value known as the **Constructible Supremum**, symbolized as **Ξ**; which is considered to be beyond absolutely **all** Ω-constructible theories altogether, regardless of their deductive/consistency strength or any properties or axioms they may have (or lack thereof). As with transhierarchical multiplicities, one can hypothetically make a higher Ω-constructible theory that encompasses Ξ and extend the notion of such even further; however, once more, this is ill-defined as it is completely arbitrary by nature. Any attempt at categorizing Ξ, and anything beyond it, into a Ω-constructible theory will fail by nature.

## Definition

**Catafinity** is defined as the Constructible Supremum of **unbounded Ω-constructible theories** that also encompass completely unformalizable systems such as metaphysics and pataphysics (as a branch of philosophy); in other words becoming completely inexpressible even through the strongest forms of both transcendental logic and metaphysics we have used so far to define even numbers such as **Zenith**. As such, it lies beyond even the principles, modes, and attributes that govern all interpretations of the Absolute Infinite within both bounded and unbounded Ω-constructible theories; with anything introduced before Catafinity being essentially illusionary and nonexistent to it's true, absolutely boundless totality and actuality.

1. ↑ The true magnitude of Absolute Totality - Sergey Aytzhanov, June 2021