Perfect Catafinity (♔)

**Perfect Catafinity** (♔) is an extension to the original Catafinity created by TheGoogologist as their official re-entry into the Hypergoogology Wiki.

## Fractal Ineffability

**Fractal ineffability** is an extension of the notion of ineffability presented within the HG Analysis Language page, primarily based on chaining ineffable statements on top of one another to produce evermore ineffable and inconceivable versions of various concepts. We will now define ▨(a, b) such that "a" is the index of ineffability that is applied on "b"; which is the input of the function. Concepts that are considered to be effable are ▨(0), ineffable concepts are ▨(1), and ineffably ineffable concepts are ▨(2); creating an infinitely extensible system with it's own limits and extensions which go beyond said limits.

### Ultimate Ineffability

**Ultimate ineffability** is the absolute highest state of fractal ineffability (to an ultimately ineffable and self-unbounded degree); existing so far removed from **any** extension, superextension, or hyperextension of such to the point that even if one were somehow able to "define" everything that can be "represented" using fractal ineffability they would still not be able to define an absolutely infinitesimal fraction of an infinitely supercollapsed version of it's true form; even if one were to maximally believe an attempt at explaining it to an ultimately self-unbounded degree. This degree of "transcendence" makes it similar to the tiras- prefix that is used within the Heir to the Stars setting for some of it's Branches of Philosophy.

A concept that is ultimately ineffable is represented as ▢(a); for example, ▢(Ω) would be the ultimately ineffable true form of Cantor's absolute infinite.

## Fractal Self-Unboundedness

**Fractal self-unboundedness** is an extension of the notion of self-unboundedness indirectly introduced within the Opux page. Similar to fractal ineffability, it can also be represented as a function ⇱(a, b) where "a" is the index and "b" is the conceptual input; with the patterns underlying the former being able to be perfectly transposed onto the latter. For example, a concept with ⇱(1) is self-unbounded and a concept with ⇱(2) is unbounded even by the notion of self-unboundedness, once more forming an infinitely extensible system.

There also exists an ultimate form of self-unboundedness that is also ultimately ineffable (and is similarly far removed from any notion or extension thereof) denoted as ⇔(a); making both notions back each other up.

## Self-Unbounded Ω-Constructibles

A **self-unbounded Ω-constructible theory** is an extension of the notion of unbounded Ω-constructibles defined within the original Catafinity article; essentially being higher versions of the aforementioned that occupy some degree of both fractal ineffability and self-unboundedness and therefore have the freedom to encompass things other than hyper-abstract notions of size (such as absolutely everything).

If a given definition is protected by "⁅⁆" brackets, it is to completely overwrite all other (contradictory) definitions to such an ultimately ineffable and self-unbounded degree that they might as well be completely nonexistent (also to an ultimately ineffable and self-unbounded degree).

## Definition

* ⁅**Perfect Catafinity** is the supremum of absolutely all self-unbounded Ω-constructibles to an ultimately ineffable and self-unbounded degree.⁆