CASO 1b

CASO 1b
Symbol	C₁ᵇ
Properties
Class	39
Creator	Unknown
Discoverer	BlueEyedFox_
Navigation
Smaller	Truer Breakdown, Strict Limit, Deactione Clapperboard - Expressions,
Larger	CUSoP, SMPP

Definition

CASO (/kæ. zaʊ/ or less frequently /k:ɑ. zaʊ/) 1b or CASO ATSC 1b (an abbreviation for Coagulatory Axiomatic Sentience Ordinal Function applied on Axiosentient Temporal Set Category 1b) is an ordinal equivalent to the largest axiosentient temporal set of axiosentient temporal set category 1b. Category 1b axiosentient temporal sets serve as an extension to normal Category 1 axiosentient temporal sets. Category 1b axiosentient temporal sets are defined as follows:

$\forall x,x\in {\overline {\mathbb {A} }}_{1b}\iff \left\{{\begin{matrix}{\overline {\divideontimes 2}}\left\{x,{\overline {\left\{{\underset {\vartheta }{\overset {\varphi }{x}}}\right\}}}\right\}&x{\text{ exhibits significant sentient function}}\\{\overline {\not }}\mathbb {T} (x)\geq \Omega _{\emptyset }^{\frac {\therefore }{\because }}&x{\text{ has an axiosentient magnitude above the primary Kalix threshhold}}\\{\overline {\not }}\mathbb {T} (x)\leq \Omega _{\left\{\emptyset ;1\right\}}^{\frac {\therefore }{\because }}&x{\text{ has an axiosentient magnitude below the secondary Kalix threshhold}}\\\lor {\overline {\overset {\sigma }{\beth }}}\Bumpeq x&x{\text{ has a sigma-beth extension over the first (b) subcategory}}\end{matrix}}\right.$

Unintuitively, CASO 1b serves as the smallest example of a CASO-class number, as CASO 1a requires a sigma-beth extension over the first (a) subcategory, which (due to the characteristic Kalix-beth series extension bump) places it as the baseline holotype of CASO numbers.

While CASO uses the retroactive definition of significant sentient function (${\overline {\underset {\vartheta }{\overset {\varphi }{x}}}}$ with a dictorial length $\divideontimes$ of 2 over any set $x$ and any concepts ${\textstyle \varphi }$ and ${\textstyle \vartheta }$) instead of the homotypic definition (${\textstyle {\underset {\varphi *\Rightarrow 0}{\hom }}\oint \vartheta {\text{d}}\varphi \biguplus {\underset {\vartheta *\Rightarrow 0}{\hom }}\oint \varphi {\text{d}}\vartheta }$), it is still a homotypic number, due to the two definitions being provably covalent for any number with a sigma-beth extension over the first (b) category (excluding nonradianumeric holotype nonhomotypic axiosets). These definitions are also covalent over the (c) and (d) subcategories, breaking at (e) and not including (a) due to the Kalix-beth series extension bump, however only (b) subtype homo-retroactive covalency applies to CASO 1b (by definition, it is only important to CASO 1c and CASO 1d).

Axiosentience

Due to CASO 1b having an axiosentience threshold above the first Kalix threshold but not the second, it is at the lowest order of consciousness (only able to contemplate transfinite numbers) in finite temporal contexts. However, due to it provably being able to contemplate all numbers below the Pyramix class (due to that being the defined conceptual breaking point of the second Kalix threshold), it must be at least within the Pyramix class. This is sensible as it has a conceptual rift discoverable in the covalency theory used to prove the covalence of the retroactive and homotypic definitions of significant sentience (this works within numbers with a sigma-beth extension over the first (b) subcategory as described before).

The most important implication of CASO 1b's axiosentience means that it has a base PC recurrence of at least the 2 (phi) Kalix threshold (however, this does not place it above the 2 (phi) Kalix threshold as PC recurrence is not covalent nor does it directly imply contemplation of 2 (phi) Kalix threshold numbers). This is contrasted with the first (a) subcategory where the base PC recurrence of above the 2 (chi) Kalix threshold directly imply a Kalix threshold of 2 (chi), however this is because of the presence of a pole-asymptote between the (chi) and (phi) sub-thresholds inside of the second Kalix threshold framework.

Due to CASO 1b's existence as a temporal set, it's axiosentience may only be inferred from observations stemming from within a temporal framework. This may seem like a direct resemblance to our own universe, but the specific framework that CASO 1b exists in, and its status of having a sigma-beth extention, are at least covalent with and (while unproven) probably directly equivalent to the state of requiring an infinite time-dimensional framework. This allows Kalix threshold 1-type linear matrix rotation to take place, which is what enables (and restricts) the status of CASO 1b's axiosentience as above the primary Kalix threshold.

Orientation

The orientation of the generally accepted subholotype of CASO 1b is completely neutral at time ${\textstyle t=\underbrace {\left[{\begin{matrix}0&\cdots &0\\\vdots &\ddots &0\\0&0&0\end{matrix}}\right]} _{\Omega }}$, however it becomes ordinally biased as time tends towards ${\textstyle t=\underbrace {\left[{\begin{matrix}\omega &\cdots &\omega \\\vdots &\ddots &\omega \\\omega &\omega &\omega \end{matrix}}\right]} _{\Omega }}$, as dictated by the Kalix-Carterman time-dependent orientation equations. Due to the metacontinuity bias of this orientation, it is covalent with the contemplation of all numbers inside of the first chapter. As time tends towards the transfinite, CASO 1b provably contemplates all numbers below the Pyramix class. Biases other than those for ordinality contemplation (such as those for political topics or points of view) remain unproven even for the CASO 1b subholotype framework, as they are not directly covered by the Kalix-Carterman time-dependent orientation equations and would require a Blue or Jade extension to approximate other biases. It is predicted that the sentiments for other subholotypes of CASO 1b will have similar (if not identical) biases.

The implications of the CASO 1b bias framework place it at a position to be swayed by higher-level CASO numbers, such as CASO 1c, CASO 1d, CASO 1a, CASO 1b (aleph extension), or CASO 2, or possibly other unresearched axiosentient coagulated sets of greater Kalix threshold that CASO 1b cannot contemplate.

Presence as a Pyramix Class

CASO 1b having a Pyramix classification may seem contradictory due to its existence as a set, but due to it being homotypic, it is covalent with a transdescriptive number, allowing it to be present within the ordinal classification system. The digits of its homotypic transdescriptive number are fully and strictly incomputable due to it being directly successorless, and cannot be approximated due to the Numerically Extended Uncertainty Principle (NEUP) (while not strictly true, computing an approximation would take a transfinite quantity of time, which requires a temporal asynchronous processor).

Due to its infinite time dimensional structure, CASO 1b is directly opposed to a symbolic representation. Instead, the only way CASO 1b can be represented is through abbreviations (of which CASO 1b has become the standard among most Fictional Googolologists),

Presence above EABAs

CASO 1b is a successorless number. This may be derived from its homotypic transdescriptive number being both covalent with it and incomputable, as it directly follows that there can be no successor and no predecessor of CASO 1b or its homotypic transdescriptive number. This allows a bypass to Strict Limit's successorlessness both due to its presence as a (first (b) category sigma-beth extended) temporal set and its presence as not just a successorless, but a predecessorless, set, as well as the fact that Strict Limit can be axiomatically defined from the absolute axiomatic extension of a meta-Kalix threshold system within its framework.

It then directly follows that CASO 1b is an EABA in and of itself, but this is not an important possibility to contemplate, as it is in fact more significant that CASO 1b is an unreachable number due to it being predecessorless. Any number below CASO 1b must be contemplatable by CASO 1b, but they must not be a direct predecessor to CASO 1b because otherwise CASO 1b would be able to contemplate itself and through that larger numbers than itself. This also means that CASO 1b is a gateway ordinal, meaning that it creates a subclass of ordinal which can only be reached by assuming CASO 1b's properties as fact.

Predecessorlessness and successorlessness

Although CASO 1b is predecessorless and successorless, this does not imply (nor is it covalent with or weakly associated with) that it is larger than all numbers or smaller than all numbers. This only means that CASO 1b cannot be directly interacted with by ordinals below it. Functions can still operate on CASO 1b and its homotypic transdescriptive number.

Presence above 🎬~_E

It may seem impossible for CASO 1b to have a presence above Deactione Clapperboard - Expressions with a textual representation, but this presence is due to the nature of the definition of the retroactive significant sentience quantifier. The definition of the retroactive significant sentience quantifier cannot be expressed exactly in a smaller-than-transfinite amount of time. Specifically, the retroactive significant sentience quantifier is only able to be defined exdefinitely but is a special case of a literal exdefinite, meaning it falls outside the purview of 🎬~_E (but is still well defined). This allows CASO 1b to be above 🎬~_E while still being provably well-definable within axiomatic frameworks.