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Verse and Dimensions: Structures
Binaryfield
The Binaryfield contains all things that either are or isn't a random object or concept. If you imagine any object or concept, for instance a potato, then the Binaryfield would contain both that potato and everything else. It is the combination of x and not x. Binaryfield can be thought of as the cosmological equivalent of the universal set, from which everything is drawn.
Binaryfield VS Box
Many may consider the Binaryfield to be equivalent to The Box. After all, anything that isn't x is not x, that is common sense. So the Binaryfield would encompass absolutely everything. However, this is not the case. The Box contains everything, including completely impossible objects. This would mean the Box would contain an object that is neither x or not x, as completely impossible as that is. This means that there are in fact objects beyond the Binaryfield which are within the Box. In fact, if objects can exist in more than two states, more possibilites for objects would exist. This means that there are actually more objects outside of the Binaryfield than inside it. Nevertheless, the vast majority of cosmological objects described on this Wiki are within the Binaryfield as "if y is not x then y is not x" is a very basic principle.
Still, it is worth clarifying that "Binaryfield contains everything that is either x or not x" is only valid when x also follows the law which defines the Binaryfield, specifically "everything is either x or not x". So while it is true to say that "Binaryfield contains everything that is either a potato or not a potato", it is not valid to say that "Binaryfield contains everything that is either in the Binaryfield or not in the Binaryfield" as things not within the Binaryfield do not follow the principle of "everything is either x or not x".
Note that the above does not mean the Binaryfield is not a Selfverse just because it does not contain some things which are larger than it. The Binaryfield is certainly larger than itself due to being larger than the Maiorverse and containing things which are themselves larger than themselves, but nevertheless there are many things it does not contain. Of course, these are highly abstract hypercosmological objects and concepts which are difficult to even model, as being neither x nor not x is completely impossible in any form of classical logic or mathematics. This is why, despite the majority of 'things' being beyond the Binaryfield, the majority of concepts will almost always be within it, as concepts beyond it are incredibly difficult for humans to conceptualise, even when models are used.
The Half-x Argument
It should be clarified that when referring to 'x' and 'not x', there is no in-between within our universe. For example, if x was a potato, and the potato was sliced in half, it could be argued that that half of a potato is neither a potato nor not a potato. This is not the case. Something is either absolutely x or it is not. If something is not absolutely and completely a potato then it is not a potato. As such, all things are either x or not x.
It can be debated that it is unknown what qualifies for being "absolutely and completely a potato", and this is true. However, what is known is that a line is drawn somewhere. On one side of that line the object is x. On the other it is not. There is nothing on this line or in-between x and not x. It does not matter where this line is drawn as the sum of x and not x is the same regardless of where the line is placed, as the only thing that has changed is which objects belong to which. All of the same objects are still within the Binaryfield.
Trivia
While it is true that the property of being in the Binaryfield cannot be used as x in "Binaryfield contains everything that is either x or not x", the Binaryfield itself can, as it is contained by the Binaryfield. This means that technically "Binaryfield contains everything that is either the Binaryfield or not the Binaryfield" is an accurate description of the Binaryfield.
Trinaryfield
The Trinaryfield is an inevitable extension of the definition of the Binaryfield. The Trinaryfield contains everything that is either in the Binaryfield or not in the Binaryfield. To phrase it another way, the Trinaryfield contains everything that is x, not x and everything that is neither.
Deep Trinaryfield
Generally, when talking about the Trinaryfield the parts worth mentioning are those not within the Binaryfield. As such, this area is known as the Deep Trinaryfield to avoid confusion with the full Trinaryfield.
Contents
It is difficult to even imagine what kind of objects and concepts exist within the Deep Trinaryfield, in which all objects are neither x nor not x for any x within the Binaryfield. As such, very little is known about the contents of the Deep Trinaryfield. However, there are logics that model things similar to logic within the Trinaryfield, which are called many-valued logics.
What is known is that the Deep Trinaryfield actually contains more than the Binaryfield, as if objects can exist in more states there will be more possibilities for different objects. Any y within the Binaryfield can only be in 2 states in relation to x (either x or not x), meanwhile in the Deep Trinaryfield y can be in 3 states in relation to x. (x, not x or neither.) As such, there are more possibilities for objects within the Deep Trinaryfield compared to the Binaryfield. However, nothing more about these objects is known.
n-aryfields
n-aryfields are a way of extending the base definitions of Binaryfield and Trinaryfield to far higher levels. Each value of n represents a different collection of objects and concepts, as n corresponds to the number of states y can be in relation to x. (y and x are two different random objects.)
If n is 1, y can only be x, meaning the Unary field contains only one object. If n is 2, y can be either x or not x, forming the Binaryfield. If n is 3, y can be either x, not x or neither, forming the Trinaryfield. If n is 4, y can be either x, not x, neither or none of the others. This continues up the chain for every number imaginable. The n-aryfield in which n is 4 is known as the Quaternaryfield.
This can progress all the way up to ω, at which point y can be in a countably infinite amount of states in relation to x. (Known as an Apeironaryfield. If there is an uncountable amount of states y can be in it is known as a Circularyfield.) Further beyond infinite numbers, you could hypothetically use post-infinites such as E_0 (the smallest number larger than itself), for which the number of states y can be in is greater than the number of states y can be in.
Another way to imagine these successive n-aryfields is expressing every single one as some form of 'x and not x' by using the previous n-aryfield as x. The Trinaryfield can be expressed as everything that either is or isn't within the Binaryfield. Furthermore, the Quaternaryfield can be expressed as everything that either is or isn't within the Trinaryfield. This can be used for every n-aryfield, though once transfinite and infinite numbers are reached, this definition is not particularly helpful and displays very little information about what the field actually contains. Nevertheless, it is still technically accurate.
Size of n-aryfields
Successive n-aryfields follow the same rule outlined previously that the more states any object can be in, the more possibilites for different objects there are. Because of this, within each n-aryfield the previous n-aryfield contains the minority of objects. This means most of the Quaternaryfield is within the Deep Quaternaryfield, most of the Quinaryfield is within the Deep Quinaryfield, and so on. Thus, each n-aryfield is exponentially larger than the previous by an unknown amount.
All n-aryfields, regardless of what is used for n, are likely within the Schemafield, though some disagree with this interpretation.
Schemafield
The schemafield is a cosmological pseudo-structure containing all means of arranging information and all means of comparing operations, functions, or concepts, including meta-comparisons such as abstraction and degeneration – but, most importantly, also containing schema that exceed these in scope. Such constructions, while not possible to elaborate on under a conventional set-theoretic view, give the schemafield dominance over close to all other defined cosmological structures and entities.
The schemafield contains not only infinitely many such ideas as "the empty set" or "the complete set", but also all the means of constructing them, anything equivalent to them, any abstractions of them, and any greater or lesser operations that can or cannot be applied to them in any way.
While it vaguely resembles a conventional cosmological structure more than it does a single function like a Prism Gate, comparing the schemafield to -verses of more concrete size or contents is not entirely proper, since even making statements about what similarities it has to lesser -verses counts only as a generalization, and neglects its full extent.
Approaches
For the sake of simplicity in understanding, it can be useful to see the schemafield as a means of generalizing over the basic processes of abstraction that allow for certain hierarchies and tools to work. While one region of the schemafield uses the conventional Prism-Gate-like functionality of taking an input and viewing it as a degenerate case of something else, other regions handle input entirely differently.
Naturally, since generalization forms only a part of the schemafield, no matter how many times one attempts to apply it, one finds the exact same problem every time – there will always be regions that generalization cannot reach and that only more powerful operations (which also exist within the schemafield) can.
Similarly, there are other "approaches" up the schemafield using functions that it itself contains, none of which can apply to its full extent. Completing a thorough definition of the schemafield using only one of these is therefore effectively impossible.
Another approach would be to define information first, and then have the Schemafield contain that information.
x is formal means x can be represented as a string in a formal language.
x is information means x is formal or can be obtained from references from information. A reference to a cosmology x is a statement which x satisfies, such as "x is red".
One may doubt if this is sufficient to describe the Schemafield. However, you can easily demonstrate that operations such as generalization and abstraction are within this interpretation, as you can write down "x is generalization" or "x is abstraction" within our universe. However this assumes the universe is a formal structure, which seems likely, given how the universe can be demonstrated to follow mathematical law, and how elementary particles can be described as solutions to equations.
Cosmological Relations
Below the schemafield exist domains and regions where only particular groups of information arrangements and concept comparisons apply. One such area, for example, is the area in which general and degenerate cases exist in the way in which we are familiar, and in which Prism Gates operate normally. Others may share this property, of course, but some ignore it altogether.
Past the schemafield, operations on information break down entirely, nothing can be compared, and no organizational pattern applies.
Prism Gate
A Prism Gate is an extracosmological mechanism, metaphorically resembling a portal of sorts, with the capability to generalize over any concept and create an infinite set of alternate instances and iterations in which variables are altered in different ways. As they are capable of what eventually can become arbitrary meta-abstraction without bound, Prism Gates are independent of most classification systems and infinitely extensible.
When given an input of some kind – physical, metaphysical, conceptual, or otherwise – the perceptible set of a Prism Gate's contents adjusts, based on whatever fundamentals it can divide the input into. It then outputs a pathway to each alternate instance of the input where each fundamental is a special case. In this way, Prism Gates generalize over any applicable property and provide access to alternate realities, philosophies, and systems.
Utility
One of the simplest uses of a Prism Gate is to give it information about a -verse (e.g. a universe) and have it output the -verse's corresponding power set (a multiverse). It accomplishes this by dissecting a universe into fundamental constants and initial values, and then listing off all relevant cases of them. Given further instruction, it will repeat the operation indefinitely, ultimately assembling the omniverse from the principles its smallest components are built on.
Cosmological hierarchies, as they are also built on orderings and permutations, are also easy enough for Prism Gates to expand on. Given enough information to define a cosmology, a Prism Gate will derive the full set of altarcae, possessing all cosmological hierarchies and concepts, all at once.
The distinctions between ultimata do not restrict a Prism Gate in any meaningful sense. Given the concept of the Box (which is either a rather small amount of information, or an inexpressibly large amount), a Prism Gate will return pathways to express all equivalent structures; everything from the Omniumverse to the Imaginarium, as expressed by the Law of Box Equivalency, is included in the results as simply alternate interpretations of the input. It is worth noting that no Prism Gate output can actually exceed these; outputs still count, in a roundabout sense, as "things" to be sorted into the ultimata once again.
With a definition of omnipotence, a Prism Gate can return all omnipotent entities; with some hierarchy or system of power, it outputs all systems that are derived from the same axioms.
A Prism Gate can work with even less specification from its input. Given no information, a Prism Gate will first derive different means of expressing information (or any other equivalent concepts), and from there construct possible logical axioms and organizational schema. The results are, of course, inexpressible, as aspects of them inevitably violate any external systematic approach to organizing them.
Recursion
A Prism Gate can also operate on itself. In doing so, it produces alternate Prism Gates operating on different conceptual backgrounds, each of which will then begin assembling arbitrary axioms and information from there.
The process can be run recursively, generating an infinite loop of Prism Gates processing increasingly abstract Prism Gates and their relevant bases, though at a certain point this is no longer cosmologically meaningful and can be surpassed by a more powerful operation.
Abfield
The Abfield is a pseudo-object surrounding the Schemafield that consists of the concepts inaccessible through informational schema. It resembles something akin to an amplified region of metempiric space, but instead of possessing no cosmological information, it possesses no information at all, nothing that can be derived from manipulating it, and so on ad infinitum.
Oddly enough, the Abfield cannot be considered the negation of the Schemafield, since the process of negating an object's contents is something entirely constructible within the Schemafield. It also cannot count entirely as an abstraction of any operation the Schemafield applies to Prism Gates and other functions within, of course, because that is simply one step higher in the Schemafield. Rather, the best analogy one can make between the two objects – at risk of invoking the Schemafield's "comparison" containment – is that it is the Deep Imaginarium to the Schemafield's Shallow. The Abfield is no longer permeable to abstraction or degeneration, or to any organizational concepts established in the cosmological domains below.
This means that it innately has a level of indescribability, perhaps not as great as that of Nothing – though, on the other hand, it is misguided to attempt to compare the two in the first place – but sufficient to disallow entry through any sort of ontological manipulation or organized informational attack.
Contents and Exploitation
It is easiest to say what the contents of the Abfield are not: they do not form a set, or any construct under mathematical or logical systems of any kind. They do exist, in some sense, but there can be no axioms formalizing which things exist or do not exist in the abfield, no axioms that establish why no axioms for determining this exist, and so on repeatedly.
It could be said that the Abfield is an ultimatum in that it is entirely detached from all structures below, and "supersedes" the Schemafield, which strictly contains everything consistently or inconsistently expressible. Identifying a single object within the Abfield is impossible, however. While unboundedly complex structures, lifeforms, civilizations, and cosmic entities may exist within it, and can even be described in great detail, any means of expressing them will bring them into the Schemafield.
This particular property – retroactive ejection of information – makes the Abfield a popular target for concept mining, similar to that in the Imaginarium. Naturally, it is both more random and infinitely more dangerous; as the Schemafield does not necessarily connect all ideas in all ways, and the Abfield's nature allows for the existence of objects with far more discrepancies than any system can enumerate, bringing in something totally unconnected to one's own base construction of reality causes total reality collapses or worse. Accessing the Abfield in the first place, of course, is only possible from certain points in the Schemafield or -fields within, points that specifically allow for the transfer of information out of a realm where information breaks down.
Nonetheless, there is the infinitesimal "chance" – disconnected from proper probability, of course – of retrieving an unschemed artifact, be that an entity, item, or -verse, that could be unbound by logic and higher metasystems. Such a procedure is speculated to allow omnipotence far beyond that of the conventional hierarchy, ultimatum-surpassing events and structures, and unthinkably more mind-bending things, if one's local set theory is not demolished in the process in the way a hydrogen bomb would break open a porcelain pot from inside.
Cosmological Relations
Rationally, it is infeasible to compare the Abfield to any other cosmological or hypercosmological structure, even an ultimatum, due to its rejection of comparability. While it is not necessarily strictly false to claim that its contents can surpass any other structures, this statement has no meaning until concepts are brought into a Schemafield, frequently destroying it in the process and rendering further comparative analysis therefore useless.
Weighing the Abfield in comparison to other concepts such as the Box or Nothing are therefore counterproductive and perhaps detrimental to any concepts involved.
Itfield
The Itfield is the cosmological object that contains everything. It is in the Itfield, for any definition of 'it'.
From that alone, it may already appear as if there is not a single object, concept or anything else (including literal nothing) not contained within the Itfield, making it completely identical to The Box. But this is not the case. The entire rest of the Box's article could be placed into this one with the word 'Box' replaced with the word 'Itfield' and the Itfield would still not contain even a fraction of what the Box contains. This is all because it is missing just four words in the first sentence of its definition.
The Itfield is identical to the Box in every way except for one: The Itfield does not ignore the internal properties of an object or any paradoxes that occur due to this. Because of this, the amount it contains is substantially less. But why is the Itfield so much smaller just because of a seemingly insignificant detail?
Explanation
True Size
Despite being tiny compared to the Box, the Itfield is a truly colossal structure. Any object or concept you can or can't think of is within the Itfield. Any object or concepts that you neither can nor can't think of or any variations there-upon are within the Itfield. Anything that is not an object nor concept, or anything that is not any of the above options or any extension of such is within the Itfield. All n-aryfields and other fields (including the Schemafield and Abfield) along with everything they contain are within the Itfield.
Anything, whether it is or isn't a thing (or any other possibility) is within the Itfield. Well, almost anything. There is a very specific group of objects that the Itfield does not, and cannot, contain, despite its definition confirming it can and does contain anything and everything.
The Weakness
Naturally, in Hypercosmology it is assumed everything exists. If there was so much as one object, concept or otherwise that did not exist, it could not be contained by the Box, and therefore the Box does not contain everything. This means we can confirm that since the Box contains everything, it would contain an object that cannot be contained because it contains everything. And this is using the word 'cannot' in its most absolute sense. There is no trick or technicality that the Itfield can abuse to contain it, it simply cannot be contained in any way. So the Itfield would not contain this object as it simply cannot be contained by any structure. The fact that the Itfield contains everything, and that object is part of 'everything', is meaningless. It simply cannot be contained, regardless of the Itfield's properties.
The Box, however, is immune to this as the Box ignores the internal properties of the object and any paradoxes that arise from this. An inability to be contained is an internal property and containing an object that can't be contained is a paradox. As such, both are ignored allowing the Box to contain this uncontainable object. It is for this reason that the Box still contains everything while the Itfield does not.
The Extent of This Weakness
The extent of just how much the Itfield does not contain may not be initially clear. After all, "things that absolutely cannot be contained" is a very specific and (presumably) small group of objects and/or concepts. However, it should be noted that there are far fewer restrictions on these objects than the ones within the Itfield. While objects within the Itfield are not required to operate within the boundaries of any form of logic, be expressed as information or not operate within the realm of paradoxicality, they still must be contained, which is a limitation. Objects beyond the Itfield, however, are truly free. They do not have to hold any property. They can, but do not have to. They are contained by nothing, and as such must abide by nothing. They even are debatably free of Nothing itself.
This allows them to be or do truly impossible things that objects within the Itfield cannot be or do. (Such as, for example, not be contained by a structure that contains anything.) Among their many abilities would be to exist in far greater numbers than the objects in the Itfield, and ignore anything that may allow any objects within the Itfield to reach remotely similar quantities.
Because of this, there are far more objects outside of the Itfield than within it, and the vast majority of objects within the Box are truly impossible. Despite the Itfield containing almost all objects and/or concepts that will ever be discussed, it does not, in fact, contain even a fraction of all objects and/or concepts that actually exist. This is a very common theme within Hypercosmology, where structures contain the majority of concepts discussed but the minority of concepts that exists. Examples include the Schemafield and Binaryfield.
Beyond the Itfield
The Itfields marks a serious turning point in terms of all Cosmology. At a universal level, all concepts can be easily comprehended by human minds. As -verses increase in size, however, the number of concepts that humans can understand decreases while the number of concepts that require models to understand increases. The Multiverse, as a low level -verse, contains mostly fully comprehensible concepts, but some which require models. Higher-level -verses, such as the Omniverse, contain more concepts that require models to comprehend than ones that do not. Once The Barrelplex is reached, all concepts beyond it require models, as they're no longer any concepts that can be comprehended without them.
Then, as we progress into Hypercosmological concepts, the amount of concepts that cannot be understood even with models increases while the amount that can decreases. Lower level Hypercosmological -verses such as Selfverses can largely be comprehended via models with a few aspects that cannot, while mid-range -verses like the Trinaryfield largely cannot be comprehended via models with only a few aspects that can.
Finally, when going beyond Itfield into high-level Hypercosmology, models become completely useless. Things beyond the Itfield cannot be properly modeled whatsoever and as such attempting to examine such concepts becomes useless and only the relationships between objects become worth examining. (Usually represented by Parafields.)
-Verses can still exist which contain things beyond the Itfield, the Box being the most notable example. To do so, they must simply hold the Box's property of ignoring all internal properties of objects it contains as well as all paradoxes that result from this. The two most basic -verses beyond the Itfield are the Nonfield and Antifield.