Perfect Conkept

## Definition

**Perfect Conkept** (Ѫ) is a Dhaxarum-Class Entity, indistinguishable from a collective known as the Nameless Horde, which live in a place called the White Desert. It is a place of complete emptiness and insanity.

**Perfect Conkept** is defined as follows:

*"For any particular Domain of Discourse, Perfect Conkept does not exist within it's jurisdiction"*

The upshot of this is that Perfect Conkept is not subject to any axioms. In a sense it also can not exist since it can nowhere be found. It ceases to exist to all domains since any Domain of Discourse must not include it as a member.

To define Perfect Conkept, we must first define a few terms. A **Mathematical Theory**, is a set of Axioms which hold true Absolutely. Theories normally live independently of each other, which means the axioms of one theory never effect the axioms of another. As long as we are clear what theory we are working in all possible axioms needn't be consistent, and in fact for every axiom we may take it's negation as an axiom in an alternative theory. A **Domain of Discourse** is the collection of all objects for which a mathematical theory has jurisdiction. **Jurisdiction** means that the axiom holds over a given object. Within a mathematical theory, only objects within its domain of discourse exist, and can exist. This is because if any objects outside it's domain of discourse did exist they would not be subject to its axioms, thus violating them. Therefore they can not be objects of that theory.

For every Domain of Discourse, there exists an "outside". The Domain of Discourse is always "bound". This property of being bound means it always represents a proper subset of all objects. The outside is said to always be "boundless". This means there is no domain of discourse that has jurisdiction of all objects lying outside the first domain. Lastly, all objects outside the Domain of Discourse are not subject to the Axioms of that theory.

A **Domain Class** is the idea of taking all axioms from all domains, within a level of comprehension. Omnifinity works on the idea that it is outside the 0th Domain Class. However once one has exhausted all the axioms of one Domain Class, new higher Domain Class axioms exist.

When exceeding any Domain Class, or any collection of Domain Classes, there is still a least Domain of Discourse that encompasses all previous Domains of Discourses as a proper subset.

There is no such thing as the "Domain of Discourse of all objects", nor is there even such a thing as "all objects". Any domain and any set of objects has an outside. For every domain, the outside has a least Domain of Discourse that is a proper superset of the original domain.

So if one simply tries to take the supremum of all objects, all one gets is the supremum of one Domain of Discourse inside a yet larger Domain of Discourse.

## Estimate of Size

Lest it be thought this is a simple extension on the idea of Axiom Classes, let's consider take a concrete look at what the definition implies. Let's say we have Domains of Discourse labeled d0,d1,d2,...etc. corresponding exactly the axiom classes. The smallest object of d(x) is ∃Ψ(x). If we take all finite theories and try to be outside all of them, we simply get d(w) which contains all lower domains of discourse and more besides, as things immediately outside all such theories must themselves have domain of discourse. It is clear we can go to the first fixed point of ∃Ψ. We denote this by ∃Ψ(0,1), and we can continue using fixed points and normal functions to continue. In each case though we can always take the supremum of all such domains of discourse up to some point. Perfect Conkept however is not the supremum of any domain of discourse, because all supremums are already contained in yet another domain of discourse. Therefore it doesn't lie outside of anything, yet it is also nowhere to be found. Since this is the case, no amount of creating domain of discourse hierarchies can be as large as Perfect Conkept, because again, Perfect Conkept is not a supremum of any axiom, set of axioms, or collection of domains of discourses. Even if we went beyond all Axiom Classes ... still that would be in another domain of discourse.

Basically there is no concept, be it recursive, inaccessibility, mahlo-ness, absolute-ness, etc. etc. that will not fall within some new domain. All these concepts already exist within smaller domains, and so this process can never help us escape it.

The All of "this" can not exist. That is because the structure we are describing is a Foundation. A foundation is a system which already contains the means to take anything and create something larger. If the all of that exists, then one could take that and make something larger, which contradicts it being the all. Much like the least Bordinal, Perfect Conkept gets around this by not taking the supremum of the whole (which is a paradox), but rather simply being sideways to each and every particular instance, that is, proper subset of the whole. It's the bordinal paradox and resolution writ large ... very very very large. Perfect Conkept is also the big brother of Conkept.

## Theoretical Ramifications

The ramifications of the definition lead to some truly bizarre conclusions. Since Perfect Conkept is not bound by any domain of discourse, it's behavior can not be restricted. It may be possible that it can behave according to any axiom and it's negation simultaneously, which would satisfy the requirement of not being in either the domain of discourse including the axiom or it's negation.

Now for a dangerous question: Is Perfect Conkept uniquely defined? Well having domain of discourse means it essentially has nothing that is true about it. Therefore it has no properties. Since it has no properties its behaviour when interacting with objects is undefined. Let us say we define an entity larger than Perfect Conkept. It too would have to possess no properties. Since both Perfect Conkept and it's Absolute Successor have no properties they are indistinguishable. Furthermore, without a domain of discourse to regulate their interactions, both can claim to be larger than the other, and there is no axiom to settle the dispute. They live under no one's jurisdiction. This leads to a vision of the lawless "White Desert". It is called the White Desert because there is nothing to see, everything is exactly alike. The entities that exist here are all the same. They live in numbers that we simply can not quantify as they lie outside all domains of discourse. Therefore it is meaningless to speak of them having an order. They are orderless. Since we have exhausted all domains of discourse this lawlessness is everything else that exists ... that ... or none of this exists.

All we can say at this point is, Ѫ < \*Ѫ , and \*Ѫ < Ѫ are equally valid with no meaningful way to distinguish them, and may in fact be both true at the same time. For this reason we may also refer to this as the Great Orderless Horde, and the place they reside as the Orderless Desert. From this point on, an a<b or a>b or a=b is not guaranteed to be a strict trichotomy. This sort of the flipside of the micro-zeroids, where the idea of less than or greater than also completely breaks down (See advanced Zeroid Theory for further details).