Conkept

## Definition

**Conkept** (symbol **Э**) is an object which is defined as follows:

"*For any property there already exists an object smaller than Conkept possessing it*"

A **property** is defined here as anything which may be said to be true about an object. Anything that is true of an object must necessarily involve it's relation to something other entity. For example it's a property of 1 that it is a successor of 0, and it's a property of ω that it is not the immediate successor of any non-negative integer, etc.

An **object** is defined here as a discrete and singular entity uniquely specified by it's definition.

## Theoretical Assumptions

Conkept operates under the theoretical assumption that **The First Existential Axiom** always holds:

"*For any property there exists a least object possessing it"*

This axiom firstly, guarantees that every property has at least one object that possesses it. This concept is heavily used to define transfinite ordinals, as limit ordinals are asserted to be the least ordinal larger than every member in a given set of monotonically increasing ordinals.

The axiom further places certain strict limits on how disorderly the objects of the theory may be. There can not be, for example an infinite descending chain of objects possessing a property. This means the number line can always be broken into before a property first occurs and after it occurs. Beyond this however, this axiom does not force any strict order on how the properties are introduced. It can not tell us whether y is larger than x if y and x possess two different properties. None the less, we can say y is less than x, if x occurs before the occurrence of a property and y occurs after it.

The axiom also implies that any desired set of properties is eventually passed ... if one goes far enough ...

## Consequences

We may say, in an informal sense, that Conkept is the "supremum of the First Existential Axiom". It should be noted that this statement is contradictory, as the supremum is the defined as the least object larger than every member than a certain set. However the property of being a supremum of "something" means its a property held by a smaller object. Because of the nature of Conkept, it can't not be thought of as a limit or supremum of anything. This would make it the least to possess a certain property! So what is precisely meant by this phrase? What it means is that it's an object that violates the axiom, and therefore can not actually be part of the theory!

It is larger than any other known number, because all other numbers are defined by possessing a certain property. For example, terminus has sometimes been described as the first cylon number. True Infinity is the first infinity that goes beyond all dimensional numbers, as any number of dimensions you could name, other than having *truly infinite dimensions* would be too small. Outerconst is the first to lie outside of A mathematics.

Conkept is not a fixed point, or a supremum, or the limit of, anything. All these objects are essentially the least object that is not bounded by something. So if for example we define the fixed point of the powerset, well, that property is possessed by a smaller object. If we define something as the first fixed point of the strong successor function, well, that property is also possessed by a smaller object. Weirdly we can not say that something like 0 x Э is meaningful, because if it were true for example that 0 x Э = Э, this would mean it possesses a property, and such a property would have to be held by a smaller object (in fact it is said that 0 x Absolute Eternal = Absolute Eternal). What we can say instead that there is some object smaller than Э, call it A, such that 0 x A = A. It must also be true that Conkept is larger than any infinity that can counteract any hyper-zero. For example, there is an object smaller than Э, call it B, such that \_ x B = B, and there must be object smaller than Э, call it C, such that 0.0 x C = C, and so on and so forth. So this lies beyond all reciprocals of hyper-zeroes as well. Conkept can not have a reciprocal because then Conkept would have a property which would have to be possessed by a smaller object!

Conkept can not be the first to possess any property, because if it was it would contradict its own definition. Instead it must be larger than any property at all, and must itself possess absolutely no properties. It is similiar to the reflection principle, except that, unlike Absolute Infinity which states that any property it possesses is possessed by a smaller object, it has no properties (not all properties) because any property we could attempt to assign it as possessing would be possessed by a smaller object.

Unfortunately, there is therefore nothing we could ever truly know about Conkept, since it lies beyond all concepts. We could ask hypothetically what exists beyond it, but this leads to paradoxes. For example if we define it as the first object with no properties, and then try to claim the second object with no properties is larger, well, then this become a property! So this means there must be an object with no properties smaller than Conkept. In fact it must be larger than the second object with no properties, or third, because these two are properties which must be possessed by a smaller object. If we claim that Conkept is the first object for which any property is posssessed by a smaller object, we still run into the same issue. So Conkept can't even be the first with the definition of Conkept. We can't even define something as the first thing that comes after Conkept as that would itself be a property, so there is a smaller object than Conkept which is the first to be greater than Conkept!

Any theory which forbids any concept whatsoever must be less than Conkept. For example, in a cyclon free theory, not only must terminus be larger, but so must Conkept. Conkept must therefore exist outside of any theory, since a theory requires axioms that limit what is possible.

Conkept is a difficult concept. Is it even a concept? Or the absence of a concept? What lies beyond all concepts? ... Conkept ...