So, thinking about how thinkable things are! Meta thinking would be thinking about thinking, so what is this?
I'm going to have to refer to my cosmology here because it partly started with some comments in relation to it. So, tu take that out of the way: We don't know if the Imaginarium is equal to the Omniumverse but we know that it both contains and is contained by it. Those two relations are not identical. To both contain and be contained at the same time is not necessarily the same as being identical.
This entire debate is predicated on some of the properties defined to objects in the Thinkability scale so let's examine those. (The link was modified to point to the old version of the page I was commenting on, seing as the page was renamed and modified.)
The scale itself is just another way of categorizing things in neat groups by this new property that defines how thinkable they are. So far so good. There are two problems though: categorizing any specific objects at any level of that scale other than thinkable and the idea that there actually is a progression of some kind through the scale. There is not.
Any and all object that we can think of is, by definition thinkable. Any object that is presented on this wiki, or anywhere else for that matter, has been thinked of by its creator and by anyone reading the article. So all those objects must be thinkable. Things like The Box and the Omniumverse that are both classified as being Beyond Unthinkable must therefore be Thinkable. So either the scale does not work or things are just being misclassified. Now, there is no reason that one needs to assume that those objects are entirely thinkable, they can have properties which we are not able to think so they could very well be only Partially Unthinkable. But they surely are at least partially thinkable as well or we couldn't have thought of them in the first place. And this doesn't say much about them either. As we cannot think about things that are Unthinkable we have no way of telling if other objects that we think are Thinkable have Unthinkable parts to them making them also partially Unthinkable. A chair could be Partially Unthinkable for all we know for it can have "hidden" properties which we might not be able to think of. So the difference between Thinkable and Partially Unthinkable is not one we can perceive.
Assuming there exists some Unthinkable object, the only thing it is possible to say about it is what as already been said on this phrase. Or to put it in other way: "Here is an Unthinkable Object. Nothing else I can say about it for I cannot think about it. Even this might be too much". A Beyond Unthinkable Object would be beyond even this. It would be beyond "I cannot think it".
Now, the idea of Beyond Unthinkable referring to something which is Unthinkable in an Unthinkable way, meaning that we can't even think about how much or in what way it is Unthinkable, is an interesting idea. But this will open an all new infinite can of worms. This just moves everything one step further away. There is always the option of thinking about how Beyond Unthinkable something is instead of just thinking about how Unthinkable something is, which we wouldn't be able to do with a Beyond Unthinkable Object, obviously. So we can define a Beyond Beyond Unthinkable, and then a Beyond Beyond Beyond Unthinkable and it is set theory all over again. Defining T as the Thinkability of some object O, T(O) can be somewhere on the Thinkability Scale. But what the Beyond Unthinkable category says is that it is ok to categorize the Thinkability of some Object on that scale as well, so we can have T(T(O)) as well, or T(T(T(O))) and so on to infinity and beyond. So maybe there is some object O for which T(O) = Unthinkable and for which T(T(O)) = Unthinkable. This would mean that T(O) actually is = Beyond Unthinkable. So this gives us a contradiction, but it is an inconsequential one and can, therefore, be ignored. To simplify the notation we can make T0 = T(O), T1 = T(T0) and so on, or more generally Tn = (Tn-1). The other thing is that we can have another object O such that T0 up to Tn (for any arbitrary n) = Beyond Unthinkable but for which Tn+1 can be Thinkable. The opposed idea is to have a type of object where Tn = Unthinkable or Beyond Unthinkable for any n which would correspond to n = Infinity which would give us a new category in the scale, which I call here Absolute Unthinkable. Off course the point of this entire scale is more or less lost on itself seing as we cannot think about most of it.
The definition of Thinkable given on the scale asserts that "with sufficient measurements, all needed properties can be determined to within the limits given by physics" and that all such objects must be self-consistent by allowing no contradictions. This definition is extremely restricted and not coherent with the given name of Thinkable at all. First there is no definition of what a "needed property" is. Second it implies that a Thinkable object must be measurable, even if only within limits of the laws of physics. This is not the case as it is perfectly possible to think about unmeasurable objects, either with or without physical laws limiting or governing such laws. Objects with non-physical laws can also be thinked about or even objects without any laws at all. Finally the self-consistency is also not required for an object to be Thinkable. It is perfectly possible to think up a contradiction as have been proved and done by many philosophers through the ages.
The definition of Unthinkable is also lacking for it is also possible to think up an object who's properties cannot be ascribed to its whole. But the following definition that the "object does not, in fact, possess any definite properties across its whole" is even more limiting. Instead of trying to imply that we cannot think of some things that we can clearly think of, it limits the possible set of Unthinkable Objects. There is no way of saying if all the objects that we cannot think about all possess no definite properties across its whole. Maybe they do? If we cannot think about a single Unthinkable Object how could we expect to provide such a generalization? It is not possible to generalize a property of all Unthinkable Objects for we cannot think of them at all.
To end this criticism with the beginning of the definition of the scale, it is stated that "If a verse has a certain level of think ability, then all of its contents have an equal or lesser level of Thinkability. A verse can not contain something less thinkable than itself." This doesn't follow from any definition of Thinkability at all. Given the Unthinkable Objects X and Y of which we know nothing about because we are not able to think about them at all, we can define a set S = {X,Y}. S is also an object but it is very much a Thinkable Object. We are able to think about S without needing to think about X and/or Y at all and we are indeed able to do much thinking about S as proved by set theory. It would, maybe, be possible to define a property of an Object that would pass up its Thinkability in the defined way but that clearly does not apply for the X or Y Unthinkable Objects mentioned before, whatever they are, so it is hard to imagine. It is a concept that seems to be itself at least Unthinkable if not Beyond it.
The conclusion is that this scale is not very useful and seems to be arbitrarily defined. Even if it where coerently defined it would still not be very useful though for everything we are able to talk about falls into a single category of the scale, possible to if we count the Partially Unthinkable.
I did not intend this article to pure criticism though so I'm purposing that the definitions given for the term Thinkability better match a property that could be called Definability. How much of an object we are able to define. A Definable Object is much more coherent with one that can be measured to some extent. How much measurable it can be relates to how much definable it can be. An Undefinable Object could correctly be defined as something for which we cannot ascribe properties to its whole. It wouldn't say anything about it having specific properties or not though, just about our capability of defining them. It is possible for an object to have concrete properties without us being able to define them. Those are two entirely different concepts. And a Beyond Undefinable Object would be an Object for which it wouldn't even be possible to define how Definable it is or is not. And then infinity again.
Definability also does nothing to prevent contradictions though. It is perfectly possible to define a self-contradictory object in all its properties.
Definability seems to be a sub-property of Thinkability. If something is Definable it surely is Thinkable but the reverse is not necessarily true.
Definability, although more limited in scope, seems to be a slightly more useful concept for it allows us to classify objects we are able to talk about into more places of the scale. Even then it seems that most things would also just fall into the Partially Undefinable position as is the case of The Box and the Imaginarium. They are both defined to some extent but not to their whole in any way. The only thing I can probably think of (which indicates it still is Thinkable) but probably cannot define is the Nothing. Despite that, there are many attempts at defining even it so maybe it is also only Partially Undefinable. If that is the case and, as with the Thinkability scale, the Definability is equally not very useful.
Both Thinkability and Definability can also be either subjective or objective. Subjective Thinkability or Definability don't define the object in essence but either apply to the relation between the object and the subject. An object that is Unthinkable for us can be Thinkable for others and vice versa. The same goes for Definability. On the objective sense these would really be a property of the objects in question though. An Objectively Unthinkable Object is Unthinkable by anyone. The same for an Objectively Definable Object.
Finally, in my Cosmology I have the related concept of Indirect Thinkability. (I was actually calling it Indirect Imaginability but its the same thing.) This indicates the possibility of something that I may not be able to Think about but maybe I'm able to Think of some entity that is, in turn, able to think of said object and so on. This scale can also grow to an infinite number of indirect references. This can be seen on the cosmology by Imaginatas that exist, not as fantasies created from the Realium, but as fantasies created from other Imaginatas, therefore being indirectly created.
It just occurred to me: If we think up these scales starting from 0 and going to infinity, what would the -1 position on the Thinkability scale look like? What would the -Infinite position on the scale be like? On the Thinkability scale or any other. What would an Athinkable (??) Object be like? Something that is neither Thinkable nor Unthinkable. This goes deep into illogical or contradictory space pretty fast. Real numbers are possible due to the Partially... category. But what about Unreal numbers? Imaginary numbers and complex numbers? The several types of infinity like surreal numbers and such are easy as it has already been shown that the scales can increase infinitely so meh.