In my previous blog post, i tried to define Oblivion. It seems that i failed. Now i'm taking a different approach on this problem - i'm extending mathematics.
Right now, we only have quantifiers, functions and relations, and that is enough for us, because it can describe everything in reality. But maybe it can't describe everything outside of reality. What if there was something that can't be defined with quantifiers and relations? What if we made a new symbol for this undefinable object? That would extend mathematics at least to that object, and probably much more. This object is what i call the 2-object (quantifiers are 1-objects and relations are 0-objects). We could use the 2-object in some new formulae, make new axiomatic systems and theories, which could be stronger than any theories we have ever made with quantifiers and relations, and they could even define these normal theories. So an axiomatic system with 2-objects in its axioms is probably what Bowers called a 2-system and the normal axiomatic systems are 1-systems.
To define Oblivion, we'd make a relatively simple 2-system that can define 1-systems, and we would use up to a gongulus symbols to define a 1-system in the 2-system and then up to a kungulus symbols to define a natural number in the 1-system (we'd first need to define gongulus and kungulus, or we could use something close to them, such as fωω100(10) and fΓ0(100))
But it doesn't have to end with 2-objects. We could make 3-objects, which can't be defined using any combination of 0-objects, 1-objects and 2-objects. We could make 3-systems, which have 3-objects in their axioms and can efficiently define 2-systems. With a simple 3-system, we could make a function K2(n), which would return 2-systems defined with n symbols in the simple 3-system. Then we could make 4-objects, undefinable with any combination of n-objects for n<4, 4-systems that use 4-objects and K3(n). Then 5-objects with 5-systems, 6-objects with 6-systems, etc. We can even continue beyond natural numbers. An α-object is an object that can't be defined with any combination of β-objects for α>β.
Now we need a general way to define some "simple α-system" that can define the previous systems define N(α,n) as the smallest natural number larger than every natural number definable using n symbols in the simple α-system. A slightly different version of Utter Oblivion is N(Oblivion+1,Oblivion). But N(α,n) gives us natural numbers. So if we did the same thing but with ordinals, we'd have O(α,n), the smallest ordinal larger than every ordinal definable using n symbols in the simple α-system.
I define a very large ordinal γ0 as the supremum of fn(2) for natural numbers n, where f(α)=O(α,Utter Oblivion) and a very large number as N(γ0,Utter Oblivion). I'm using Utter Oblivion in those definitions to ensure that it doesn't get stuck because of not having enough symbols.