BAO (beyond axioms ordinal) or BA(⍵) is a number that is based on a finding of higher axioms that are needed to be able to compare omnifinity with smaller numbers. We can then diagonalize over those to get BA(⍵), which we then call the beyond axioms ordinal.
Definition[]
Informal definition (of class-n):[]
Although omnifinity is beyond all axioms, we can still tell it is larger than other numbers. We can tell that by the axiom that if a>b and b>c then a>c. Where b acts as a limit of axioms while c is a smaller number, say true infinity, and a is omnifinity. This implies that there are class 2 axioms, which can be used to compare things beyond class one axioms. Then we can get a number beyond class 2 axioms. We can use the same argument as before to stumble upon class 3 axioms. We can continue this process until we get to an arbitrary class of axioms.
Slightly more formal definition (of class-n):[]
Axioms can still be used to compare numbers that are beyond all axioms, which implies that there are more powerful axioms, denoted as class-2 axioms. Then a number beyond all class-2 axioms can still be compared, which implies class-3 axioms. Each class-n axiom has numbers outside of them (by their own TEA), yet they can still be compared, which implies the existence of class-(n+1) axioms which can be used to compare these numbers. Please note that class-n axioms are not axioms, but rather different objects that are functionally similar to regular axioms.
BAO definition[]
BA(n) is defined as the first number x such that any axioms in class-n can't prove any statements regarding x.
I define BAO (beyond axioms ordinal) as BA(⍵) = sup{BA(1),BA(2),BA(3),..}
Size[]
We can see that class-2 axioms are a lot more powerful than class-1 axioms. The class 2 axioms can definitely work on BA(1) and (most) functions that can be applied to BA(1). So using C2A (class-2 axioms) we can tell that BA(1)<BA(1)+1 or something like that. This means an (admittedly bad) lower bound for BA(n) is P(BA(n-1)) where P is powerset, but this can be extended to (nearly) any function.