[Question] What are ranks? In your opinion.

[Catullus]

1) What do you think when you ear the term rank in the context of hyperoperations?
2) What do you think is the deep meaning and importance of the rank parameter?
3) What is, in your opinion, the main obstacle to solving the problem of non-integer ranks?
4) How do you see the mathematician of the future surpass this obstacle, if they ever manage to do it at all. In which field of mathematics do you see the key to the solution residing in?

1) One plus the amount of times the super function operation is applied to addition.
2) The deep meaning and importance of the rank parameter is the hyper recursion being used that iterates the super function operation itself, and how chains of repeatedly applying one operator form the next. In fact, the recursion is so strong that \(3[n]3\)\([>]\)\(\text{PRA}\). This hyper recursion can be used to generate super huge numbers!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! The rank parameter is also important because diagonalizing over the rank parameter could be used to define fundamental sequences for ordinals such as \(\def\ {\omega}\varphi(\ ,0,0)\), which is equal to \(\ [\ ]\ \). Via ordinal hierarchies such as Hardy Hierarchy, defining fundamental sequences for ordinals can be used to generate super huge numbers!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Please see this webpage for how ordinal tetration could be defined. The method used on that webpage for working with ordinal tetration could be generalized to higher ordinal hyper-operations.
3) The hyper recursion that is used to form the hyper-operation sequence.
4) It was pointed out that a uniqueness criterion similar to this one for tetration could be used to do continuous iteration on lots of functions. With that method, it could be possible to do continuous iteration on the super function function.