Exactly what I was thinking. I was trying to think of functions that grow fast; extremely fast.
The first problem is getting convergence. I do think it's possible as well. The main problem was that my function could be broken into a quotient: \( \vartheta_n(s) = \frac{\psi(s)}{\psi(n)} \) This was a technique I used to ensure \( \vartheta_n(n) = 1 \) but I think it backfires now.
I'm going to think about possible functions for \( \vartheta \). I think it only needs double exponential: \( e^{-e^t} \) since the series is the logarithm of the hyper operators.
The second problem is seeing if we can recover recursion at natural x and y. I think I have a technique at accessing this. I'll have to think about it. The beauty is that we are dealing with natural numbers so I'm thinking we'll be able to do some kind of proof by induction. This is difficult to vocalize without strictly going through the process of what I mean but I'll get to it probably tomorrow. I'm still trying to think hard about convergence.
The third problem is getting a right hand identity. Proving non commutativity and non associativity. Then I'll say I have a solution.
The first problem is getting convergence. I do think it's possible as well. The main problem was that my function could be broken into a quotient: \( \vartheta_n(s) = \frac{\psi(s)}{\psi(n)} \) This was a technique I used to ensure \( \vartheta_n(n) = 1 \) but I think it backfires now.
I'm going to think about possible functions for \( \vartheta \). I think it only needs double exponential: \( e^{-e^t} \) since the series is the logarithm of the hyper operators.
The second problem is seeing if we can recover recursion at natural x and y. I think I have a technique at accessing this. I'll have to think about it. The beauty is that we are dealing with natural numbers so I'm thinking we'll be able to do some kind of proof by induction. This is difficult to vocalize without strictly going through the process of what I mean but I'll get to it probably tomorrow. I'm still trying to think hard about convergence.
The third problem is getting a right hand identity. Proving non commutativity and non associativity. Then I'll say I have a solution.
