09/24/2021, 04:42 PM
(07/08/2021, 05:11 AM)JmsNxn Wrote: I believe William Paulsen and Samuel Cogwill's uniqueness condition is quite beautiful (Which is largely just based around Henryk and Dmitrii's work). Personally though; I'm very opposed to using fixed points, it just feels unnatural to me--sort of, arbitrary; like "why that fixed point and not this one?". "A kind of" Fixed point at infinity seems a bit more natural to me. Also, tetration diverging to infinity as we increase the imaginary argument also seems more anomalous--I think it represents well just how whacky tetration is. If anything, I've just thrown a wrench in the gears; but I think it's a good thing. Have you seen what proposed pentations/hexations/septations look like with kneser?--they look less than desirable.
I think at this point, in the quest for "the right tetration"; which ever one runs faster and simpler and solves the storage of large numbers in a better way will probably win out. It's definitely Kneser's at the moment. I still feel Kneser is the superior tetration, simply because it's much better behaved, and taylor series are much easier to grab. I'm still having trouble making a non glitching program. God damn overflow errors. Need a perfect turing machine with geometric convergence speeds.
Regards, James
Oh. I think "goodness tetration" must uniquely determine Tetration, and also pent/hex/sept/... all of the Ackermann-like function, (until \( f\omega^\omega (n) \) growth rate)
It's hard to argue that a solution other than kneser could have done better.
However, in cases where it is difficult to calculate L/L*, like base 0/1/Shell-Thron-region , it does make sense to use a solution other than kneser. (But I don't see you starting any work on it
)