Question about tetration methods

[Daniel]

Gottfried claimed that there were about fifteen approaches to tetration. Could someone list the more important techniques? Any additional information would be great.

Hi Daniel - there must have been a lightning in the air... frizzeling the electronic of our silicium cpus... I know of a handful of methods, but I think my hand has only 5 fingers ... ;-)) 
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Gottfried

Tommy claimed your comment, Daniel. Tommy claimed about 10 of his formula's converge. Spoiler alert, they don't. But they are attempts.

As Gottfried spoke, he meant that he can think of less than 5 solutions to tetration which are intrinsically unique. This is true. There is Kneser. There is Sheldon. There is Carlemann (programmed by Gottfried). There is Kouznetsov. There is Paulsen and Cowgill.


The Only one that is really proven, which I referred to is, Kneser and Paulsen and Cowgill. This was pointed out by Gottfried, who said that even Sheldon's fatou.gp isn't mathematically proven. Even though we all believe it is true, it is not proven.
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What Gottfried is saying, Daniel. Is that there are about 4-5 tetration formulas that hold up to scrutiny. As algorithms. And as mathematical ideas. And myself and Tommy are excluded from that list.

Thanks JmsNxn. I appreciate everybody's feedback as it gives me much to study and think about. 

My tetration research involves the range of complex numbers. Do all the techniques listed here deal with the range in the real numbers?

My work is experimentally consistent with Bell matrices. Are Bell matrices considered along with Carlemann matrices in people's research?