A more consistent definition of tetration of tetration for rational exponents

[Gottfried]

Well, in my opinion we are still in a phase of definition. Note, that also a discussion of "the best" way to define the gamma-function can still be led (although I think, that the known alternatives are not as power- and /or useful). It is a matter of which properties seem to be the most useful for number theory, I think. Also I think, that the methods (at least implicitely) based on matrix-diagonalization and/or -logarithm will come out to be the most useful ones - but that's still just an assumtion, with good arguments, though.

Another point is, that the three mentioned methods are not different, but essentially the same, as I understand things now - so there is no majority in this competition of methods (just a remark).

Kind regards -

Gottfried