Uniqueness

[Daniel]

Thanks for the feedback. The concern I have about uniqueness is a consequence of the logarithm being infinitely valued. Tetration is just as much iterated logarithm as it is iterated exponentiation. The technique I have developed for extending tetration is based on fixed points, but there are an infinite number of fixed points. Say you have an extension for tetration. What is the value of \( ^{-1} a \)? It is usually assumed to be 0, but it can actually be \( 2 \pi k \) where k is an integer; it has an infinite number of values. Then \( ^{-2} a \) also has an infinite number of values \( \omega^2 = \omega \). It seems to me that this leads to \( ^{-\omega} a \) having as many values as the continuum has points. This results in \( ^n a \) consisting on an \( \aleph _1 \) family of solutions. It is my understanding that there are \( \aleph _2 \) number of curves, but holomorphic functions are much more restricted. I wonder if it isn’t true that any holomorphic function agreeing with the values of \( ^k a \), where k is a natural number, comes arbitrarily close to one of the infinite family of tetration solutions.
Daniel