What is the geometric, physical or other meaning of the tetration or more high hyperoperations?
Is it exists in general or it has only math concept?
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.Sign up to join this community
There is an entry in citizendium, where D. Kousnetzov describes his proposed general solution for the tetration-function. He links to some papers of his own where he gives more examples (there are only few so far) of physical applications. (Something with light transmission in glass-fibers, increasing mass of a downwards rolling snowball).
Also I came once across an article called "wexzal" where the authors use the inverse of $\ ^2x$ to solve for aeroplane propulsion, and for dynamics in the explosion "chamber" of a gun-shot. (See here, I made it a pdf; don't know whether this link to tetration-forum's literature-database gives open access)
Sorry I can only give that vague hints, hope they help for a first step anyway...
See Wikipedia's article on Goodstein's theorem. Perhaps the statement that this thing is unprovable in Peano arithmetic is more interesting that its bare statement.