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What is a geometric, physical, or other meaning of the tetration or more high hyperoperations?

Is it exist in general or it's just a math conception?

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    $\begingroup$ Could you elaborate on this distinction you make between existing in general and having only math concept? $\endgroup$
    – joriki
    Sep 20, 2012 at 22:06
  • $\begingroup$ @joriki, sorry for my language. I mean is there physics applications of tetration? Or it's only have theoretical meaning. For example, first and secod derivatives of function is velocity and acceleration in terms of kinematics. Fermat's Last Theorem has no applications but has a great theoretical value. $\endgroup$ Sep 21, 2012 at 11:20

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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,
  • something with increasing mass of a downwards rolling snowball).

Moreover once I came 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.

(For the latter 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...

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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.

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$2 \uparrow \uparrow (n-1)$ is the number of sets of rank $n$. Intuitively, the rank of a set is the depth of the nesting braces needed to write it. Recursively, the rank of the empty set is zero, and the rank of a nonempty set is the one plus the maximum rank of its elements. It is also the depth of the stack needed by a pushdown automaton to recognize the braces are balanced. In short, $2 \uparrow \uparrow (n-1) = |V_n|$ where

$$V_\alpha = \bigcup_{\beta < \alpha} \mathcal{P}(V_\beta)$$

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