Let's say $ f(p) $ is a number defined as shown below:  



$ \hspace {10 mm}f(p) = {\sqrt {2} ^ {{\sqrt {2} ^ \sqrt {2}} ^ {... \hspace {1 mm} p\hspace {1 mm} times } }} $


What I understand is:   

We normally assume, presume or impose a restriction on $p$ that $p$ should be a positive integer.
  

  

My question is:   

**Can we allow $p$ to be any number in general?**



```
      e.g. Can $p$ be irrational? 
```

If *yes*, how to interpret such a number?
  

If *no*, why?
  



Thanks.



How to interpret this class of numbers?
*As suggested in meta, I'll turn my comment into an answer (CW).*


At this time there is no commonly accepted solution to the general problem of extending tetration to the real or complex values of n. Various approaches are mentioned in <https://en.wikipedia.org/wiki/Tetration#Extension_to_real_heights> .


