10/01/2007, 06:24 PM

Quote:*IF* we assume that tetration is defined as . If we don't assume that (in particular if we assume that ) ...... then we have a more serious problem.

If you define to be different from 1, say , then . And in common use of the word tetration (hopefully you dont want to change this for the sake of continuity of your construction).

So if you accept this then any (real) value of different from 1 poses the contradiction . You see that it is necessary to define .

I mean there is no need to define it for , it rather shows the pattern. Your solution is also not continuous at exponent 1. Consider the sequence , by your definition . Then .