09/30/2007, 07:33 PM

bo198214 Wrote:Nobody denies that the tetraroot is the inverse of (by definition) however to define as being the tetraroot is quite arbitrary

Why? What is "arbitrary" about it? And if it's "arbitrary", in what "sense" is it "arbitrary"? Again, if the definition of the n-th order tetraroot as is "arbitrary", surely the tetraroot can be defined better as for some m,n. Do you care to define m and n in some other consistent way for the tetraroot?

bo198214 Wrote:and additionally does not coincide with our other methods.

So? Has a divine judge decided on the validity of any of the proposed methods so far?

bo198214 Wrote:Sorry Ioannis, but this rather proves that it is the wrong definition. As should be a function continuous in it must

Sorry, I am not convinced that should even *be* continuous (even whether it exists), despite the agreement between all the current methods. All the methods so far (including mine), exhibit a certain "artificiality" if you wish, which is apparent from the complexity which reveals itself when one asks a very simple question: HOW do you define the tetration function for RATIONAL values.

If you cannot tell me how the tetration function is defined at the rationals, then you *cannot* tell me how it's defined at the reals.

If you want to debate the above, then I will ask you the following:

how do you define for example

? or

?

Sorry, definitions via decimal expansions won't cut it, because decimal expansions suffer from non-uniqueness. So, if you tell me for example, take Andrew's or Gottfried's or your method and "input" 0.666... or 0.6363..., and then see what the function outputs, this is already suffering badly as a definition.