(05/04/2009, 01:06 AM)Tetratophile Wrote: Yes. x can, in principle, be any function*; interpreting the "iterants" (the n in f [It_k] n) as constant functions rather than as simply numbers is what makes the hierarchy possible (since a function outputs numbers we can use as iterant). I should have written the iterant as g(x) to emphasize that.

But anyway its no *definition*.

See have your original equation:

and lets add the initial condition:

Interpreting x as constant function, with the above lines we can derive the function

for any constant function .

And that is all! We can no derive what means for any non-constant function . Just because on the left side there are only constant functions in the second argument.

Well if we allow any function for , not only constant functions, then it is still no definition, because we have no initial condition, which stops the recursion.

For a constant function , the right side needs to evaluate in the second argument, to evaluate this it must be evaluated at and so on until one derives at m=1. This the initial condition and the recursion is finished.

If you put however any function there then on the right side in the second argument needs to be evaluated then then and so on but this recursion does never stop because never becomes a constant function for which we know how to evaluate.

Think of writing a computer program to evaluate your operator. It can not guess what your intention was for non-constant second arguments. But I can guess and thatswhy I proposed the definition I gave.