I am posting this as an excerpt of a private e-mail from GFR, who hopefully will join our forum soon:

Gianfranco

- – We can build a “Theory of Numbers”, starting from the positive integers and trying to give them a “group” structure via the definition of a binary operation “+”, and a unit/neutral element “0”. We immediately (so to say) discover that, for justifying the group structure, we need to include within the postulates, the definition of “negative integers”, for allowing the inverse operation. If we go ... further, we may proceed defining the “multiplication group” structure around operation “x” and unit element “1” that, automatically, generates the exigency of defining the rational numbers (including the Z set), for allowing the inverse operation. For similar (but ... more complicated) reasons, the introduction of the “^” operation obliges us to define the set of real and complex numbers, even without reaching the “group structure”, for allowing the two inverse operations. In fact, the “^” operation is non-commutative and the “unit element” cannot be found.

- – Having said that (in a .... very informal way), we may observe that the implementation of any ... step (rank) in the hyper-operation hierarchy, dramatically increases the set of postulates of the theory, for justifying the return (inverse) operations. This postulate set is probably infinite-countable and Prof. Goedel would have been happy to read this.

- – My problem is that I was always convinced that hyper-operation rank s=3 (exponentiation), not only opened the door to complex (including real) numbers, but also opened ... the window to something that we might call “multiple” or “fibrated(?)” numbers. In fact, I am convinced that the square root of 4 is, indeed, a “double number”: -2 and +2. The fact of indicating it by plus/minus-sqrt(4) is orthodox (and perhaps also ... catholic, i.e. universal ) but does not satisfies my ... dirty mind. Moreover, we might write cbrt(8) = 2 and cbrt(-8) = -2, but we also know that we have, in both cases, two other well defined complex numbers, as solutions of this operation. In fact, the square roots of a number are two and the cubic roots of a number are three. Of course, this is very well known (the problem of the multiple complex roots of number 1). But this is often the subject of some superficially-deep specialist considerations, which we may very well leave out of the ... main window, if we limit ourselves to the ... reality.

- – Actually, equation x^n - a = 0 is a particular case of a general algebraic equation of n degree. The “General Theorem of Algebra” says that the multiplicity of solutions of an algebraic equation of n degree is n. In other words, the number of solutions of x^n = a must exactly be n. For example, the four solutions of the fourth root of 16 are: +2, +i2, -2, -i2, which we might put as 4th-rt(16) = {2,i2,-2,-i2}.The way of writing it is not relevant. The tricky (but not ... fuzzy!) idea behind is that, if the fourth root of 16 must be a number (one number), well, this number has to be “multiple”, with multiplicity 4. It is not a question of “choosing” (I know, this is the official point of view), there is nothing to be chosen, it’s just like that. But, this point of view implies the definition of multiple (fibrated? matrix?) numbers. Well ..., you know, a Civilization that was able to swallow number “i” can do much more than that. We just have to check if it is worth while!

- – Now (and here I risk to become ... crazy), we know that sqrt(4) = 4^(1/2). Then, we might put 4^(1/2) = {-2,2}, with multiplicity 2. But, how about 4^(1/e)? Or, better (worse?) we know that y = 4^x is a very civilized exponential function, with base 4, very “near” to the shape of y = e^x, mother and sister of all of us (analytic, fully derivable and with y = y’ = y”, and so on). Nevertheless (there is always a ... never-the-less), how to represent its complete behaviour, showing its multiplicities for x = 1/n? Or (again), if x is irrational transcendent (for instance, but not exclusively, < 1), what can we do? Ggggghhhrrrrrhhhoooofff ... ! ;-)

- – Last but not least, let us consider y = [x->+oo] lim (sin(x)). Of course, it doesn’t exist, it is indeterminate. Nevertheless (ops ...) it must be a value limited between -1 and +1, i.e. something like y = [-1, 1], a ... continuou-multiple (or limited-undetermined number. What a mess, ... in my mind!

Gianfranco