I know that the simplest number system is the natural numbers ($0, 1, 2, …$). And while we can easily define addition and multiplication for naturals, subtraction doesn’t work. Because what would $2-3$ be?
And so we define the integers, which include negatives so that subtraction (the inverse of addition) works consistently. Except now we run into another problem: division doesn’t always work. After all, what integer could $1/2$ be?
So we define rationals, which include fractions so that division (the inverse of multiplication) works. But now we have another problem: roots don’t always work.
So we define the real numbers, which include nth roots so that roots consistently works. Except that roots of negatives still don’t work. So we invent complex numbers. Now, taking roots (the inverse of exponentiation) has consistent results. So do logarithms (ignoring the whole $+2ik\pi$ thing).
Now we come along and invent a new thing: tetration, which is like repeated exponentiation. Specifically, 3^^4 is equal to $3^{3^{3^3}} = 3^{3^{27}} = 3^{7625597484987}.$
We also define an inverse function: the super-root. So the 4th super-root of $3^{7625597484987}$ is 3.
Oh boy. If inverting addition led to the discovery of negatives, and inverting multiplication (repeated addition) led to the rationals, and inverting exponentiation (repeated multiplication) led to the real and complex numbers, then inverse tetration (repeated exponentiation) should also lead to one or two new number systems.
I’m wondering what properties these systems might have, and if they could exist at all.
IN ADVANCE: I’m not looking for quaternions. I know they exist, but they’re just a mindless 4d version of complex numbers; they don’t have a unique idea. The new system should have an original idea, like the density of the rationals, or the multi-dimensionality of the complex numbers.
For those of you who think tetration only makes sense when the super-exponent is a positive non-zero integer, you should check again. $2^{1.5}$ is perfectly well defined; it’s $2\sqrt{2}$. So please don’t write that tetration and its inverses only work on integers, because that’s not what I believe.
I know that was a lot, but I hope somebody will be able to help me. I’ve been wondering about this for a few days, and I have no idea where to begin in order to create or try to find the rules of such a new number system.
P.S. Since tetration can be repeated (to make pentation), and that can be repeated (creating sextation), shouldn’t we need an infinite number of new number systems. Just a thought.