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does tetration bring up a new number domain?
#1
quickfur Wrote:It makes one wonder if the inverse of tetration would also create new numbers... I suspect it must've come up in this forum before, right?
It wasnt topic yet, so it is time to make it a topic!

In analogy to the demand that every polynomial shall have at least one zero, which led us to the algebraic numbers (or more sloppy to the complex numbers). The question here is whether every equation has at least one solution in the complex numbers.

By the striking correspondence of the function on the reals compared with the function on the positive reals, I would restrict the question to (in similarity to whether there is a complex solution of for ):

Is there a (complex) solution of for (which is the minimum of at )?

The answer to this can be given with the lambertW function:

let


We know that the Lambert W function (which is the inverse of ) is defined on any complex number, and so

.

Not only for every but for every complex the equation has a solution.

I guess that is equally possible with the other functions.
So, sorry guys, but tetration will not bring up any new numbers but is contained in the complex numbers (though it of course extends the algebraic numbers).
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#2
I think the topic must be studied more carefully. In my opinion, structures such as numbers are fairly well studied (hopefully) as opposed to operations which remain the stumbling block of mathematics.

Their dynamics, interaction makes the structure of mathematics- invisible, so far, with few glimpses of a system existing behind the scenes.

In my opinion, hyperoperations does create new numbers as does any operation. Tetration is the original source of complex numbers, not sqrt(-1) and need to solve x^2+1.

So most likely, going up and in between in operations will reveal linkages to all already known numbers ( quaternions, octonions, sedenions, in all formats, other structures like groups etc.) . Some new hyperstructures may appear as well.

An illustration: if I and complex numbers are resulting from infinite tetration leading to infinite dimensional hypervolumes and hypersurfaces, then I+I will actually produce pentation, while I*I*I*I ..... will be hexation, I^I^I^I - heptation and I[4]n - octation of reals. And so on.

Most likely, infinite octation of reals leads to all kinds of quaternions, quaternion+quaternion = [9], quaternion*quaternion=[10], quaternion^quaternion=[11], quaternion[4]n= real[12]n leads to octonions.

And so on.

The trick would be to start from another end [infinity] and see what structure naturally would fit that purpose. In my opinion, it has to purely imaginary number, without any real element with infinite number of infinite orthogonal components.The speed of this operation will be the fastest possible which of course raises aquestion by itself- what does that mean- it can only mean that the speed of this operation is such that all infinite numbers of structures react to it simultaneously . It is omnipresent, that operation. That would be than totally undifferentiated number, which, when operation [infinity] or [infinity-1] is applied to it, will produce first real component.

In normal world, the higher opertations above tetration will then cycle with a cycle of 4, meaning in most cases 4 operations/interactions is good enough approximation,as all operations with increased speed work on structures with increaesed mathematical inertia.However, there is always pentation, [9], [13], [17],[21], etc. working behind summation, etc.

The existance of higher operations leads to fine structure of operations/interactions, which reveals itself only clearly if these opertations can be separated. The total value of fine structure constant is a result of some kind of summing of these smaller influences of higher operations/structures on the basic operations/interactions.

To close this, there is a need for organizational math, that regulates ( maps?) relations between structures ( numbers, number functions etc) and operations and operational mathematics. The prime candidates for this purpose are functions similar to Riemann Z.

One more thing is to accept that integers are not the simplest , but most complex number structures there are.

Ivars
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#3
Ivars, this thread is in the rubrik "mathematical discussion" and not "esoterics"! Though what you write sounds quite cool and visionful it lacks any mathematical background, doesnt it? The interesting thing about mathematics is that you can *verify* speculations, raising it to a substantial level (or of course dismiss it).
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#4
hej Henryk

The question asked was rather wide, so I thought I may contribute since I have been thinking about it most of the time for last years.
So if there are no interesting ideas, just dismiss it.

The FACT that operations are not studied compared to structures is OBVIOUS even to me. So things remain rather open.There is no difference how You derive i- it will work the same way.
I am not esoteric, but I simply see there has to be one way to explain nature, and it is mathematics and logic - nothing more should be needed. The trick is to go around Godel in a logical way, which is a separate topic.

I remember Andy coming up with -2,-4,-6 etc for negative infinite n-tations, and negative infinite pentation of base e produced "decent" fine structure constant approximation by me effortlessly-because I knew I should be looking for it in hyperoperations- and leaves room for further improvements if higher operations can be added.
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