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Infinite tetration and superroot of infinitesimal
Thanks, GFR,

I know that , of course. I am deliberately trying to find a true imaginary infinitesimal, not i *epsilon, sorry, not i *dx.

Well if we begin with i as infinitely rotated into higher dymensions length (e^pi/2) so that -i = h(e^pi/2) is a hypervolume of an infinite hyperdimension. OK? Totally unclear , no physical content, but that is what it could be.

If we than remove one lenghts e^pi/2 , back tetrate, so to say, what will happen with that hypervolume? Will it stay the same? or will it be one hyperdimension smaller?

Would it than be the same -i or, perhaps, -i - di? So that difference between 2 such hyperdimensions is di, meaning that to get -i, You have to integrate over all hyperdimensions there is of di-> which equals infinite tetration of e^(pi/2).

I know i is considered to be constant, but I do not see why that has to be accepted. i, pi, e are just symbols, and their content is in fact, infinite of unknown since they have no physical analoques.

di = 0 is an axiomatic assumption that works well in math. The same for d(pi)=0. Fine. And than we have Godels law and happily watch exponential growth and specialization of mathematics because we have limited ourselves to the usage of well defined symbols and application of well defined operations.The number of finite symbols and operations will grow infinitely. The unification theories can not be based on such approach as by definition the deeper You go, the bigger diversity You will get. That is just obvious, so following the same logic and hoping that suddenly string theory will unite everything is just impossible- the complexity of picture will just increase. Today we have 10^500 string theories that might fit- than we will develop even deeper understanding- and there will be 10^500^500 superstring or whatever theories appearing. There is no end to this, and Godel's theorem say exactly that. The same applies to mathematics as such - the more You apply the principles and axioms, the more diverse and incoherent picture You will get although there will be visible similarities in many branches, the proofs that these similarities are rigorous will grow exponentially bigger and more difficult to find.

But in fact, using ill defined or undefined symbols like infinity and applying logical rules to them
we can go around Godels theorem and build a logically coherent system with undefined or not so well defined inputs-which is a small price to pay.

e.g. I is imaginary unit and equals infinite hyperspace volume. dI is a change of dimension of this hyperspace by one while still close to infinity. Very ill defined-no understandable content as infinity is involved.

Then we apply the rules of differentiation/integration or may be invent new ones and see what happens. i appears in so many mathematical formulas that it will not take long before some pattern might emerge or might not. If not, we may change the rules, or intial bad defintions to even worse, and start again.

It should not take long before the right rules can be found, or, their inexistance proved. If inexistance is unprovable because of the same Godel's theorem- than it is possible.


Messages In This Thread
RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/23/2008, 05:13 PM

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