01/02/2008, 08:36 PM

Absolutely great!

But allow me a comment as a conclusion of what I read:

- mathematics is not finished without analytical theory on tetration, especially infinite since it is natural completion of infinite sums and products, and basically should not be more complex than other operations

-since mathematics is not finished no physical theory which does not imply usage of infinite tetration can be considered finished, especially in part where Andrew talks about rotation through dimensions. Nature is much more complex than addition and multiplication and simple exponents.

-the problem of analytical theory of tetration can not be solved without stepping outside real number line, limits and understanding the scale dependency of imaginary unit i.

An example: We say h(e^pi/2) = i; -i is not tetration of e^pi/2, since real number > e^1/e infinitely tetrated can not become complex number as a limit of convergent series of real numbers. Also we say that limit must have 1 value. Full stop.

Why don't we consider that since h(s) where s> e^(1/e) are all complex, and multiple, that it is tetration, but something is wrong with real number line and definition of limit as applied to tetration? It is obvious that tetration is an operation that WANTS to push real numbers into complex plane, but this obvious fact is dismissed as garbage since holy real number line as it is constructed today would not allow it. Tetration is yelling at us that something is wrong with basic numbers since they do not accomodate such a natural thing as tetration, and thus in 230 years since Euler no big advances has been made. And he did not care much about rigorous use of either limits or sets.

So why not change it-discard and rebuild - the real number line , complex numbers so that they fit tetration? That would be a million dollar worth achievement, and solve all other prized problems on a way. Hyperreals fall short. Scaled infinitesimals and infinities do not - but that is my personal opinion.

I have not even heard about such a suggestion, so probably there is nothing to read about it either- which just offers more opportunities for people who really understand tetration-like You ( not me).

Happy New Year!

But allow me a comment as a conclusion of what I read:

- mathematics is not finished without analytical theory on tetration, especially infinite since it is natural completion of infinite sums and products, and basically should not be more complex than other operations

-since mathematics is not finished no physical theory which does not imply usage of infinite tetration can be considered finished, especially in part where Andrew talks about rotation through dimensions. Nature is much more complex than addition and multiplication and simple exponents.

-the problem of analytical theory of tetration can not be solved without stepping outside real number line, limits and understanding the scale dependency of imaginary unit i.

An example: We say h(e^pi/2) = i; -i is not tetration of e^pi/2, since real number > e^1/e infinitely tetrated can not become complex number as a limit of convergent series of real numbers. Also we say that limit must have 1 value. Full stop.

Why don't we consider that since h(s) where s> e^(1/e) are all complex, and multiple, that it is tetration, but something is wrong with real number line and definition of limit as applied to tetration? It is obvious that tetration is an operation that WANTS to push real numbers into complex plane, but this obvious fact is dismissed as garbage since holy real number line as it is constructed today would not allow it. Tetration is yelling at us that something is wrong with basic numbers since they do not accomodate such a natural thing as tetration, and thus in 230 years since Euler no big advances has been made. And he did not care much about rigorous use of either limits or sets.

So why not change it-discard and rebuild - the real number line , complex numbers so that they fit tetration? That would be a million dollar worth achievement, and solve all other prized problems on a way. Hyperreals fall short. Scaled infinitesimals and infinities do not - but that is my personal opinion.

I have not even heard about such a suggestion, so probably there is nothing to read about it either- which just offers more opportunities for people who really understand tetration-like You ( not me).

Happy New Year!