01/24/2021, 06:53 AM

I have the intuition that tetration requires the introduction of numbers with real dimension (non integer, like complex or quaternion sets).

A problem is that those numbers have to include hypercomplex as a subset, but, for example a set with the dimension of a serpinsky triangle, would be a subset of the complex, and it cannot be closed under addition/multiplication, without including all of the complex numbers, but then it would just be equal to the complex.

So, closure has to be abandoned, or the set has to be a set of sets with all the possible dimensions.

Also, sometimes tetration leads to step functions or dirac delta functions. A diract delta is similar to a segment of line with unit length. It is one dimensional, meanwhile an hypercomplex number is a point, which is zero-dimensional.

It is analogous to surreal numbers, where a segment of line would be like a number infinitely larger than a point.

So maybe the serpinsky triangle should be taken as something in between a point and a surface, instead of a subset of the complex.

Instead of thinking of numbers as points, we should think of then as fractal sets.

An exponentiation to a real exponent, in general, has infinite solutions (because a root has infinite solutions in the complex field). Those infinite solutions are a set.

Probably we should think of tetration as operations between sets instead of between numbers. Probably fractal sets.

Possibly singularities in functions are just numbers with a different dimension. A pole in a complex function may be a line number with length equal to the pole's residue.

Possibly Taylor series require numbers with a different dimension outside of his radius of convergence, to be able to calculate pass singularities.

A problem is that those numbers have to include hypercomplex as a subset, but, for example a set with the dimension of a serpinsky triangle, would be a subset of the complex, and it cannot be closed under addition/multiplication, without including all of the complex numbers, but then it would just be equal to the complex.

So, closure has to be abandoned, or the set has to be a set of sets with all the possible dimensions.

Also, sometimes tetration leads to step functions or dirac delta functions. A diract delta is similar to a segment of line with unit length. It is one dimensional, meanwhile an hypercomplex number is a point, which is zero-dimensional.

It is analogous to surreal numbers, where a segment of line would be like a number infinitely larger than a point.

So maybe the serpinsky triangle should be taken as something in between a point and a surface, instead of a subset of the complex.

Instead of thinking of numbers as points, we should think of then as fractal sets.

An exponentiation to a real exponent, in general, has infinite solutions (because a root has infinite solutions in the complex field). Those infinite solutions are a set.

Probably we should think of tetration as operations between sets instead of between numbers. Probably fractal sets.

Possibly singularities in functions are just numbers with a different dimension. A pole in a complex function may be a line number with length equal to the pole's residue.

Possibly Taylor series require numbers with a different dimension outside of his radius of convergence, to be able to calculate pass singularities.

I have the result, but I do not yet know how to get it.