A few more ideas involving tetration as missing operation in the end. It has nothing to do with Cantor, sorry. I am not a fan of neither limits, nor real number line. I hope some one will point to some obvious mistake I am making in logic, not in rigor

What would You say about Pi being infinitely infinite? And i being scalable imaginary infinitesimal with internal linear wave structure (basically, organized frequencies).

It goes like this:

Look at this link:

Continuous variable transmission with flexible shaft
1) From it we can see, that , if the circle is measured from inside with a disc with radius= 1/2 we get ratio circumference/radius of the big gear = pi/2. If it is measured by rotating the same disc in a plane perpendicular to big gear, ratio circumference/radius of the big gear = pi; if measured from outside, ratio is 3pi/2. When it is turned back downside, You get pi again, and if turned back inside ( 4th turn by 90 degrees= totally 360 degrees) , You get Pi/2 again.

2) So in fact, PI is multidimensional, or, infinitely dimensional number as any infinitesimal change of measuring angle will lead to change in ratio. That is why circle is not squarable in the plane, You can only do it in infinite dimensions, each of which is represented by infinitesimal angle.

3) Other infinity involved in PI is the size of measuring disc. If we use 1/2 of radius for measuring disc, we get Pi values I described above. If we use smaller disc, the ratios will be different- if we use infinitesimal disc, the ratios would be the same from all sides=pi , while if we use infinite disc, ratio will always be 1/2.

4) So there is 2 infinite spectra's of values pi can have , based on angle and measuring disc side ( there is no other exact way to measure circumference of a circle with infinitesimally thin boundary as by rotating a tangent disc with a same thickness boundary along it).

5) So we have Pi as function of 2 infinities. They may sometimes cancel out, giving pi, but in most cases, they would not. This leaves Pi as a number of dimension say infinity^infinity.

6) When we differentiate anything involving pi in a continuous way, we reduce the scale (that is the length part of uncertainty in PI) of PI , the dimensionality by 1 infinity, so it becomes infinity^infinity-1. For all practical purposes, differentiation or integration would leave PI infinite compared to any other number, but in fact, its size will be reduced/increased with each such operation. The angle part remains infinitely complicated ( if differentiation is done in Cartesian coordinates, not e.g. in polar and bu angle).

7) Now how to get a finite rotation out of infinitely dimensional number? We know that rotations in each scale are performed by i, so that

e^i* pi/2 always means 90 degrees to the left, anticlockwise. That means that while Pi, i, and perhaps also 2 change in each scale with each differentiation/integration by coordinate, e^I*pi/2 is always the same, finite.So that d(e^i*pi/2)/dxdangled something) = e^ipi/2

As Pi has 2 uncertainties involved ( see above), we can make a conjecture that in each scale ( after each differentiation/integration) e takes care of change in scale (since de^x/dx = e^x) so that e is simpler infinitely dimensional number than Pi, i takes care of angle uncertainty ( so I is also simpler than Pi, but have different dimensionality in each scale as well ) and I am not sure what, but 1/2 must also take care of some thing.

9) So the final conjecture is that Pi is perhaps dimensional in a way (infinity^infinity) infinite times - a tetration of (infinity^infinity). That is not so far fetched, since h( e^pi/2) = i-1/i+1 ; -i +1/-i-1 as can be easily verified.

10) There is no way PI can stay on real axis, since reals are 1 dimensional infinite numbers, working in 1 scale only. The same applies to 1/2 and e. i becomes a hyperreal infinitesimal with structure that is based on symmetries of infinitesimal linear angles in each scale.