Hi,
After some pause, I have found a solution why generalized logarithms will work, despite the proofs by bo, Andy that real number line does not support such extension. I can give only a general confirmation, without details yet.
Briefly, because they will take values from INSIDE real number line , where the organization of Real number line will be changed by each level generalized logarithm. From the representation of numbers as ordered lengths of Euclidean straight lines to much more complex ways to organize the Numbers.
Each of these INTEGER level number organization ways will be linked to each other via transformation groups of respective projective spaces to spinor spaces (e.g. h(CP1)=Spin(2)). NON-INTEGER levels will be presented by a non-integer iteration point in process a continuous transformation ( like tetration) that maps Real number line with one organization to Real number line with different ordering.
Thus, also non-integer operations and dimensions, iterations will obtain their geometric meaning and intuitive understanding will be possible.
Here is a link to more detailed first ideas about how tetration transforms +RP1, -RP1, CP1 etc. with comments from Tony Smith about group structure of such transformations.
http://math.eretrandre.org/tetrationforu...hp?tid=216
Plus some ideas from Tony Smith about this transformation ( still working on it):
Ivars
After some pause, I have found a solution why generalized logarithms will work, despite the proofs by bo, Andy that real number line does not support such extension. I can give only a general confirmation, without details yet.
Briefly, because they will take values from INSIDE real number line , where the organization of Real number line will be changed by each level generalized logarithm. From the representation of numbers as ordered lengths of Euclidean straight lines to much more complex ways to organize the Numbers.
Each of these INTEGER level number organization ways will be linked to each other via transformation groups of respective projective spaces to spinor spaces (e.g. h(CP1)=Spin(2)). NON-INTEGER levels will be presented by a non-integer iteration point in process a continuous transformation ( like tetration) that maps Real number line with one organization to Real number line with different ordering.
Thus, also non-integer operations and dimensions, iterations will obtain their geometric meaning and intuitive understanding will be possible.
Here is a link to more detailed first ideas about how tetration transforms +RP1, -RP1, CP1 etc. with comments from Tony Smith about group structure of such transformations.
http://math.eretrandre.org/tetrationforu...hp?tid=216
Plus some ideas from Tony Smith about this transformation ( still working on it):
Quote:CP1 = SU(2) / U(1) = S3 / S1 = S2 by the Hopf fibration S1 -> S3 -> S2 = 2-sphere
Since SU(2) = S3 = Spin(3) = 3-sphere
and U(1) = S1 = Spin(2) = circle
I think that maybe when you write Spin(1) you should be writing U(1) = Spin(2) = circle
Since we are talking about a complex projective space CP1
the 2 in Spin(2) may refer to the 2-real-dim nature of 1-complex-dim space.
What the Hopf fibration S1 -> S3 -> S2 = CP1 means geometrically
is
that the 3-sphere S3 looks like a 2-sphere S2 = CP1
with a little circle = S1 = U(1) = Spin(2) attached to each of its points.
Tony
Ivars