Well I wanted to extend this while I still have the ideas in my mind.
Probably, we have to start with tetra-integers n(4) , meaning n in a[4]n(4). These integers will have :
-Non associative, non-commutative partitions (since 2(4)+3(4) = 5(4)? will not be the same as 3(4)+2(4) in terms of the result of operation a[4]5 = a[4](2(4)+3(4)) is not the same as a[4](3(4)+2(4))
-correspondingly different combinatorics, ordered
-different notion of primes , etc.
- once these rules are established, it should be possible to create :
tetra rationals Q(4)
tetra reals R(4)
tetra negative numbers and Z(4)
now, with this, it would be possible to extend the notion of equation to e.g.:
x[4]n+x[4]m = C
Which would allow to define tetra imaginary unit perhaps as solution to :
x[4]2=-1 (x^x=-1) and lead to properties of tetra complex numbers C(4).
if we have function a[4]n = f(a,4,n) obviously taking n-> infinity ends up with principially unreachable scale of infinity (tetra-infinity) if we take f(a,4,n) to infinity via taking a->infinity.
This allows to look for another notion of tetra - infinitesimal which would allow to establish tetra-differentiation and tetra-integration.
Now we can also answer the question about h(z) moving things to complex numbers from reals.
The result h(z) = complex means that from the math point of view we use, the next scale(s) of infinity ( tetra and above) shows up like imaginary for C(3) .They are however real in R(4). However, from equation x^x=-1 and generally x[n]2=-1 it is obvios that there is a structure behind this. However, what the speed of tetration does, is to push the result over the limits of infinity scale we work in with normal, exponential functions and normal integers.
Obviously, the number of discrete scales of infinity thus corresponds to the number of operations- tetra, penta, etc. We may again form operation integers based on this and ask what are their partitions ,combinatorics etc and extensions to rational, real, etc values and calculus.
So we should end up with 3 types of integers
1) Integers for numbers
2) Integers for operations
3) integers for application times of operations to numbers
Each of them will have different partition laws and combinatorics; integers for NUMBERS is clear and well known, properties of integers for operations is not known, properties for integres for number of application of operations will depend on the number of operations. If operation number is fixed, e.g tetra= 4 these combinatorics/partitions of INTEGER n for number of applications of operations will be a function of operation so, n(4).
Now we can define pentation as exponentiation in scale based on tetration and move on with finding out the laws of arithmetics of n(5), n(6) etc. and corresponding Q(5), R(5), Z(5), C(5) and look for patterns how they develop, as we have N(n), Q(n), R(n), Z(n), C(n) etc. and corresponding calculus.
If we find them, we can extend these PATTERNS of laws of arithmetics/combinatorics to rational, real, imaginary operation numbers and see what happens.
Then we look back and try to understand the whole thing together.
The first instructive thing would be to develop artithmetics and combinatorics of tetration integers n(4) = application times of tetration to some number.
I got these ideas from Ramanujans f(x) mentioned above and Hardy's "Orders of infinity", combined. I have these in djvu.
Ivars
P.S. The question about the physical essence of these things are what really interests me, so please excuse for non-rigid notions. The fine structure of imaginary unit appearing as a result of cumulative action of all scales of infinities defined above could be related to alpha, which I have tried to catch earlier, but without understanding why and what it would be shooting in the dark. "Jumping" over infinity, as tetration does , is also a very useful idea for phase changes in general (for me things always happen in general, somehow...
).
Probably, we have to start with tetra-integers n(4) , meaning n in a[4]n(4). These integers will have :
-Non associative, non-commutative partitions (since 2(4)+3(4) = 5(4)? will not be the same as 3(4)+2(4) in terms of the result of operation a[4]5 = a[4](2(4)+3(4)) is not the same as a[4](3(4)+2(4))
-correspondingly different combinatorics, ordered
-different notion of primes , etc.
- once these rules are established, it should be possible to create :
tetra rationals Q(4)
tetra reals R(4)
tetra negative numbers and Z(4)
now, with this, it would be possible to extend the notion of equation to e.g.:
x[4]n+x[4]m = C
Which would allow to define tetra imaginary unit perhaps as solution to :
x[4]2=-1 (x^x=-1) and lead to properties of tetra complex numbers C(4).
if we have function a[4]n = f(a,4,n) obviously taking n-> infinity ends up with principially unreachable scale of infinity (tetra-infinity) if we take f(a,4,n) to infinity via taking a->infinity.
This allows to look for another notion of tetra - infinitesimal which would allow to establish tetra-differentiation and tetra-integration.
Now we can also answer the question about h(z) moving things to complex numbers from reals.
The result h(z) = complex means that from the math point of view we use, the next scale(s) of infinity ( tetra and above) shows up like imaginary for C(3) .They are however real in R(4). However, from equation x^x=-1 and generally x[n]2=-1 it is obvios that there is a structure behind this. However, what the speed of tetration does, is to push the result over the limits of infinity scale we work in with normal, exponential functions and normal integers.
Obviously, the number of discrete scales of infinity thus corresponds to the number of operations- tetra, penta, etc. We may again form operation integers based on this and ask what are their partitions ,combinatorics etc and extensions to rational, real, etc values and calculus.
So we should end up with 3 types of integers
1) Integers for numbers
2) Integers for operations
3) integers for application times of operations to numbers
Each of them will have different partition laws and combinatorics; integers for NUMBERS is clear and well known, properties of integers for operations is not known, properties for integres for number of application of operations will depend on the number of operations. If operation number is fixed, e.g tetra= 4 these combinatorics/partitions of INTEGER n for number of applications of operations will be a function of operation so, n(4).
Now we can define pentation as exponentiation in scale based on tetration and move on with finding out the laws of arithmetics of n(5), n(6) etc. and corresponding Q(5), R(5), Z(5), C(5) and look for patterns how they develop, as we have N(n), Q(n), R(n), Z(n), C(n) etc. and corresponding calculus.
If we find them, we can extend these PATTERNS of laws of arithmetics/combinatorics to rational, real, imaginary operation numbers and see what happens.
Then we look back and try to understand the whole thing together.
The first instructive thing would be to develop artithmetics and combinatorics of tetration integers n(4) = application times of tetration to some number.
I got these ideas from Ramanujans f(x) mentioned above and Hardy's "Orders of infinity", combined. I have these in djvu.
Ivars
P.S. The question about the physical essence of these things are what really interests me, so please excuse for non-rigid notions. The fine structure of imaginary unit appearing as a result of cumulative action of all scales of infinities defined above could be related to alpha, which I have tried to catch earlier, but without understanding why and what it would be shooting in the dark. "Jumping" over infinity, as tetration does , is also a very useful idea for phase changes in general (for me things always happen in general, somehow...
).