A better measure of the natural logarithm

Start with sets and define the cartesion   product of multiple sets making another set. This is basic set theory. Then define the decomposition of sets into a unique set of prime sets, show the prime sets are ordered, then all sets are uniquely ordered. And the ordering can be found from the sequence of prime sets that make up any set. We include cartision division, the removal from some cartesian product all elements having members from the divisor.

Having done that, then define the following:

The thing to the left maps any set x to its unique position onto a ruler.  As the number of prime sets grows the thing on the left retains it mapping properties, and the ruler is notched with markers that are logarithm spaced. Logarithm is defined to mean the spacing between notches that keeps the laws of sets valid.

Then you have it.  From there, let the primes go to infinity and you have calculus.  Find the inverse mapping (the exponential) and you have the standard number line.  Thus, everything is ultimately combinatorics, and the rules of combinatorics remain valid as the number of combinations go to infinity.

From there we use the science of yardsticks.  The notch length and the space between notches are both defined, the notch length has to be ordered opposite the spacing..  Exchanging the notch with the spacing gives us a dual yardstick defining one as the dividing line between them.  The dual yardstick has the same ordering, but in the opposite direction.

The basic idea is to simply a lot of proofs using the minimum redundancy. For example, what is the least number of operations required to find the position of a particular set on the ruler?  Use its prime decomposition. Then as you let the primes go to infinity a lot of prime theory is maintained.

All the transcendentals are simply the definition of a set formula as the number of primes gets larger. Irrationals are defined as the value of some integer fraction as the number of primes gets larger.

So I am using this approach since my head is getting worn out when I have to constantly ask, how does nature know the value of some transcendental,