In yesterday's blog entry, I was approaching the relationship as between the primes and the natural numbers from the Type 1 perspective, where ultimately we view the primes and natural numbers in cardinal terms.
However what is vital to appreciate, in showing how the same fundamental relationship as between the primes and natural numbers is reiterated without limit throughout the number system through the definition of different order primes, is that (i) the cardinal notion of number necessarily implies the ordinal; (ii) the very notion of the primes necessarily implies the natural numbers.
So properly understood - which inherently requires holistic dynamic appreciation - both cardinal and ordinal notions are ultimately completely interdependent; likewise the notions of prime and natural numbers are likewise ultimately completely interdependent with each other.
When one examines the approach I adopted in deriving 2nd Order, 3rd Order,....Nth Order Primes, it is clear that we keep switching as between both cardinal and ordinal notions and also as between prime and natural number notions.
So for example in moving from the 1st Order Primes i.e. 2, 3, 5, 7, 11, .... to the 2nd Order Primes 3, 5, 11, 17, 31,...., we initially rank the 1st Order Primes sequentially in natural number order (i.e. 1, 2, 3, 4, 5,...).
We have moved here therefore to natural number notions of an ordinal kind. Indeed even momentary reflection on the issue will indicate that the very ability to order the primes, implicitly implies the natural numbers (in an ordinal manner). So this begs the key question of how possibly the primes can be unambiguously seen as the building blocks of the natural number system (in cardinal terms), when the very recognition of the primes already requires the natural numbers (in an ordinal manner).
Having ranked the 1st Order Primes sequentially in natural number fashion, we then choose from among these rankings, those corresponding sequentially with the primes (again in an ordinal manner).
So from our list of 1st Order Primes we choose the 2nd, 3rd, 5th, 7th, 11th,.... members on the list.
We then identify these ordinally derived members with the original cardinal primes on the list thus giving us the 2nd Order Primes 3, 5, 11, 17, 31,......
We then express the frequency of these cardinal primes in relation to the original list of primes (also expressed in a cardinal manner).
Therefore though the ultimate relationship is expressed as between both prime and natural numbers expressed in a cardinal manner, their very derivation entails switching as between both prime and natural number notions, and also as between cardinal and ordinal notions.
There is another aspect to all this that might not appear immediately obvious.
I have listed below first 20 natural numbers and the corresponding list of primes (up to 20).
So we are expressing here the relationship of the 0th Order Primes (i.e. the natural numbers) and the 1st Order Primes (the full list of primes).
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
2, 3, 5, 7, 11, 13, 17, 19
Now in the 2nd case, I will provide a list of the first 20 1st Order Primes with the corresponding list of 2nd Order Primes.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71
3, 5, 11, 17, 31, 41, 59, 67
In some ways, though obvious, we can point to a remarkable finding here, in that the relationship as between the 1st Order and 2nd Order Primes exactly replicates the same relationship as between the the starting list of the natural numbers (0th Order) and primes (1st Order).
Thus in the first 20 natural numbers we have a frequency of 8 primes. Likewise in the first 20 (1st Order) Primes, we have a frequency of 8 (2nd Order) Primes.
And no matter how large our starting value n, the frequency in both cases would be exactly similar.
Thus there is not just one relationship governing the primes (1st Order) and natural numbers (0th Order).
We have in fact a never-ending iteration of subsequent relationships as between (N – 1)th Order and Nth Order primes where the same fundamental pattern is exactly replicated.
In fact, in all these cases the Nth Order can be seen as playing the role of the natural numbers in replicating the original order of the primes with the natural numbers.
So in fact we have two ways in general of expressing all these relationships.
We can express each relationship as - I have been doing so far - as that between (N – 1)th Order and Nth Order primes, So the first standard formulation (i.e. primes and natural numbers) expresses therefore the relationship as between the 1st Order and 0th Order Primes. So the natural numbers are here revealed as expressing a special form of primes!
We can express each relationship as between the Nth Order Primes and the Nth Order Natural Numbers. So now the primes are equally revealed as representing a special form of natural number!
So the first standard formulation (again of the primes and natural numbers) can, be equally expressed as that between 1st Order Primes and the 1st Order Natural Numbers.
Thus all subsequent formulations, from this perspective, entails a relationship as between both higher Order Primes and Natural Numbers.