Primes as Figure

In the Last post, we saw how it was not possible to capture primes in a formal system with their definition as via negative of composites. Let’s continue our endeavour to capture Primes and Primeness in a formal system, but this time using the positive definition of non-divisibility of primes i.e. a prime number is not divisible by any number from two to one less than itself.

For easy reading I am going to use ‘%’ symbol as a symbol for hyphen.

We represent the primeness of 2 as our axiom, our Formal system for prime numbers is P.

Axiom: P%% ;

But first we capture the property of non-divisibility of any two numbers through another Formal system DND.

Axiom Schema: xyDNDx, where x and y are hyphen strings.

e.g. %%%DND%% where x is ‘%%’ and y is ‘%’. The idea is simple, no number greater than x divides x, is our axiom schema. Remember, however that the interpretation we give to the system is outside of the system (that is what the game is about).

Rule: If xDNDy is a theorem, then so is xDNDxy (x and y stand for hyphen string)

The rule is simple enough, If x doesn’t divide y, it doesn’t divide x+y either.

e.g. %%%DND%% where x is ‘%%%’ and y is ‘%%’, is a theorem. So is, %%%DND%%%%%, by the rule of production.

Now, we are going to use the DND formal system, to create a formal system which captures the property of a number n1 being divisor free till another number n2.We construct a framework where each theorem bears witness to the unbroken chain of divisor-free existence.This formal system is DF (divisor free).

Rule: If %%DNDz is a theorem, so is zDF%%.

Rule: If zDFx is a theorem and also x%DNDz is a theorem, then zDFx% is a theorem.

e.g. %%DND%%%%% is a theorem of DND, so %%%%%DF%% is a theorem of DF.

Now, following the 2nd rule, :

since, %%%DND%%%%% is a theorem, so is %%%%%DF%%%.

iteratively, %%%%DND%%%%% implies %%%%%DF%%%%.

Here comes our Formal system P.

Axiom Schema: P%%

Rule: If z%DFz is a theorem, then Pz% is a theorem.

e.g. the above derivation makes P%%%%% a theorem.

So, the task to represent primeness in a formal system involves constructing a formal system DND to represent non-divisibility, a formal system DF to represent numbers which are not divisible till a number and finally using them to construct P.