Number partitions, prime numbers, and Godel Incompleteness approach to partition theorem

Numbers as a formal system: axioms

This is an attempt to prove that there can never be an exact formula for the say nth prime or for that matter the no. of partitions for number n. 

• 0 and 1 exists
•  addition is a way of adding 1 to an existing number
• 0 is the identity element for repeated addtion operation
• Primes is special type of number, which has exactly one partition where all the constituent numbers are equal(in this case 1).

I might be missing something, but i think next step would be to prove that having an “exact closed form formula” for prime would be the equivalent of proving one of the axioms.

Then from Godel’s incompleteness theorem we have proved our premise.