number partitions — part 2

So in the previous post,  I outlined the assumptions involved from a formal systems (and/or Godel’s theorem centric) approach for a proof of “There’ll never be an exact specific formula for predicting number of partitions for number ‘n'”.

This is a follow-up post, trying to get into more detailed nature and define what we mean my “exact specific formula”. 

• “formula” — We refer to a mathematical function   To re-phrase, it is a mapping/pairing from set X to set Y. There can be one-to-one pairings, one-to-many, many-to-many, directed/undirected pairings, cyclic/a-cyclic pairings. (Yes i just made a comparison of it with graphs). Anyways for our case, we add the following constraints.        it will take natural numbers and produce natural numbers.  It is acyclic.  It cannot be one-to-many, but it can be many-to-one. 
• “exact specific” — We refer to the symbol between LHS and RHS and are trying to say it must be the equals operator.
• Another way to put this same is a closed-form solution.

 

So as per the previous post, the thesis/theorem goal becomes to prove that we don’t have a graph subject to above conditions, if we restrict the operations to those defined (as per previous post). 

Formally, a function f from a set X to a set Y is defined by a set G of ordered pairs (x, y) such that x ∈ X, y ∈ Y, and every element of X is the first component of exactly one ordered pair in G.