Am Samstag, 25. August 2012 20:56:31 UTC+2 schrieb (unbekannt): > In order to try and understand the Godel incompleteness proofs, I am trying to fill in the following gap in my understanding. > > > > Let f be a function: N ^ k -> N where N denotes the natural numbers and ^ k means to take the n-fold Cartesian product. > > > > Suppose f(0, x_1,...,x_k-1) is a recursive function from N ^ k-1 to N. > > Suppose also that for x_1 > 0, f(x_1, ..., x_k) can be defined in terms of recursive functions and in terms of f(x_1 - 1, ..., x_k).
To avoid ambiguity (such as accidentally allwoing to chose a different recursive function for each value of x_1), I'd prefer for the second condition:
There is a recursive function g: N^(k+1) -> N such that f(x_1+1, x_2, ..., x_k) = g(x_1, ..., x_k, f(x_1, ..., x_k)) holds for all (x_1, .., x_k) in N^k