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Topic: Basics of recursive functions
Replies: 3   Last Post: Aug 26, 2012 1:49 PM

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hagman

Posts: 1,923
Registered: 1/29/05
Re: Basics of recursive functions
Posted: Aug 26, 2012 1:49 PM
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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.
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> Let f be a function: N ^ k -> N where N denotes the natural numbers and ^ k means to take the n-fold Cartesian product.
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> Suppose f(0, x_1,...,x_k-1) is a recursive function from N ^ k-1 to N.
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> 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

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> Why is f recursive?
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