Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In the theory of computation in computer science, a programming system is a Godel numbering of the set mathcal{T} of all Turing-computable functions from mathbb{N} to mathbb{N}. The name derives from the numbering of mathcal{T} induced by a numbering of the programs of a Turing-complete programming language. A programming system phi_1, phi_2, phi_3, dots is said to be universal if its (partial) universal function, u: mathbb{N}^2 rightharpoonup mathbb{N}: forall n, x in mathbb{N}, u(n,x) = phi_n(x), is Turing-computable. An acceptable programming system (also called an admissible Godel numbering of mathcal{T}), is a programming system that is universal and has a total Turing-computable composition function c: mathbb{N}^2 to mathbb{N}: forall i,j in mathbb{N}, phi_{c(i,j)} = phi_i circ phi_j . Equivalently, an acceptable programming system is a programming system that is universal and satisfies the s-m-n theorem.

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