Partial recursive, or computable functions, may be defined in a number of equivalent ways. This is what Church's thesis is about: all definitions of computability turn out to be equivalent. Church's thesis justifies some confidence in ‘semi-formal’ arguments, used to show that a given function is computable. These arguments can be accepted only if at any moment, upon request, the author of the argument is able to fully formalize it in one of the available axiomatizations.

In this summary, functions are always partial, unless otherwise specified.

Partial recursive functions

The most basic way of defining computable functions is by means of computing devices of which Turing machines are the most well known. A given Turing machine defines, for each n, a partial function f : ωn → ω. More mathematical presentations are by means of recursive program schemes, or by means of combinations of basic recursive functions.

Theorem A1.1.1 (Gödel-Kleene)For any n, the set of Turing computable functions from ωn to ω) is the set of partial recursive functions from ωn to ω, where by definition the class of partial recursive (p.r.) functions is the smallest class containing:

• : ω → ω defined by 0(x) = 0.

• succ : ω → ω (the successor function).

• Projections πn,i : ωn → ω defined by πn,i(x1,…,xn) = xi, and closed under the following constructions:

• Composition: If f1 : ωm → ω, …, fn : ωm → ω and g : ωn → ω are partial recursive, then g ∘ 〈f1, …, fn〉 : ωm → ω is partial recursive.

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