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→Example: Definition for functionals |
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:# For any computable enumeration operator Φ there is a recursively enumerable set ''F'' such that Φ(''F'') = ''F'' and ''F'' is the smallest set with this property.
:# For any recursive operator Ψ there is a partial computable function φ such that Ψ(φ) = φ and φ is the smallest partial computable function with this property.
The first recursion theorem is also called Fixed point theorem (of recursion theory)<ref>{{Cite book |last=Cutland |first=Nigel |title=Computability: an introduction to recursive function theory}}</ref>. There is also a definition which can be applied to [[Primitive recursive functional|recursive functionals]] as follows:
Let <math>\Phi: \mathbb{F}(\mathbb{N}^k) \rightarrow (\mathbb{N}^k)</math> be a recursive functional. Then <math>\Phi</math> has a least fixed point <math>f_{\Phi}: \mathbb{N}^k \rightarrow \mathbb{N}</math> which is computable i.e.
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