nLab recursive subset

Idea

A set $S$ of natural numbers is recursive if there is an algorithm which will decide in finitely many steps whether a given natural number belongs to $S$.

Definition

A subset $S$ of the set of natural numbers $\mathbb{N}$, or more generally of $\mathbb{N}^k$ with $k$ finite, is recursive if there is a computable function (a total recursive function) $f: \mathbb{N}^k \to \mathbf{2} = \{0, 1\} \subseteq \mathbb{N}$ such that $S = f^{-1}(1)$. Recursive subsets are a proper subclass of the class of recursively enumerable? sets, which are domains of partial recursive functions $f: \mathbb{N}^k \to \mathbb{N}$, or equivalently images of total recursive functions.

Properties

Recursive sets form a Boolean subalgebra of the power set algebra $P(\mathbb{N})$ (whereas recursively enumerable subsets do not form a Boolean subalgebra). Even in constructive mathematics (where the power set algebra may be only a Heyting algebra), the recursive sets form a Boolean algebra (a Boolean subalgebra of the algebra of decidable subsets).

Examples and counterexamples

• One can encode proofs in the formal theory $PA$ (Peano arithmetic) as natural numbers, via a process of Gödel-numbering. The set of codes of such formal proofs is a recursive set. In colloquial language: it is possible to program a computer to detect whether or not a string of symbols represents a valid proof in $PA$.

• It follows that the set of codes of theorems (provable propositions) is recursively enumerable. However, it is not recursive. This is one way of saying that provability in PA cannot be decided by an algorithm, which is closely related to Gödel’s incompleteness theorems.

Related concepts

• definable set, constructible set