Decidable set

A set of constructive objects of some fixed type which admits an algorithm for checking whether an element belongs to it. In fact one can restrict oneself to the concept of a decidable set of natural numbers, since the more general case can be reduced to this case by enumerating the objects under consideration. A set $M$ of natural numbers is said to be decidable if there exists a general recursive function $f$ such that $M = \{ n : f(n) = 0 \}$. In this case $f$ is an algorithm for checking whether a natural number belongs to $M$: indeed, $n \in M$ is equivalent to $f(n) = 0$. A decidable set of natural numbers is also often called a general recursive set or a recursive set.

In many well-known mathematical problems (such as the word identity problem, the homeomorphism problem, Hilbert's 10th problem, the "Entscheidungsproblem" in mathematical logic) one is required to prove or refute the assertion that a certain concrete set is decidable. Well-known (negative) solutions to the problems listed above consist of establishing that the sets corresponding to them are undecidable (see also Algorithmic problem).

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Decidable sets are also referred to as recursive or solvable. ([a3], p.11)

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