Decidable set

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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).


[1] E. Mendelson, "Introduction to mathematical logic" , v. Nostrand (1964)


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


[a1] H. Rogers jr., "Theory of recursive functions and effective computability" , McGraw-Hill (1967) pp. 164–165
[a2] S.C. Kleene, "Introduction to metamathematics" , North-Holland (1951) pp. 288
[a3] William I. Gasarch, Georgia Martin, "Bounded Queries in Recursion Theory", Progress in Computer Science and Applied Logic 16. Springer (1999) ISBN 0817639667
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