  We say that a set, \(S\), is decidable if and only if there is a total Turing machine, \(M\), that decides for all \(n \in \mathbf{N}\) whether or not \(n \in S\). Think of “1” as “yes” and “0” as “no”. For all \(n\in \mathbf{N}\), if \(n \in S\), then \(M(n) = 1\), i.e., \(M\) on input \(n\) eventually halts and outputs “yes”, whereas if \(n \not\in S\), then \(M(n) = 0\), i.e., \(M\) on input \(n\) eventually halts and outputs “no”. Synonyms for decidable are: computable, solvable, and recursive.