13. Computability and Decidability

A function f is computable if there exist an algorithm that produces the value of f correctly for each possible argument of f. The computation of f for a particular argument x just involves following the instructions in the algorithm, and it produces the result f(x) in a finite number of steps if x is in the domain of f. If x is not in the domain of f then the algorithm may produce an answer saying so or it might run forever never halting. The concept of computability may be made precise in several equivalent ways such as Church’s lambda calculus, recursive function theory or by the theoretical Turing machines. Church and Turing independently showed in 1936 that mathematics is not decidable: i.e. there is no mechanical procedure (i.e. algorithm) to determine whether an arbitrary mathematical proposition is true or false, and so the only way is to determine the truth or falsity of a statement is, try to solve the problem.