We say that a set is recursive (decidable) if there exists an algorithm that, given an element determines whether .
Example: the set of even numbers is recursive. The algorithm takes represented in binary and checks whether its least signifficant digit is 0.
Example: the set of all pairs such that is a representation of Turing machine and is an input such that terminates on input , is not recursive.
Example: the set of encodings of all valid first-order logic formulas is not recursive.
Example: the set of all valid formulas of Presburger arithmetic is recursive.