# Recursive Set

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.