  An impredicative, and potentially “dangerous”, definition would be to define an element to be a number if it satisfies all inductive properties. It is then easy to show that this property “to be a number” is itself inductive. Indeed, 0 is a number since it satisfies all inductive properties, and if \(x\) satisfies all inductive properties then so does \(x+1\). Similarly it is easy to show that \(x+y\) is a number if \(x,y\) are numbers. Indeed the property \(Q(z)\) that \(x+z\) is a number is inductive: \(Q\)(0) holds since \(x+0=x\) and if \(x+z\) is a number then so is \(x+(z+1) = (x+z)+1\). This whole argument however is circular since the property “to be a number” is not predicative and should be treated with suspicion. 