This paper answers the open problem of finding a type system that characterizes hereditary permutators.??First this paper shows that there does not exist such a type system by showing that the set of hereditary permutators is not recursively enumerable.??The set of positive primitive recursive functions is used to prove it.??Secondly this paper gives a best-possible solution by providing a countably infinite set of types such that a term has every type in the set if and only if the term is a hereditary permutator.??By the same technique for the first claim, this paper also shows that a set of normalizing terms in infinite lambda-calculus is not recursively enumerable if it contains some term having a computable infinite path,and shows the set of streams is not recursively enumerable.