Recursively Enumerable Set - Examples

Examples

• Every recursive set is recursively enumerable, but it is not true that every recursively enumerable set is recursive.
• A recursively enumerable language is a recursively enumerable subset of a formal language.
• The set of all provable sentences in an effectively presented axiomatic system is a recursively enumerable set.
• Matiyasevich's theorem states that every recursively enumerable set is a Diophantine set (the converse is trivially true).
• The simple sets are recursively enumerable but not recursive.
• The creative sets are recursively enumerable but not recursive.
• Any productive set is not recursively enumerable.
• Given a Gödel numbering of the computable functions, the set (where is the Cantor pairing function and indicates is defined) is recursively enumerable. This set encodes the halting problem as it describes the input parameters for which each Turing machine halts.
• Given a Gödel numbering of the computable functions, the set is recursively enumerable. This set encodes the problem of deciding a function value.
• Given a partial function f from the natural numbers into the natural numbers, f is a partial recursive function if and only if the graph of f, that is, the set of all pairs such that f(x) is defined, is recursively enumerable.