In computability theory, a set S of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable...

every computably enumerable set is many-one reducible to the halting problem, and thus the halting problem is the most complicated computably enumerable set...

countable sets. However it is also often used for computably enumerable sets, which are the countable sets for which an enumeration function can be computed with...

the computable ones consists of the computably enumerable (c.e.) sets, also called semidecidable sets. For these sets, it is only required that there is...

if n is in the set. Thus a set is computably enumerable if and only if it is the domain of some computable function. The word enumerable is used because...

recursively enumerable (also recognizable, partially decidable, semidecidable, Turing-acceptable or Turing-recognizable) if it is a recursively enumerable subset...

states that a set of integers is Diophantine if and only if it is computably enumerable. A set of integers S is computably enumerable if and only if...

{\mathcal {O}}} ; and given any notation for an ordinal, there is a computably enumerable set of notations which contains one element for each smaller ordinal...

recognize. The domain of any universal computable function is a computably enumerable set but never a computable set. The domain is always Turing equivalent...

S-reducibility states that a computably enumerable real set A {\displaystyle A} is s-reducible to another computably enumerable real set B {\displaystyle B} if...

Saul Kripke. Boolos's proof proceeds by constructing, for any computably enumerable set S of true sentences of arithmetic, another sentence which is true...

complete, consistent extension of even Peano arithmetic based on a computably enumerable set of axioms. A theory such as Peano arithmetic cannot even prove...

In computability theory, a subset of the natural numbers is called simple if it is computably enumerable (c.e.) and co-infinite (i.e. its complement is...

As 0 ′ {\displaystyle 0'} is a [computably enumerable] set, it must be computable in the limit itself as the computable function can be defined r ^ ( x...

run with oracle B, computes a partial function with domain A, then A is said to be B-recursively enumerable and B-computably enumerable. We say A {\displaystyle...

work is highly cited in the fields of vector spaces, including computably enumerable sets and vector spaces. Robbins, Gary (October 6, 2017). "Renowned...

for a non-computable, computably enumerable set which the halting problem could not be Turing reduced to. As he could not construct such a set in 1944,...

about the worthlessness of such formulas. Because the set of primes is a computably enumerable set, by Matiyasevich's theorem, it can be obtained from a...

mathematics, the power set (or powerset) of a set S is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory (as developed...

Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any...

but unprovable statement. That is, for any computably enumerable set of axioms for arithmetic (that is, a set that can in principle be printed out by an...

In computability theory, two disjoint sets of natural numbers are called computably inseparable or recursively inseparable if they cannot be "separated"...

{\displaystyle W_{e}} be a computable enumeration of all c.e. sets. Let A {\displaystyle {\mathcal {A}}} be a class of partial computable functions. If A = {...

degree is called recursively enumerable (r.e.) or computably enumerable (c.e.) if it contains a recursively enumerable set. Every r.e. degree is below...

input x} represents the halting problem. This set is recursively enumerable, which means there is a computable function that lists all of the pairs (i, x)...

Robert I. (1987). Recursively enumerable sets and degrees. A study of computable functions and computably generated sets. Perspectives in Mathematical...

semi-decidable, solvable, or provable if A is a recursively enumerable set. In computability theory, the halting problem is a decision problem which can...

In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations...

especially when its elements are themselves sets. Roster or enumeration notation defines a set by listing its elements between curly brackets, separated...

solvable, or provable if the set of inputs (or natural numbers) for which the answer is yes is a recursively enumerable set. Problems that are not decidable...