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

both computably enumerable(c.e.). The preimage of a computable set under a total computable function is computable. The image of a computable set under...

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...

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

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...

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...

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...

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,...

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...

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

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

}}0} is defined to be x {\displaystyle x} . Because the set of primes is a computably enumerable set, by Matiyasevich's theorem, it can be obtained from a...

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...

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

specifying a set, one has either to list its elements or to provide a property that uniquely characterizes the set elements. Roster or enumeration notation...

the empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories...

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

Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are...

In set theory, the complement of a set A, often denoted by A c {\displaystyle A^{c}} (or A′), is the set of elements not in A. When all elements in the...

making the notion of recursive enumerability perfectly rigorous. It is evident that Diophantine sets are recursively enumerable (also known as semi-decidable)...

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...

recursion theory, the mathematical theory of computability, a maximal set is a coinfinite recursively enumerable subset A of the natural numbers such that...

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

{\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 = {...

showing that the computable numbers are subcountable. The set S {\displaystyle S} of these Gödel numbers, however, is not computably enumerable (and consequently...

In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in...

such as the set of integers or rationals, but not possible for example if S is the set of real numbers, in which case we cannot enumerate all irrational...