of steps. A set is noncomputable (or undecidable) if it is not computable. A subset S {\displaystyle S} of the natural numbers is computable if there exists...

of computably enumerable sets). Every computable set is computably enumerable, but it is not true that every computably enumerable set is computable. For...

of computability that can be imagined can compute only functions that are computable in the above sense. Before the precise definition of computable functions...

set is computable if and only if the set and its complement are both computably enumerable. Infinite c.e. sets have always infinite computable subsets;...

the recursive numbers, effective numbers, computable reals, or recursive reals. The concept of a computable real number was introduced by Émile Borel...

set is Turing equivalent to its complement. Every computable set is Turing reducible to every other set. Because any computable set can be computed with...

inseparable if they cannot be "separated" with a computable set. These sets arise in the study of computability theory itself, particularly in relation to Π...

axiomatic set theory. Topos theory can interpret various alternatives to that theory, such as constructivism, finite set theory, and computable set theory...

sets. A set may be finite or infinite. There is a unique set with no elements, called the empty set; a set with a single element is a singleton. Sets...

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

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...

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...

verification that g is computable relies on the following constructs (or their equivalents): computable subprograms (the program that computes f is a subprogram...

numbering of partial computable functions. Let φ e {\displaystyle \varphi _{e}} be a computable enumeration of all partial computable functions, and W e...

delimits a set interleaves with previous curves, starting with the three-circle diagram. Venn's construction for four sets (use Gray code to compute, the digit...

of computable functions. It states that a function on the natural numbers can be calculated by an effective method if and only if it is computable by...

numbers) to the enumerated set must be computable. The set being enumerated is then called recursively enumerable (or computably enumerable in more contemporary...

mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set. For example, given a set called A containing the first four...

specifically computability and set theory, an ordinal α {\displaystyle \alpha } is said to be computable or recursive if there is a computable well-ordering...

In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by dom ⁡ ( f ) {\displaystyle \operatorname...

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...

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...

mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related...

days of set theory; see Skolem's paradox for more. The minimal standard model includes all the algebraic numbers and all effectively computable transcendental...

a set-theoretic paradox published by the British philosopher and mathematician, Bertrand Russell, in 1901. Russell's paradox shows that every set theory...

It is possible to invent a single machine which can be used to compute any computable sequence. If this machine U is supplied with the tape on the beginning...

function problem of computing the characteristic function of the set associated to the decision problem. If this function is computable then the associated...

No instruction set computing (NISC) is a computing architecture and compiler technology for designing highly efficient custom processors and hardware...

In set theory, the intersection of two sets A {\displaystyle A} and B , {\displaystyle B,} denoted by A ∩ B , {\displaystyle A\cap B,} is the set containing...