Synonymer & Information om | Engelska ordet ENUMERABLE

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• In mathematics, logic and computer science, a formal language is called recursively enumerable (also recognizable, partially decidable, semidecidable, Turing-acceptable or Turing-recognizable) if it is a recursively enumerable subset in the set of all possible words over the alphabet of the language, i.
• If S is indexed as above and R recursively enumerable, then the presentation is a recursive presentation and the corresponding group is recursively presented.
• Each of these was further subdivided into three orders: enumerable (lowest, intermediate, and highest), innumerable (nearly innumerable, truly innumerable, and innumerably innumerable), and infinite (nearly infinite, truly infinite, infinitely infinite).
• A set S of natural numbers is called computably enumerable if there is a partial computable function whose domain is exactly S, meaning that the function is defined if and only if its input is a member of S.
• However, there is no recursively enumerable scheme for systematically naming all ordinals less than the Church–Kleene ordinal, which is a countable ordinal.
• Basic results are that all recursively enumerable classes of functions are learnable while the class REC of all computable functions is not learnable.
• Because the set of primes is a computably enumerable set, by Matiyasevich's theorem, it can be obtained from a system of Diophantine equations.
• There is no complete, consistent extension of even Peano arithmetic based on a computably enumerable set of axioms.
• Harvey Friedman (1974) proved that in any recursively enumerable extension of intuitionistic arithmetic, the disjunction property implies the numerical existence property.
• Because each finitary relation on the natural numbers can be identified with a corresponding set of finite sequences of natural numbers, the notions of computable relation and computably enumerable relation can be defined from their analogues for sets.
• Wicked problems do not have an enumerable (or an exhaustively describable) set of potential solutions, nor is there a well-described set of permissible operations that may be incorporated into the plan.
• This numbering will be surjective (like all numberings) but not injective: there will be distinct numbers that map to the same recursively enumerable set under W.
• On the other hand, we can "enumerate" any recursively enumerable set (see also its complexity class RE) by a primitive-recursive function in the following sense: given an input (M, k), where M is a Turing machine and k is an integer, if M halts within k steps then output M; otherwise output nothing.
• Gödel's incompleteness theorems show that Hilbert's program cannot be realized: if a consistent computably enumerable theory is strong enough to formalize its own metamathematics (whether something is a proof or not), i.
• Every recursively enumerable (or even hyperarithmetic) nonempty subset of this total ordering has a least element.
• Recursively enumerable languages are closed under Kleene star, concatenation, union, and intersection, but not under set difference; see Recursively enumerable language#Closure properties.
• The effective version of the lemma's statement, "every consistent computably enumerable theory can be extended to a complete consistent computably enumerable theory," fails (provided Peano arithmetic is consistent) by Gödel's incompleteness theorem.
• In mathematical logic, Craig's theorem (also known as Craig's trick) states that any recursively enumerable set of well-formed formulas of a first-order language is (primitively) recursively axiomatizable.
• Post was searching for a non-computable, computably enumerable set which the halting problem could not be Turing reduced to.
• In recursion theory, the mathematical theory of computability, a maximal set is a coinfinite recursively enumerable subset A of the natural numbers such that for every further recursively enumerable subset B of the natural numbers, either B is cofinite or B is a finite variant of A or B is not a superset of A.

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