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Mathematical logic concept
In computability theory, a set S of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable
Computably_enumerable_set
Ordered listing of items in collection
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
Enumeration
Study of computable functions and Turing degrees
every computably enumerable set is many-one reducible to the halting problem, and thus the halting problem is the most complicated computably enumerable set
Computability_theory
Set with algorithmic membership test
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
Computable_set
{\mathcal {O}}} ; and given any notation for an ordinal, there is a computably enumerable set of notations that contains one element for each smaller ordinal
Kleene's_O
Solution of some Diophantine equation
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
Diophantine_set
Mathematical function that can be computed by a program
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
Computable_function
Formal language
recursively enumerable (also recognizable, partially decidable, semidecidable, Turing-acceptable or Turing-recognizable) if it is a recursively enumerable subset
Recursively enumerable language
Recursively_enumerable_language
In computability theory, a maximal set is a coinfinite computably enumerable subset A of the natural numbers such that for every further computably enumerable
Maximal set (computability theory)
Maximal_set_(computability_theory)
In computability theory, the assignment of natural numbers to a set of objects
same computably enumerable set under W. A numbering is total if it is a total function. If the domain of a partial numbering is computably enumerable then
Numbering (computability theory)
Numbering_(computability_theory)
Theorem in computability theory
{\displaystyle \Sigma _{n+1}^{0}} if and only if B {\displaystyle B} is computably enumerable by an oracle Turing machine with an oracle for ∅ ( n ) {\displaystyle
Post's_theorem
In computability theory, a Friedberg numbering is a computable numbering (enumeration) of the set of all computably enumerable sets that has no repetitions:
Friedberg_numbering
Whether a decision problem has an effective method to derive the answer
consequence of, and thus a member of, the theory. Every complete computably enumerable first-order theory is decidable. An extension of a decidable theory
Decidability_(logic)
Solovay). Informally, a computably enumerable real set A {\displaystyle A} is s-reducible to another computably enumerable real set B {\displaystyle B} if
Enumeration_reducibility
Concept in computability theory
In computability theory, two disjoint sets of natural numbers are called computably inseparable or recursively inseparable if they cannot be "separated"
Computably_inseparable
Limit of a uniformly computable sequence of functions
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
Computation_in_the_limit
Attempt to formalize all of mathematics, based on a finite set of axioms
complete, consistent extension of even Peano arithmetic based on a computably enumerable set of axioms. A theory such as Peano arithmetic cannot even prove
Hilbert's_program
axiomatic set theory by the axiom of infinity, which asserts the existence of the set N of natural numbers. Every infinite set which can be enumerated by natural
Paradoxes_of_set_theory
Method of comparing problems by transforming one into another in computability theory
for a non-computable, computably enumerable set that the halting problem could not be Turing reduced to. As he could not construct such a set in 1944,
Reduction (computability theory)
Reduction_(computability_theory)
Limitative results in mathematical logic
Saul Kripke. Boolos's proof proceeds by constructing, for any computably enumerable set S of true sentences of arithmetic, another sentence which is true
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Halting probability of a random computer program
recognize. The domain of any universal computable function is a computably enumerable set but never a computable set. The domain is always Turing equivalent
Chaitin's_constant
Theorem that arithmetical truth cannot be defined in arithmetic
arithmetic defining the set of codes for arithmetic sentences, and for provable arithmetic sentences (a computably enumerable set). The undefinability theorem
Tarski's undefinability theorem
Tarski's_undefinability_theorem
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
Simple_set
Branch of mathematics that studies sets
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any
Set_theory
Yes-or-no question that cannot ever be solved by a computer
semi-decidable, solvable, or provable if A is a recursively enumerable set. In computability theory, the halting problem is a decision problem which can
Undecidable_problem
Functions in computability theory
Grzegorczyk hierarchy. This implies in particular that every computably enumerable set is enumerable by some E 0 {\displaystyle {\mathcal {E}}^{0}} -function
Grzegorczyk_hierarchy
Collection of mathematical objects
of a possibly larger set. Roster or enumeration notation is a notation introduced by Ernst Zermelo in 1908 that specifies a set by listing its elements
Set_(mathematics)
Informal set theories
Naive set theory is any of several set theories used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined
Naive_set_theory
Mathematical set containing all objects
In set theory, a universal set is a set that contains all of the objects in the theory, including itself. In set theory as usually formulated, it can
Universal_set
Classes of partial recursive functions
{\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 = {
Index_set_(computability)
American mathematician
proof of it, Tennenbaum had also studied Suslin's problem and computably enumerable sets. Over his academic career in the 1960s and 1970s, he switched
Stanley_Tennenbaum
On solvability of Diophantine equations
making the notion of recursive enumerability perfectly rigorous. It is evident that Diophantine sets are recursively enumerable (also known as semi-decidable)
Hilbert's_tenth_problem
Axioms for the natural numbers
Hilbert's tenth problem, whose proof implies that all computably enumerable sets are diophantine sets, and thus definable by existentially quantified formulas
Peano_axioms
Standard system of axiomatic set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in
Zermelo–Fraenkel_set_theory
Formula whose values are the prime numbers
matter of curiosity than of practical use. Because the set of primes is a computably enumerable set, by Matiyasevich's theorem, it can be obtained from a
Formula_for_primes
Maximal proper filter
In the mathematical field of set theory, an ultrafilter on a set X {\displaystyle X} is a maximal filter on the set X . {\displaystyle X.} In other words
Ultrafilter_on_a_set
Set of elements in any of some sets
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
Union_(set_theory)
Mathematical set that can be enumerated
vary and care is needed respecting the difference with recursively enumerable. A set S {\displaystyle S} is countable if: Its cardinality | S | {\displaystyle
Countable_set
Set of the elements not in a given subset
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
Complement_(set_theory)
Russian mathematician and computer scientist (born 1947)
Hilary Putnam had shown that this suffices to prove that every computably enumerable set is Diophantine, a result which solves Hilbert's tenth problem
Yuri_Matiyasevich
Mathematician and philosopher (1906–1978)
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
Kurt_Gödel
Mathematical set of all subsets of a set
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
Power_set
Mathematical set containing no elements
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
Empty_set
Low (computability) Soare, R. I. (1987). Recursively enumerable sets and degrees : a study of computable functions and computably generated sets. Berlin:
High_(computability)
System of mathematical set theory
of mathematics, Morse–Kelley set theory (MK), Kelley–Morse set theory (KM), Morse–Tarski set theory (MT), Quine–Morse set theory (QM) or the system of
Morse–Kelley_set_theory
Set of elements common to all of some sets
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
Intersection_(set_theory)
Fundamental theorem in mathematical logic
that it is possible to computably enumerate the semantic consequences of any computably enumerable first-order theory, by enumerating all the possible formal
Gödel's_completeness_theorem
Class of mathematical set whose elements are all subsets
In set theory, a branch of mathematics, a set A {\displaystyle A} is called transitive if either of the following equivalent conditions holds: whenever
Transitive_set
System of mathematical set theory
connections between KP, computability theory, and the theory of admissible ordinals. KP can be studied as a constructive set theory by dropping the law
Kripke–Platek_set_theory
Yes/no problem in computer science
semidecidable, solvable, or provable if the set of inputs for which the answer is YES is a recursively enumerable set. Problems that are not decidable are undecidable
Decision_problem
Axiomatic set theories based on the principles of mathematical constructivism
constitute sets, the next level being the computably enumerable ones at Σ 1 0 {\displaystyle \Sigma _{1}^{0}} . There is a large corpus of computability theory
Constructive_set_theory
recursively enumerable set is productive. The complement of the set T will not be recursively enumerable, and thus T is an example of a productive set whose
Creative_and_productive_sets
Measure of unsolvability
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
Turing_degree
Ability to solve a problem by an effective procedure
which are recursively enumerable, but not recursive? And, furthermore, are there languages which are not even recursively enumerable? The halting problem
Computability
Type of set in mathematics
Kučera and Terwijn. They built a computably enumerable set that is low for Martin-Löf-randomness but not computable. Their cost function was adaptive
K-trivial_set
Concept in mathematical logic
In set theory, a hereditary set (or pure set) is a set whose elements are all hereditary sets. That is, all elements of the set are themselves sets, as
Hereditary_set
Paradox in set theory
a set-theoretic paradox published by the British philosopher and mathematician, Bertrand Russell, in 1901. Russell's paradox shows that every set theory
Russell's_paradox
American mathematician (1948–2017)
work is highly cited in the fields of vector spaces, including computably enumerable sets and vector spaces. Robbins, Gary (October 6, 2017). "Renowned
Jeffrey_B._Remmel
Set of all true first-order statements about the arithmetic of natural numbers
{R}}} ) of the recursively enumerable Turing degrees, in the signature of partial orders. In particular, there are computable functions S and T such that:
True_arithmetic
Concept in computability theory
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
Turing_reduction
Any one of the distinct objects that make up a set in set theory
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
Element_of_a_set
Problem in computer science
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)
Halting_problem
American mathematician
analytic sets then x# exists for all reals x, and proving with Saharon Shelah that the first-order theory of the partially ordered set of computably enumerable
Leo_Harrington
Fractal named after mathematician Benoit Mandelbrot
set is not computable, but its complement is computably enumerable. Many simple objects (e.g., the graph of exponentiation) are also not computable in
Mandelbrot_set
Set theory concept
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by V, is the class of hereditary
Von_Neumann_universe
Robert I. (1987). Recursively enumerable sets and degrees. A study of computable functions and computably generated sets. Perspectives in Mathematical
Low_(computability)
Diagram that shows all possible logical relations between a collection of sets
between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships
Venn_diagram
Theorem in set theory
In set theory, Kőnig's theorem states that if the axiom of choice holds, I is a set, κ i {\displaystyle \kappa _{i}} and λ i {\displaystyle \lambda _{i}}
Kőnig's_theorem_(set_theory)
Generalization of Turing computability
a set X of natural numbers: in the definition of an ordinal notation, the clause for limit ordinals is changed so that the computable enumeration of
Hyperarithmetical_theory
Finite collection of distinct objects
(or the cardinal number) of the set. A set that is not a finite set is called an infinite set. For example, the set { 1 , 2 , 3 , … } {\displaystyle
Finite_set
Automata that lists elements of some given set
times. An Enumerable Language is Turing Recognizable It's very easy to construct a Turing Machine M {\displaystyle M} that recognizes the enumerable language
Enumerator_(computer_science)
Size of a set in mathematics
different sizes of infinity. They defined three major classes of number: enumerable (finite numbers), unenumerable (asamkhyata, roughly, countably infinite)
Cardinality
Named set of data type values
data type consisting of a set of named values called elements, members, enumeral, or enumerators of the type. The enumerator names are usually identifiers
Enumerated_type
Set that is not a finite set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. The set of natural numbers (whose existence
Infinite_set
Collection of sets in mathematics that can be defined based on a property of its members
In set theory and its applications throughout mathematics, a class is a collection of mathematical objects (often sets) that can be unambiguously defined
Class_(set_theory)
Theory that allows sets to be elements of themselves
Non-well-founded set theories (sometimes unhyphenated, as nonwellfounded; or poorly founded) are variants of axiomatic set theory that allow sets to be elements
Non-well-founded_set_theory
System of mathematical set theory
Tarski–Grothendieck set theory (TG, named after mathematicians Alfred Tarski and Alexander Grothendieck) is an axiomatic set theory. It is a non-conservative
Tarski–Grothendieck set theory
Tarski–Grothendieck_set_theory
System of mathematical set theory
Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel–choice set theory (ZFC). NBG introduces
Von Neumann–Bernays–Gödel set theory
Von_Neumann–Bernays–Gödel_set_theory
Algorithm that outputs all solutions to a problem
recursively enumerable problems. This is the class of sets for which there exist an enumeration algorithm that will produce all elements of the set: the algorithm
Enumeration_algorithm
Being equally consistent
show that Hilbert's program cannot be realized: if a consistent computably enumerable theory is strong enough to formalize its own metamathematics (whether
Equiconsistency
Infinite set that is not countable
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
Uncountable_set
Unicode character block
computers from the 1970s and 1980s, extending the set of characters provided by the Symbols for Legacy Computing block. It includes characters from Amstrad CPC
Symbols for Legacy Computing Supplement
Symbols_for_Legacy_Computing_Supplement
are computably isomorphic. Let A , B ⊆ N {\displaystyle A,B\subseteq \mathbb {N} } be two sets and assume that there are injective, total computable functions
Myhill_isomorphism_theorem
Part of a machine instruction
In computing, an opcode (abbreviated from operation code) is an enumerated value that specifies the operation to be performed. Opcodes are employed in
Opcode
Operation in computability theory
Soare, R.I. (1987). Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets. Springer. ISBN 3-540-15299-7
Turing_jump
Identities and relationships involving sets
mathematics, particularly in the study of set theory, the algebra of sets defines the properties and laws of sets, the set-theoretic operations of union, intersection
Algebra_of_sets
Real number that can be computed within arbitrary precision
showing that the computable numbers are subcountable. The set S {\displaystyle S} of these Gödel numbers, however, is not computably enumerable (and consequently
Computable_number
Category of mathematical proof
Turing: "... the set of solvable Diophantine equations is an example of a computably enumerable but not decidable set, and the set of unsolvable Diophantine
Proof_of_impossibility
Mathematical logic hierarchy
a computably enumerable sequence of basic open sets. A code for such a set is a pair (0,e), where e is the index of a program enumerating the sequence
Borel_hierarchy
Hierarchy of classes of formal grammars
recursive language is recursively enumerable. These are all proper inclusions, meaning that there exist recursively enumerable languages that are not context-sensitive
Chomsky_hierarchy
American logician (1933–2019)
Robert I. (1987), Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets, Perspectives in Mathematical
Gerald_Sacks
Abstract data type for storing distinct values
a given value is in the set, or enumerating the values in some arbitrary order. Other variants, called dynamic or mutable sets, allow also the insertion
Set_(abstract_data_type)
Function computable with bounded loops
functions. For example, the set of provably total functions (in Peano arithmetic) is also recursively enumerable, as one can enumerate all the proofs of the
Primitive_recursive_function
Mathematical result on infinite trees
Robert I. (1987), Recursively Enumerable Sets and Degrees: A study of computable functions and computably generated sets, Perspectives in Mathematical
Kőnig's_lemma
Mathematical set formed from two given sets
In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is an
Cartesian_product
Russian mathematician (1934–2019)
I. Soare, Recursively Enumberable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets. Springer-Verlag, 1999, ISBN 3-540-15299-7;
Albert_Muchnik
Every set is smaller than its power set
theorem can be seen to be true by simple enumeration of the number of subsets. Counting the empty set as a subset, a set with n {\displaystyle n} elements has
Cantor's_theorem
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 recursively axiomatizable
Craig's_theorem
Computation model defining an abstract machine
recursively enumerable language. The Turing machine can equivalently be defined as a model that recognises valid input strings, rather than enumerating output
Turing_machine
Subfield of automated reasoning and mathematical logic
semantically valid well-formed formulas, so the valid formulas are computably enumerable: given unbounded resources, any valid formula can eventually be
Automated_theorem_proving
COMPUTABLY ENUMERABLE-SET
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