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Set with algorithmic membership test
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
Computable_set
Mathematical function that can be computed by a program
of computability that can be imagined can compute only functions that are computable in the above sense. Before the precise definition of computable functions
Computable_function
Mathematical logic concept
of computably enumerable sets). Every computable set is computably enumerable, but it is not true that every computably enumerable set is computable. For
Computably_enumerable_set
Real number that can be computed within arbitrary precision
the recursive numbers, effective numbers, computable reals, or recursive reals. The concept of a computable real number was introduced by Émile Borel
Computable_number
Study of computable functions and Turing degrees
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;
Computability_theory
Branch of mathematics that studies sets
axiomatic set theory. Topos theory can interpret various alternatives to that theory, such as constructivism, finite set theory, and computable set theory
Set_theory
Concept in computability theory
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
Turing_reduction
Diagram that shows all possible logical relations between a collection of sets
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
Venn_diagram
Problem in computer science
verification that g is computable relies on the following constructs (or their equivalents): computable subprograms (the program that computes f is a subprogram
Halting_problem
Set of all things that may be the input of a mathematical function
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
Domain_of_a_function
Computation model defining an abstract machine
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
Turing_machine
Collection of mathematical objects
In mathematics, a set is a collection of different things; the things are called elements or members of the set and are typically mathematical objects:
Set_(mathematics)
hyperhypersimple and r-maximal; the latter property says that every computable set R contains either only finitely many elements of the complement of A
Maximal set (computability theory)
Maximal_set_(computability_theory)
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
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
Thesis on the nature of computability
definition of computable function, mathematicians often used the informal term effectively calculable to describe functions that are computable by paper-and-pencil
Church–Turing_thesis
Countable ordinal that is the order type of a computable well-ordering of natural numbers
specifically computability and set theory, an ordinal α {\displaystyle \alpha } is said to be computable or recursive if there is a computable well-ordering
Computable_ordinal
Ordered listing of items in collection
numbers) to the enumerated set must be computable. The set being enumerated is then called recursively enumerable (or computably enumerable in more contemporary
Enumeration
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 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
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
Concept in computability theory
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
Computably_inseparable
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)
Type of set in mathematics
of a computable set. Solovay proved in 1975 that a set can be K-trivial without being computable. The Schnorr–Levin theorem says that random sets have
K-trivial_set
Proof by Alan Turing
proof by Alan Turing, first published in November 1936 with the title "On Computable Numbers, with an Application to the Entscheidungsproblem". It was the
Turing's_proof
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)
Set whose elements all belong to another set
In mathematics, a set A is a subset of a set B if and only if all elements of A are also elements of B; B is then a superset of A. It is possible for A
Subset
Processor executing one instruction in minimal clock cycles
reduced instruction set computer (RISC) chips. Explicitly parallel instruction computing No instruction set computing One-instruction set computer Very long
Reduced instruction set computer
Reduced_instruction_set_computer
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
Classes of partial recursive functions
numbering of partial computable functions. Let φ e {\displaystyle \varphi _{e}} be a computable enumeration of all partial computable functions, and W e
Index_set_(computability)
Axiomatic set theories based on the principles of mathematical constructivism
establish a bijection of two such sets is by relating them through a computable isomorphism, which is a computable permutation of all the naturals. The
Constructive_set_theory
Mathematical concept
Arithmetical hierarchy Computable set Computable number Hartley Rogers Jr. (1967). Theory of recursive functions and effective computability. McGraw-Hill. OCLC 527706
Arithmetical_set
Study of mathematical analysis seen through computability theory
norm operator is also computable. This implies the computability of Riemann integration. The Riemann integral is a computable operator: In other words
Computable_analysis
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
Attempt to formalize all of mathematics, based on a finite set of axioms
In his first theorem, Gödel showed that any consistent system with a computable set of axioms which is capable of expressing arithmetic can never be complete:
Hilbert's_program
Mathematical set that can be enumerated
days of set theory; see Skolem's paradox for more. The minimal standard model includes all the algebraic numbers and all effectively computable transcendental
Countable_set
Halting probability of a random computer program
can be used to simulate any computable function of one variable. Informally, w represents a "script" for the computable function f, and F represents
Chaitin's_constant
Type of computing architecture
No instruction set computing (NISC) is a computing architecture and compiler technology for designing highly efficient custom processors and hardware
No_instruction_set_computing
Branch of mathematical logic
"computable", as in computable function. This name is used because RCA0 corresponds informally to "computable mathematics". In particular, any set of
Reverse_mathematics
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
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
Mathematical-logic system based on functions
usual for such a proof, computable means computable by any model of computation that is Turing complete. In fact computability can itself be defined via
Lambda_calculus
Hierarchy of complexity classes for formulas defining sets
returns whether it is in S; so S is computable. The Turing computable sets of natural numbers are exactly the sets at level Δ 1 0 {\displaystyle \Delta
Arithmetical_hierarchy
Process of repeating items in a self-similar way
scenario that does not use recursion to produce an answer A recursive step — a set of rules that reduces all successive cases toward the base case. For example
Recursion
Impossible task in computing
intuitive notion of "effectively calculable" is captured by the functions computable by a Turing machine (or equivalently, by those expressible in the lambda
Entscheidungsproblem
Processor with instructions capable of multi-step operations
instruction computing Minimal instruction set computer Reduced instruction set computer One-instruction set computer Zero instruction set computer Very
Complex instruction set computer
Complex_instruction_set_computer
Subfield of mathematics
called computability theory, studies the properties of computable functions and the Turing degrees, which divide the uncomputable functions into sets that
Mathematical_logic
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
Statement that is taken to be true
Zermelo–Fraenkel set theory with choice, abbreviated ZFC, or some very similar system of axiomatic set theory like Von Neumann–Bernays–Gödel set theory, a conservative
Axiom
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
Limit of a uniformly computable sequence of functions
computability theory, a function is called limit computable if it is the limit of a uniformly computable sequence of functions. The terms computable in
Computation_in_the_limit
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
Infinite cardinal number
geometric sense), the set of all algebraic numbers, the set of all computable numbers, the set of all computable functions, the set of all binary strings
Aleph_number
Size of a set in mathematics
In mathematics, cardinality is an inherent property of sets, roughly meaning the number of individual objects they contain, which may be infinite. The
Cardinality
Yes/no problem in computer science
function problem of computing the characteristic function of the set associated to the decision problem. If this function is computable then the associated
Decision_problem
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
Fractal named after mathematician Benoit Mandelbrot
present, it is unknown whether the Mandelbrot set is computable in models of real computation based on computable analysis, which correspond more closely to
Mandelbrot_set
Theorem that arithmetical truth cannot be defined in arithmetic
being a formula, being a sentence, etc.), these sets are computable. Moreover, any computable set of numbers can be defined by some arithmetical formula
Tarski's undefinability theorem
Tarski's_undefinability_theorem
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
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)
Non-contradiction of a theory
\lnot \varphi } are elements of the set of consequences of T {\displaystyle T} . Let A {\displaystyle A} be a set of closed sentences (informally "axioms")
Consistency
Measure of algorithmic complexity
2^{*}} be a computable function mapping finite binary strings to binary strings. It is a universal function if, and only if, for any computable f : 2 ∗ →
Kolmogorov_complexity
Function computable with bounded loops
closely with our intuition of what a computable function must be. Certainly the initial functions are intuitively computable (in their very simplicity), and
Primitive_recursive_function
Symbolic description of a mathematical object
powerful definition of 'well-defined' that is able to capture both computable and 'non-computable' statements. All statements characterised in modern programming
Expression_(mathematics)
Symbol representing a mathematical object
often numbers. More specifically, the values involved may form a set, such as the set of real numbers. The object may not always exist, or it might be
Variable_(mathematics)
Yes-or-no question that cannot ever be solved by a computer
said to be undecidable if there is no computable function that correctly answers every question in the problem set. The connection between these two is
Undecidable_problem
Target set of a mathematical function
codomain or set of destination of a function is a set into which all of the outputs of the function are constrained to fall. It is the set Y in the notation
Codomain
Mathematical operation with two operands
specifically, a binary operation on a set is a binary function that maps every pair of elements of the set to an element of the set. Examples include the familiar
Binary_operation
Basic notion of sameness in mathematics
of naive set theory, it was shown that the parallel postulate cannot be proved, the existence of mathematical objects that cannot be computed or explicitly
Equality_(mathematics)
Relationship where one statement follows from another
Undecidable, Basic Papers on Undecidable Propositions, Unsolvable Problems And Computable Functions, New York: Raven Press, ISBN 9780486432281. Papers include those
Logical_consequence
Mathematical theory of data types
to compute the value. The axiom of choice is less powerful in type theory than most set theories, because type theory's functions must be computable and
Type_theory
Characteristic of some logical systems
is a consistent theory. Gödel's incompleteness theorem shows that any computable system that is sufficiently powerful, such as Peano arithmetic, cannot
Completeness_(logic)
Axioms for the natural numbers
model of PA in which either the addition or multiplication operation is computable. This result shows it is difficult to be completely explicit in describing
Peano_axioms
Theorem in computability theory
contradiction that P {\displaystyle P} is a non-trivial, extensional and computable set of natural numbers. Since P {\displaystyle P} is non-trivial, there
Rice's_theorem
Function in mathematical logic
be constructed by invertibly mapping this set of symbols (through, say, an invertible function h) to the set of digits of a bijective base-K numeral system
Gödel_numbering
Reasoning for mathematical statements
models of a given intuitive concept, based on alternate sets of axioms, for example axiomatic set theory and non-Euclidean geometry. As practiced, a proof
Mathematical_proof
Ordered list of whole numbers
definable integer sequences that are not computable, such as sequences that encode the Turing jumps of computable sets. For some transitive models M {\displaystyle
Integer_sequence
Proposition in mathematical logic
specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: There is no set whose
Continuum_hypothesis
Structure of a formal language
A formal grammar is a set of symbols and the production rules for rewriting some of them into every possible string of a formal language over an alphabet
Formal_grammar
In logic, a statement which is always true
formula (one formula per propositional variable). The set of such formulas is a proper subset of the set of logically valid sentences of predicate logic (i
Tautology_(logic)
Size of a possibly infinite set
measures the cardinality of a set, i.e., how many elements there are in a set. The cardinal number associated with a set A {\displaystyle A} is generally
Cardinal_number
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
Complexity class used to classify decision problems
time) is a complexity class used to classify decision problems. NP is the set of decision problems for which the problem instances, where the answer is
NP_(complexity)
Limitative results in mathematical logic
said to be undecidable if there is no computable function that correctly answers every question in the problem set (see undecidable problem). Because of
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Ability of a computing system to simulate Turing machines
guaranteed to complete and halt cannot compute the computable function produced by Cantor's diagonal argument on all computable functions in that language. A computer
Turing_completeness
Basic framework of mathematics
discipline called mathematical logic that includes set theory, model theory, proof theory, computability and computational complexity theory, and more recently
Foundations_of_mathematics
Value indicating the relation of a proposition to truth
has only two possible values (true or false). Truth values are used in computing as well as various types of logic. In some programming languages, any
Truth_value
One-to-one correspondence
function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Given
Bijection
Symbol representing a property or relation in logic
functions (i.e., functions from a set element to a truth value). Set-builder notation makes use of predicates to define sets. In autoepistemic logic, which
Predicate_(logic)
μ {\displaystyle \mu } (of the same set of objects) are called computably isomorphic if there exists a computable bijection f {\displaystyle f} so that
Computable_isomorphism
On solvability of Diophantine equations
Hilary Putnam). Because there exists a recursively enumerable set that is not computable, the unsolvability of Hilbert's tenth problem is an immediate
Hilbert's_tenth_problem
In mathematics, a statement that has been proven
modern axiomatized set theory, the set of all sets cannot be expressed with a well-formed formula. More precisely, if the set of all sets could be expressed
Theorem
Subfield of automated reasoning and mathematical logic
running on a JOHNNIAC, the Logic Theorist constructed proofs from a small set of propositional axioms and three deduction rules: modus ponens, (propositional)
Automated_theorem_proving
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
Axiom of set theory
an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, one can identify another set containing one
Axiom_of_choice
Syntactically correct logical formula
set of variables is finite: <alpha set> ::= p | q | r | s | t | u | ... (the arbitrary finite set of propositional variables) <form> ::= <alpha set>
Well-formed_formula
Sequence of words formed by specific rules
science, and linguistics, a formal language is a set of strings whose symbols are taken from a set called "alphabet". The alphabet of a formal language
Formal_language
priority method. They give a construction for a set that is simple (and thus non-computable), but fails to compute the halting problem. In what follows, W e
Simple_set
Fundamental theorem in mathematical logic
completeness theorem is that it is possible to computably enumerate the semantic consequences of any computably enumerable first-order theory, by enumerating
Gödel's_completeness_theorem
Models of computation
any "computable" function that can be computed by a mathematician with a pen and paper using a finite set of simple algorithms, can be computed by a Turing
Hypercomputation
COMPUTABLE SET
COMPUTABLE SET
Surname or Lastname
English
English : habitational name from a place in Shropshire, so named from Welsh mynydd ‘hill’ + Old English tūn ‘enclosure’, ‘settlement’.
Male
Greek
(Σήθος) Greek form of Egyptian Sutekh, possibly SETHOS means "one who dazzles." In mythology, this is the name of an ancient evil god of Chaos, storms, and the desert, who slew Osiris.Â
Surname or Lastname
English
English : habitational name from Mitcham in Surrey, so named from Old English micel ‘big’ + hÄm ‘homestead’, ‘settlement’.
Female
Japanese
(節å) Japanese name SETSUKO means "temperate child."
Girl/Female
Indian
Sacred, Pure, Comparable to the ganges, Another name for Durga, ***, Another name for Durga
Girl/Female
Tamil
Sacred, Pure, Comparable to the ganges, Another name for Durga, ***, Another name for Durga
Boy/Male
Afghan, Arabic, Celebrity, German, Indian, Muslim, Sindhi
Observer; Supervisor; Little; Insignificant; Warner; Similar; Comparable; Another Name for the Quran; One who Preaches
Boy/Male
Muslim
Similar. Comparable.
Surname or Lastname
English
English : occupational name for a stone- or bricklayer, from Middle English setter ‘one who lays stones or bricks in building’ (agent derivative of setten ‘to set’).English : occupational name from Old French saietier ‘silk weaver’ (an agent derivative of sayete, a kind of silk).English : from an agent derivative of Middle English setten ‘to place (decoration, on a garment or metal surface)’, probably an occupational name for an embroiderer.German : unexplained.Norwegian : unexplained.
Boy/Male
Muslim
Similar. Comparable.
Boy/Male
Arabic, Australian, Muslim
Similar; Comparable; One who Warns
Surname or Lastname
English
English : habitational name from a place in North Yorkshire, so named from Old English setl ‘seat’, ‘dwelling’.
Male
Greek
(Σήθι) Greek form of Egyptian Seti, SETHI means "of Seth."Â
Male
Hindi/Indian
(सेठ) Hindi name derived from the Sanskrit word setu, SETH means "bridge." Compare with other forms of Seth.
Male
Italian
Italian form of Roman Latin Septimus, SETTIMIO means "seventh."
Male
English
Anglicized form of Hebrew Sheth, SETH means "buttocks." In the bible, this is the name of the third son of Adam and Eve. Compare with other forms of Seth.
Girl/Female
Indian
Sacred, Pure, Comparable to the ganges, Another name for Durga, ***, Another name for Durga
Surname or Lastname
English
English : patronymic from Setter.
Girl/Female
Tamil
Sacred, Pure, Comparable to the ganges, Another name for Durga, ***, Another name for Durga
Surname or Lastname
English
English : topographic name for someone who lived in the center of a village, from Middle English midde ‘mid’ + toun ‘village’, ‘town’.English : habitational name from places in Lancashire, Worcestershire, and West Yorkshire, so named in Old English as ‘farmstead at a river confluence’, from (ge)m̄ðe ‘river confluence’ + tūn ‘farmstead’, ‘settlement’.
COMPUTABLE SET
COMPUTABLE SET
Female
Welsh
Variant spelling of Welsh Ceridwen, CERRIDWYN means "fair poetess."
Girl/Female
German
Noble; Kind
Surname or Lastname
German
German : variant of Ulrich.English : probably from the Old English personal name Wulfrīc (see Wooldridge).Scottish : reduced form of McUrich, itself a reduced form of McCurrach.
Surname or Lastname
English
English : variant of Fern 1.Norwegian : habitational name from a farm so named, from far ‘road’, ‘track’ + nes ‘headland’, ‘promontory’.
Girl/Female
American, Australian, Christian, Danish, Dutch, French, German, Hebrew
Bitter; Wished-for Child; Combination of Mary and Lou
Girl/Female
Hindu, Indian, Marathi
Sprout; Bud
Female
English
Pet form of English Coral, CORALIE means "coral" or "small pebble."
Boy/Male
Arabic, Australian, Muslim
One who Loves; Friend
Boy/Male
Tamil
Intelligent, Wise, Prudent, Learned
Boy/Male
Indian, Punjabi, Sikh
Honour of the People
COMPUTABLE SET
COMPUTABLE SET
COMPUTABLE SET
COMPUTABLE SET
COMPUTABLE SET
n.
The quality of being commutable.
a.
Capable of bending or yielding; apt to yield; compliant.
a.
Compatible; suitable; consistent.
a.
Such as can be, or is liable to be, combated; as, combatable foes, evils, or arguments.
a.
Not confutable.
a.
Capable of being computed, numbered, or reckoned.
n.
Quality of being imputable.
a.
Not commutable; not capable of being exchanged with, or substituted for, another.
a.
Capable of existing in harmony; congruous; suitable; not repugnant; -- usually followed by with.
a.
Comparable.
a.
Not compliable; not conformable.
a.
Capable of being commuted or interchanged.
a.
Capable of being attributed; ascribable; imputable.
a.
Suitable; consistent.
a.
Correspondent; conformable; hence, comparable.
a.
Not computable.
n.
The quality of being imputable; imputableness.
adv.
In a compatible manner.
a.
That may be confuted.
n.
The quality of being commutable; interchangeableness.