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Mathematical concept
arithmetical set (or arithmetic set) is a set of natural numbers that can be defined by a formula of first-order Peano arithmetic. The arithmetical sets
Arithmetical_set
Hierarchy of complexity classes for formulas defining sets
certain sets based on the complexity of formulas that define them. Any set that receives a classification is called arithmetical. The arithmetical hierarchy
Arithmetical_hierarchy
Branch of elementary mathematics
types of arithmetic. Modular arithmetic operates on a finite set of numbers. If an operation would result in a number outside this finite set then the
Arithmetic
Model that describes the programmable interface of a computer processor
of memory, but most RISC instruction sets include SIMD or vector instructions that perform the same arithmetic operation on multiple pieces of data at
Instruction_set_architecture
Branch of mathematical logic
arithmetical formula (one with no bound set variables, although possibly containing set parameters). An arithmetical formula is a formula where set variables
Reverse_mathematics
Type of average of a collection of numbers
collection. The collection is often a set of results from an experiment, an observational study, or a survey. The term "arithmetic mean" is preferred in some contexts
Arithmetic_mean
Computation modulo a fixed integer
In mathematics, modular arithmetic is a system of arithmetic operations for integers, differing from the usual ones in that numbers "wrap around" when
Modular_arithmetic
Mathematical system
variables (that is, no quantifiers over set variables) is called arithmetical. An arithmetical formula may have free set variables and bound individual variables
Second-order_arithmetic
Sequence of equally spaced numbers
interpreting the arithmetic progression as a set of equally probable outcomes. The product of the members of a finite arithmetic progression with an
Arithmetic_progression
Axioms for the natural numbers
Schröder. The Peano axioms define the arithmetical properties of natural numbers, usually represented as a set N or N . {\displaystyle \mathbb {N} .}
Peano_axioms
Combinational digital circuit
operation has completed, the ALU inputs may be set up for the next ALU operation. A number of basic arithmetic and bitwise logic functions are commonly supported
Arithmetic_logic_unit
Real number uniquely specified by description
definable in the language of arithmetic is called analytical. Every computable real number is arithmetical, and the arithmetical numbers form a subfield of
Definable_real_number
Operations on ordinals that extend classical arithmetic
In the mathematical field of set theory, ordinal arithmetic includes binary operations on ordinal numbers such as addition, multiplication, and exponentiation
Ordinal_arithmetic
Limitative results in mathematical logic
its proof, is an arithmetical relation between two numbers. Therefore, there is a statement form Bew(y) that uses this arithmetical relation to state
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Axiomatic logical system
In mathematics, Robinson arithmetic is a finitely axiomatized fragment of first-order Peano arithmetic (PA), first set out by Raphael M. Robinson in 1950
Robinson_arithmetic
Haar null set Convex set Balanced set, Absolutely convex set Fractal set Recursive set Recursively enumerable set Arithmetical set Diophantine set Hyperarithmetical
List_of_types_of_sets
Branch of mathematics that studies sets
example, the set {1} is both a member and a proper subset of the set {1, {1}}. Just as arithmetic features binary operations on numbers, set theory features
Set_theory
Model of (first-order) Peano arithmetic that contains non-standard numbers
models is known. However, the arithmetical operations are much more complicated. It is easy to see that the arithmetical structure differs from ω + (ω*
Non-standard model of arithmetic
Non-standard_model_of_arithmetic
Set of all true first-order statements about the arithmetic of natural numbers
or lower in the arithmetical hierarchy. Post's theorem shows that, for each n, Thn( N {\displaystyle {\mathcal {N}}} ) is arithmetically definable, but
True_arithmetic
Function in mathematical logic
Gödel mapping and r is an inference rule, then there should be some arithmetical function gr of natural numbers such that if formula C is derived from
Gödel_numbering
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)
Generalization of Turing computability
classified into a hierarchy extending the arithmetical hierarchy; the hyperarithmetical sets are exactly the sets that are assigned a rank in this hierarchy
Hyperarithmetical_theory
Non-contradiction of a theory
describe a strong enough fragment of arithmetic—including set theories such as Zermelo–Fraenkel set theory (ZF). These set theories cannot prove their own
Consistency
Progression-free set of numbers
particular in arithmetic combinatorics, a Salem-Spencer set is a set of numbers no three of which form an arithmetic progression. Salem–Spencer sets are also
Salem–Spencer_set
Decidable first-order theory of the natural numbers with addition
decidability of Presburger arithmetic can be shown using quantifier elimination, supplemented by reasoning about arithmetical congruence. The steps used
Presburger_arithmetic
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
Standard system of axiomatic set theory
independence proof by forcing automatically proves independence from arithmetical statements, other concrete statements, and large cardinal axioms. Some
Zermelo–Fraenkel_set_theory
Term in mathematical logic
numbers and their usual arithmetic operations and order relation. The sets definable in this structure are known as the arithmetical sets, and are classified
Definable_set
Concept in mathematical logic and set theory
In mathematical logic and descriptive set theory, the analytical hierarchy is an extension of the arithmetical hierarchy. The analytical hierarchy of formulas
Analytical_hierarchy
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)
Number used for counting
natural number c where a + c = b. This order is compatible with the arithmetical operations in the following sense: if a, b and c are natural numbers
Natural_number
IEEE standard for floating-point arithmetic
floating-point units use the IEEE 754 standard. The standard defines: arithmetic formats: sets of binary and decimal floating-point data, which consist of finite
IEEE_754
Mathematical model for deduction or proof systems
the axioms. An example of a logical system is Peano arithmetic. The standard model of arithmetic sets the domain of discourse to be the nonnegative integers
Formal_system
Arithmetic operation
1709 with a calculating clock made of wood that could perform all four arithmetical operations. These early attempts were not commercially successful but
Addition
Branch of mathematics
formulas that define them. Any set that receives a classification is called "arithmetical". More formally, the arithmetical hierarchy assigns classifications
Effective descriptive set theory
Effective_descriptive_set_theory
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
Theorem that arithmetical truth cannot be defined in arithmetic
computable set of numbers can be defined by some arithmetical formula. For example, there are formulas in the language of arithmetic defining the set of codes
Tarski's undefinability theorem
Tarski's_undefinability_theorem
Type of arithmetic where output is limited to a fixed range of values
SSE2 and AVX2 integer instruction sets. It is also available in the ARM NEON instruction set. Saturation arithmetic for integers has also been implemented
Saturation_arithmetic
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)
Finite union of linear sets of integer vectors
with finitely many period vectors. Semilinear sets are a higher-dimensional analogue of an arithmetic progression: where a progression is generated by
Semilinear_set
Branch of mathematics
on that set. It is a generalization of elementary and linear algebra since it allows mathematical objects other than numbers and non-arithmetic operations
Algebra
Sets big enough to assert the existence of arithmetic progressions with common difference
Ronald; Landman, Bruce (1999). "On the Set of Common Differences in van der Waerden's Theorem on Arithmetic Progressions". Canadian Mathematical Bulletin
Large_set_(Ramsey_theory)
Arithmetical calculations using only the human brain
Mental calculation (also known as mental computation) consists of arithmetical calculations made by the mind, within the brain, with no help from any supplies
Mental_calculation
Shift operator in computer programming
number of compilers. For example, in the x86 instruction set, the SAR instruction (arithmetic right shift) divides a signed number by a power of two, rounding
Arithmetic_shift
Shape containing unit line segments in all directions
In mathematics, a Kakeya set, or Besicovitch set, is a set of points in Euclidean space which contains a unit line segment in every direction. For instance
Kakeya_set
Method for bounding the errors of numerical computations
problems of low dimension. Interval arithmetic can be used in various areas (such as set inversion, motion planning, set estimation, or stability analysis)
Interval_arithmetic
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
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
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
Branch of algebraic geometry
ring of integers. The classical objects of interest in arithmetic geometry are rational points: sets of solutions of a system of polynomial equations over
Arithmetic_geometry
Study of computable functions and Turing degrees
the Turing degree of a set of natural numbers and the difficulty (in terms of the arithmetical hierarchy) of defining that set using a first-order formula
Computability_theory
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)
Function whose domain is the positive integers
In number theory, an arithmetic, arithmetical, or number-theoretic function is generally any function whose domain is the set of positive integers and
Arithmetic_function
Smallest ordinal number that, considered as a set, is uncountable
Epsilon numbers (mathematics) Large countable ordinal Ordinal arithmetic "Set Theory > Basic Set Theory (Stanford Encyclopedia of Philosophy)". plato.stanford
First_uncountable_ordinal
Number taken as representative of a list of numbers
9] is generally considered to be (2+3+4+7+9)/5 = 25/5 = 5. The arithmetic mean of a set of numbers x1, x2, ..., xn is typically denoted using an overhead
Average
Logical quantification that ranges over a subset of the universe of discourse
elementary, context-sensitive, and primitive recursive. In the arithmetical hierarchy, an arithmetical formula that contains only bounded quantifiers is called
Bounded_quantifier
Numbers and the basic operations on them
While elementary arithmetic mainly operates under the set of natural numbers (sometimes including 0), multiplication under other number sets can satisfy more
Elementary_arithmetic
Family of RISC-based computer architectures
and 64-bit arithmetic with its new 32-bit fixed-length instruction set. Arm Holdings has also released a series of additional instruction sets for different
ARM_architecture_family
Type of numeric sequence
set generalizes this idea to multiple dimensions – it is a set of vectors of integers, rather than a set of integers. A finite generalized arithmetic
Generalized arithmetic progression
Generalized_arithmetic_progression
Integers have unique prime factorizations
possesses arithmetical properties similar to those of the multiplicative semigroup of positive integers. Fundamental Theorem of Arithmetic is, in fact
Fundamental theorem of arithmetic
Fundamental_theorem_of_arithmetic
Computer approximation for real numbers
the discretization error and is limited by the machine epsilon. The arithmetical difference between two consecutive representable floating-point numbers
Floating-point_arithmetic
Computer arithmetic error
saturation arithmetic, overflowed results may be clamped, i.e. set to the minimum value in the representable range if the result is below the minimum and set to
Integer_overflow
Overview of and topical guide to logic
theory that is still being actively researched. Alpha recursion theory Arithmetical set Church–Turing thesis Computability logic Computable function Computation
Outline_of_logic
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
Mathematical logic concept
a partial computable function. The set S is Σ 1 0 {\displaystyle \Sigma _{1}^{0}} (referring to the arithmetical hierarchy). There is a partial computable
Computably_enumerable_set
Identities and relationships involving sets
of sets can be interpreted as the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union
Algebra_of_sets
Property of large sets
the reciprocals of the members of a set A of positive integers diverges, then A contains arbitrarily long arithmetic progressions. Formally, the conjecture
Erdős conjecture on arithmetic progressions
Erdős_conjecture_on_arithmetic_progressions
Statistical amount
The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points
Weighted_arithmetic_mean
Number in {..., –2, –1, 0, 1, 2, ...}
the various laws of arithmetic. In modern set-theoretic mathematics, a more abstract construction allowing one to define arithmetical operations without
Integer
theory Diophantine set Matiyasevich's theorem Word problem for groups Arithmetical hierarchy Subrecursion theory Presburger arithmetic Computational complexity
List of mathematical logic topics
List_of_mathematical_logic_topics
Statement that is taken to be true
set of non-logical axioms Σ {\displaystyle \Sigma } of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement
Axiom
Mathematical set that can be enumerated
mathematical set is countable if either it is finite or it can be put in one to one correspondence with the set of natural numbers. Equivalently, a set is countable
Countable_set
Set of points on a line segment with certain topological properties
The complement of the Cantor ternary set is an example of a fractal string. In arithmetical terms, the Cantor set consists of all real numbers of the unit
Cantor_set
Size of a set in mathematics
formulated many advanced theorems of set theory, and helped establish set-theoretic foundations of algebra and arithmetic. Dedekind's The Nature and Meaning
Cardinality
Theorem about prime numbers
k} , the set A {\displaystyle A} contains infinitely many arithmetic progressions of length k {\displaystyle k} . In particular, the entire set of prime
Green–Tao_theorem
Computer operation which manipulates invidual bits of data
- * / % + - << >> & ^ | Arithmetic logic unit Bit manipulation Bitboard Bitwise operations in C Double dabble Find first set Karnaugh map Logic gate Logical
Bitwise_operation
Size of a possibly infinite set
infinite sets. Therefore, cardinal numbers are not usually thought of in terms of their formal definition, but immaterially in terms of their arithmetic/algebraic
Cardinal_number
Set of integers whose sum of reciprocals diverges
a large set; this statement is equivalent to the divergence of the harmonic series. More generally, any arithmetic progression (i.e., a set of all integers
Large_set_(combinatorics)
Basic framework of mathematics
system – such as necessary to axiomatize the elementary theory of arithmetic on the (infinite) set of natural numbers – a statement that formally expresses its
Foundations_of_mathematics
3-volume treatise on mathematics, 1910–1913
It purports to reveal the fundamental basis for arithmetic. However, it is our everyday arithmetical practices such as counting which are fundamental;
Principia_Mathematica
Problem in computer science
enumerable, but not recursive, sets of numbers, or similarly Σ 1 0 {\displaystyle \Sigma _{1}^{0}} of the arithmetical hierarchy. Rice's theorem establishes
Halting_problem
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)
Disorder affecting learning arithmetic
"Brain Activity during a Visuospatial Working Memory Task Predicts Arithmetical Performance 2 Years Later". Cerebral Cortex. 22 (5): 1078–1085. doi:10
Dyscalculia
Form of entropy encoding used in data compression
digits, and recover the string. In general, arithmetic coders can produce near-optimal output for any given set of symbols and probabilities. (The optimal
Arithmetic_coding
Binary representation for signed numbers
compatible with two's complement representation. Continuity of binary arithmetical and bitwise operations in 2-adic metric also has some use in cryptography
Two's_complement
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
Set with algorithmic membership test
only if it is at level Δ 1 0 {\displaystyle \Delta _{1}^{0}} of the arithmetical hierarchy. A is computable if and only if it is either the image (or
Computable_set
Puzzle of reconstructing equations that have been enciphered into words
verbal arithmetic puzzle include the alphametic, the digimetic, and the skeletal division. Alphametic A type of verbal arithmetic puzzle in which a set of
Verbal_arithmetic
Field of mathematics
articles and books covering a wide range of arithmetical dynamical topics. Arithmetic geometry Arithmetic topology Combinatorics and dynamical systems
Arithmetic_dynamics
Concept in number theory
also referred to as asymptotic density or arithmetic density, is a measure of how "large" a subset of the set of natural numbers is. It relies chiefly
Natural_density
Axiomatic set theories based on the principles of mathematical constructivism
not merely express arithmetical sets, while all sets of naturals particular such theories prove to exist are just computable sets. Theorems therein can
Constructive_set_theory
N-th root of the product of n numbers
classical Pythagorean means, together with the arithmetic mean and the harmonic mean. For all positive data sets containing at least one pair of unequal values
Geometric_mean
Digit transferred from one column to another
In elementary arithmetic, a carry is a digit that is transferred from one column of digits to another column of more significant digits. It is part of
Carry_(arithmetic)
Concept in mathematics
Affine arithmetic (AA) is a model for self-validated numerical analysis. In AA, the quantities of interest are represented as affine combinations (affine
Affine_arithmetic
Microcontroller machine language
Set when an arithmetic result is zero, and cleared when it is non-zero. N Negative flag. Set to a copy of the most significant bit of an arithmetic result
Atmel_AVR_instruction_set
1999 American TV series or program
The Devil's Arithmetic is a 1999 historical fantasy TV movie based on the novel of the same name by Jane Yolen. It stars Kirsten Dunst as Hannah Stern
The_Devil's_Arithmetic_(film)
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
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
Function equal to the product of its values on coprime factors
shown in the article on Dirichlet series. An arithmetical function f is said to be a rational arithmetical function of order ( r , s ) {\displaystyle (r
Multiplicative_function
Long dense subsets of the integers contain arbitrarily large arithmetic progressions
Erdős and Turán conjectured that every set of integers A with positive natural density contains a k-term arithmetic progression for every k. Endre Szemerédi
Szemerédi's_theorem
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