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Field of mathematical analysis
mathematical analysis, constructive function theory is a field which studies the connection between the smoothness of a function and its degree of approximation
Constructive_function_theory
Topics referred to by the same term
investigates functions of complex numbers Constructive function theory, the study of the connection between the smoothness of a function and its degree
Function_theory
Soviet mathematician
Soviet mathematician known for contributions to real analysis and constructive function theory, in particular, for his textbooks on these subjects. His son
Isidor_Natanson
Axiomatic set theories based on the principles of mathematical constructivism
Axiomatic constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. The same first-order language
Constructive_set_theory
Philosphical view that existence proofs must be constructive
Constructivism also includes the study of constructive set theories such as CZF and the study of topos theory. Constructivism is often identified with
Constructivism (philosophy of mathematics)
Constructivism_(philosophy_of_mathematics)
Alternative foundation of mathematics
Intuitionistic type theory (also known as constructive type theory, or Martin-Löf type theory (MLTT)) is a type theory and an alternative foundation of
Intuitionistic_type_theory
Base of natural logarithms
M. (1983). "The Computation of Logarithms by Huygens" (PDF). Constructive Function Theory: 254–257. The reference is to a problem which Jacob Bernoulli
E_(mathematical_constant)
Soviet mathematician
The Constructive Theory of Functions (1905–1930), translated: Atomic Energy Commission, Springfield, Va, 1958 vol. 2, The Constructive Theory of Functions
Sergei_Bernstein
Mathematical theory of data types
Intuitionistic Type Theory. The logical framework of a type theory bears a resemblance to intuitionistic, or constructive, logic. Formally, type theory is often
Type_theory
Analytic function in mathematics
elsewhere. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics
Riemann_zeta_function
Type theory in logic and mathematics
theoretic aspects of constructive type theory" in 2008. At about the same time, Vladimir Voevodsky was independently investigating type theory in the context
Homotopy_type_theory
Function that preserves distinctness
In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements of its domain to distinct
Injective_function
Standard system of axiomatic set theory
set theories: Morse–Kelley set theory Von Neumann–Bernays–Gödel set theory Tarski–Grothendieck set theory Constructive set theory Internal set theory At
Zermelo–Fraenkel_set_theory
Quickly growing function
theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function
Ackermann_function
Symbol representing a mathematical concept
systems particularly mathematical logic, a function symbol is a non-logical symbol which represents a function or mapping on the domain of discourse, though
Function_symbol
Set of all things that may be the input of a mathematical function
domain of the unknown function(s) sought. For example, it is sometimes convenient in set theory to permit the domain of a function to be a proper class
Domain_of_a_function
Function computable with bounded loops
In computability theory, a primitive recursive function is, roughly speaking, a function that can be computed by a computer program whose loops are all
Primitive_recursive_function
Soviet mathematician
was a Soviet mathematician. He is known for his work in the constructive function theory, in particular, for the Remez algorithm and the Remez inequality
Evgeny_Yakovlevich_Remez
Branch of mathematical logic
relative to constructive theories, (2) combinatorial independence results, and (3) classifications of provably total recursive functions and provably
Proof_theory
Logical system in mathematics
foundations (also known as homotopy type theory). In cubical type theory, function extensionality and univalence are not postulated as axioms, but rather
Cubical_type_theory
3-volume treatise on mathematics, 1910–1913
that a satisfactory solution is yet obtainable. Dr Leon Chwistek [Theory of Constructive Types] took the heroic course of dispensing with the axiom without
Principia_Mathematica
Mathematical analysis
{\mathbb {N} }^{\mathbb {N} }} , constructive second-order arithmetic, or strong enough topos-, type- or constructive set theories such as C Z F {\displaystyle
Constructive_analysis
Type of polynomial used in Numerical Analysis
Encyclopedia of Mathematics, EMS Press Natanson, I.P. (1964). Constructive function theory. Volume I: Uniform approximation. Translated by Alexis N. Obolensky
Bernstein_polynomial
from classical analysis. Constructive function theory a branch of analysis that is closely related to approximation theory, studying the connection between
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Mathematical function such that every output has at least one input
surjective function (also known as surjection, or onto function /ˈɒn.tuː/) is a function f such that, for every element y of the function's codomain, there
Surjective_function
In approximation theory, a converse to Jackson's theorem
where the function φ(x) has a bounded r th derivative which is α-Hölder continuous. Bernstein's lethargy theorem Constructive function theory Achieser
Bernstein's theorem (approximation theory)
Bernstein's_theorem_(approximation_theory)
Area of mathematical logic
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing
Model_theory
expressions, shapes, functions, and sets. Mathematical objects can be very complex; for example, theorems, proofs, and even formal theories are considered as
Mathematical_object
Branch of mathematics that studies sets
that it does reflect an iterative conception of set. Systems of constructive set theory, such as CST, CZF, and IZF, embed their set axioms in intuitionistic
Set_theory
Mathematical function that can be computed by a program
Computable functions are the basic objects of study in computability theory. Informally, a function is computable if there is an algorithm that computes
Computable_function
Axiom of set theory
varieties of constructive mathematics avoid the axiom of choice, others embrace it. A choice function (also called selector or selection) is a function f, defined
Axiom_of_choice
Mathematical inequality
No. 7. ISSN 1443-5756. Zbl 1060.30003. Natanson, I.P. (1964). Constructive function theory. Volume I: Uniform approximation. Translated by Alexis N. Obolensky
Bernstein's theorem (polynomials)
Bernstein's_theorem_(polynomials)
Study of computable functions and Turing degrees
computable function. The c.e. sets, although not decidable in general, have been studied in detail in computability theory. Beginning with the theory of computable
Computability_theory
Interpretation of quantum mechanics
in de Broglie–Bohm theory is not a postulate. Rather, in this theory, the link between the probability density and the wave function has the status of
De_Broglie–Bohm_theory
Mathematical models of strategic interactions
game theory, which has developed novel representations, e.g. surreal numbers, as well as combinatorial and algebraic (and sometimes non-constructive) proof
Game_theory
Theories in mathematical logic
structure there is a signature σ listing the constants, functions, and relations of the theory together with their arities, so that the object is naturally
List_of_first-order_theories
Axiom
In constructive mathematics, Church's thesis C T {\displaystyle {\mathrm {CT} }} is the principle stating that all total functions are computable functions
Church's thesis (constructive mathematics)
Church's_thesis_(constructive_mathematics)
Proof in set theory
S\leq S} , also in constructive set theory. It is however harder or impossible to order ordinals and also cardinals, constructively. For example, the Schröder–Bernstein
Cantor's_diagonal_argument
Academic journal
function theory, functional analysis, interpolation spaces and interpolation of operators, numerical analysis, space of functions, special functions,
Constructive_Approximation
Mathematical construction of a set with an equivalence relation
or the equality on the quotient set). In proof theory, particularly the proof theory of constructive mathematics based on the Curry–Howard correspondence
Setoid
Natanson (1906–1964), real analysis and constructive function theory Melvyn Nathanson (born 1944), number theory Caryn Navy (born 1953), set-theoretic topology
List_of_Jewish_mathematicians
Millennium Prize Problem
physics, in particular constructive quantum field theory, and The mass of all particles of the force field predicted by the theory are strictly positive
Yang–Mills existence and mass gap
Yang–Mills_existence_and_mass_gap
Infinite cardinal number
In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. They
Aleph_number
Function, homomorphism, or morphism
analysis and representations in group theory. In the theory of dynamical systems, a map denotes an evolution function used to create discrete dynamical systems
Map_(mathematics)
Theorem in set theory
In set theory, the Schröder–Bernstein theorem states that, if there exist injective functions f : A → B and g : B → A between the sets A and B, then there
Schröder–Bernstein_theorem
The vast majority of positive results about computational problems are constructive proofs, i.e., a computational problem is proved to be solvable by showing
Non-constructive algorithm existence proofs
Non-constructive_algorithm_existence_proofs
Theoretical framework in physics
Perspective on Constructive Quantum Field Theory". arXiv:1203.3991v2 [math-ph]. Simon, Barry (1974). The P(phi)_2 Euclidean (quantum) field theory. Princeton
Quantum_field_theory
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)
Target set of a mathematical function
not part of a function f if f is defined as just a graph. For example, in set theory it is desirable to permit the domain of a function to be a proper
Codomain
Basic framework of mathematics
(link) Martin-Löf, Per (1998). An intuitionistic theory of types, Twenty-five years of constructive type theory (Venice,1995). Oxford Logic Guides. Vol. 36
Foundations_of_mathematics
Thesis on the nature of computability
In computability theory, the Church–Turing thesis is a thesis about the nature of computable functions. It states that a function on the natural numbers
Church–Turing_thesis
Theory of evolution by changes at the molecular level
groundworks for the theory of constructive neutral evolution (CNE) was laid by two papers in the 1990s. Constructive neutral evolution is a theory which suggests
Neutral theory of molecular evolution
Neutral_theory_of_molecular_evolution
Paradox in set theory
the problem in terms of both logic and set theory, and in particular in terms of Frege's definition of function: There is just one point where I have encountered
Russell's_paradox
Mathematical-logic system
languages Explicit substitution – The theory of substitution, as used in β-reduction Harrop formula – A kind of constructive logical formula such that proofs
Lambda_calculus
Branch of mathematical logic
previous work in constructive analysis and proof theory. The use of second-order arithmetic also allows many techniques from recursion theory to be employed;
Reverse_mathematics
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 of excluded
Kripke–Platek_set_theory
Generalization of the real numbers
they form an ordered field. If formulated in von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field in the sense that all
Surreal_number
Function returning one of only two values
Boolean functions are the subject of Boolean algebra and switching theory. A Boolean function takes the form f : { 0 , 1 } k → { 0 , 1 } {\displaystyle f:\{0
Boolean_function
First article on transfinite set theory
numbers. Both constructive and non-constructive proofs have been presented as "Cantor's proof." The popularity of presenting a non-constructive proof has
Cantor's first set theory article
Cantor's_first_set_theory_article
therefore this constant function is a realizer. If instead the realizability interpretation is used in a constructive meta-theory, then it is not justified
Markov's_principle
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
intuitionistic (i.e. constructive logic) theory, its content being clarified by the existence of a free topos. That is a set theory, in a broad sense, but
History_of_topos_theory
Axiom of set theory
theories that have weak comprehension and the capability to encode functions. This is the case, for example, in some weak constructive set theories or
Axiom_of_non-choice
Value indicating the relation of a proposition to truth
form a Boolean algebra, in intuitionistic logic, and more generally, constructive mathematics, the truth values form a Heyting algebra. Such truth values
Truth_value
discussed below are provably independent of ZFC (the canonical axiomatic set theory of contemporary mathematics, consisting of the Zermelo–Fraenkel axioms plus
List of statements independent of ZFC
List_of_statements_independent_of_ZFC
Size of a possibly infinite set
exponential function is non-decreasing. The notion of cardinality, as now understood, was formulated by Georg Cantor, the originator of set theory, in 1874–1884
Cardinal_number
model — application: replacing a function that is hard to evaluate by a simpler function Constructive function theory — field that studies connection between
List of numerical analysis topics
List_of_numerical_analysis_topics
Process of repeating items in a self-similar way
step). In set theory, this is a theorem guaranteeing that recursively defined functions exist. Given a set X, an element a of X and a function f: X → X, the
Recursion
Mapping of mathematical formulas to a particular meaning
algebra than it is in model theory. For example, the class of groups, in the signature consisting of the binary function symbol × and the constant symbol
Structure (mathematical logic)
Structure_(mathematical_logic)
Basic notion of sameness in mathematics
groupoid interpretation of type theory". In Sambin, Giovanni; Smith, Jan M. (eds.). Twenty Five Years of Constructive Type Theory. Oxford Logic Guides. Vol
Equality_(mathematics)
Mathematical set containing no elements
zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced
Empty_set
Type of infinite structure
for the exponential function by Wilkie's theorem. More generally, the complete theory of the real numbers with Pfaffian functions added. The last two
O-minimal_theory
Tarski's exponential function problem concerns the extension of this theory to another primitive operation, the exponential function. It is an open problem
Decidability of first-order theories of the real numbers
Decidability_of_first-order_theories_of_the_real_numbers
Concerned with the notion of stability in model theory
classification theory were theorems on the possible spectrum functions of a theory, counting the number of models of cardinality κ as a function of κ. Shelah's
Stable_theory
Subset of a function's codomain
the range of a function may refer to either of two closely related concepts: the codomain of the function, or the image of the function. In some cases
Range_of_a_function
Subfield of mathematics
mathematical logic into four areas: set theory model theory recursion theory, and proof theory and constructive mathematics (considered as parts of a single
Mathematical_logic
Mathematical term
classical mathematics, microcontinuity (or S-continuity) of an internal function f at a point a is defined as follows: for all x infinitely close to a,
Microcontinuity
Problem in computer science
any arbitrary computable function f cannot be the halting function h. In computability theory and computational complexity theory, a reduction is an algorithm
Halting_problem
System of mathematical set theory
John von Neumann introduced classes into set theory in 1925. The primitive notions of his theory were function and argument. Using these notions, he defined
Von Neumann–Bernays–Gödel set theory
Von_Neumann–Bernays–Gödel_set_theory
Size of a set in mathematics
for some ordinal α. This allows one to use a constructive definition of the cardinality function, by assigning each set to its equinumerous aleph
Cardinality
System of mathematical set theory
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 Quine
Morse–Kelley_set_theory
Branch of mathematics
also rejected in constructive mathematics, a branch of mathematics that insists that proofs of the existence of a number, function, or other mathematical
Calculus
Measure of algorithmic complexity
Inductive reasoning Kolmogorov structure function Levenshtein distance Manifold hypothesis Solomonoff's theory of inductive inference Sample entropy Rayo's
Kolmogorov_complexity
Large countable ordinal
several mathematical theories, such as Kripke–Platek set theory (with the axiom of infinity) and the system CZF of constructive set theory. It was introduced
Bachmann–Howard_ordinal
concepts in set theory is a major facet of the formalization of mathematics. Here "set theory" is axiomatic set theory, not the naive set theory commonly used
Implementation of mathematics in set theory
Implementation_of_mathematics_in_set_theory
Mathematical use of "there exists"
existential statement about "some" object may be achieved either by a constructive proof, which exhibits an object satisfying the "some" statement, or by
Existential_quantification
Axiomatization of quantum field theory
scattering theory by Rudolf Haag and David Ruelle affirmed their significance. The axioms exist in the context of constructive quantum field theory and are
Wightman_axioms
Axioms for the natural numbers
multiplication are often added as axioms. The respective functions and relations are constructed in set theory or second-order logic, and can be shown to be unique
Peano_axioms
Function used in computer cryptography
computational complexity theory, specifically the theory of polynomial time problems. This has nothing to do with whether the function is one-to-one; finding
One-way_function
Input to a mathematical function
of a function is a value provided to obtain the function's result. It is also called an independent variable. For example, the binary function f ( x
Argument_of_a_function
Theorem in mathematical logic
assumed. The proof below is therefore given using the means of a constructive set theory. It is evident from the proof how the theorem relies on the axiom
Diaconescu's_theorem
Non-contradiction of a theory
In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T {\displaystyle T} is consistent if there is no
Consistency
French mathematician (1789–1857)
on mathematics (on the related topics of symmetric functions, the symmetric group and the theory of higher-order algebraic equations). He attempted admission
Augustin-Louis_Cauchy
Yes-or-no question that cannot ever be solved by a computer
In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct
Undecidable_problem
Mathematical concept
mathematics, every injective function f with a nonempty domain necessarily has a left inverse; however, this may fail in constructive mathematics, where the
Inverse_function
Number of arguments required by a function
science, arity (/ˈærɪti/ ) is the number of arguments or operands taken by a function, operation or relation. In mathematics, arity may also be called rank,
Arity
Mathematical notion of infinitesimal difference
rigorous footing, such as infinitesimal differences and the derivatives of functions. The term is used in various branches of mathematics such as calculus
Differential_(mathematics)
Inequality on approximations of a function by algebraic or trigonometric polynomials
given by Bernstein's theorem. See also constructive function theory. Achiezer (Akhiezer), N.I. (2013) [1956]. Theory of approximation. Translated by Hyman
Jackson's_inequality
Existence of values making formula true
to an arbitrary theory or set of formulas: a theory is satisfiable if at least one interpretation makes every formula in the theory true, and valid if
Satisfiability
Psychological theory
In Gestalt psychology and social psychology, field theory is a theory that examines patterns of interaction between the individual and the total field
Field_theory_(psychology)
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