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The Complete class theorems is a class of theorems in decision theory. They establish that all admissible decision rules are equivalent to the Bayesian
Complete_class_theorem
Statistics term
models have a sufficient statistic which is not complete. This is important because the Lehmann–Scheffé theorem cannot be applied to such models. Galili and
Completeness_(statistics)
Fundamental theorem in mathematical logic
Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability
Gödel's_completeness_theorem
Middle quantile of a data set or probability distribution
187–193. Brown, L. D.; Cohen, Arthur; Strawderman, W. E. (1976). "A Complete Class Theorem for Strict Monotone Likelihood Ratio With Applications". Ann. Statist
Median
When a finite set S of relations yields polynomial-time or NP-complete problems
or is NP-complete, as opposed to one of the classes of intermediate complexity that is known to exist (assuming P ≠ NP) by Ladner's theorem. Special cases
Schaefer's_dichotomy_theorem
Limitative results in mathematical logic
and in philosophy of mathematics. The theorems are interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
The proof of Gödel's completeness theorem given by Kurt Gödel in his doctoral dissertation of 1929 (and a shorter version of the proof, published as an
Original proof of Gödel's completeness theorem
Original_proof_of_Gödel's_completeness_theorem
Complexity class
smaller class than polynomial-time reductions. The concept of NP-completeness was introduced in 1971 (see Cook–Levin theorem), though the term NP-complete was
NP-completeness
Boolean satisfiability is NP-complete and therefore that NP-complete problems exist
complexity theory, the Cook–Levin theorem, also known as Cook's theorem, states that the Boolean satisfiability problem is NP-complete. That is, it is in NP, and
Cook–Levin_theorem
Statement in mathematical combinatorics
large complete graph. As the simplest example, consider two colours (say, blue and red). Let r and s be any two positive integers. Ramsey's theorem states
Ramsey's_theorem
Impossible task in computing
to be impossible by Alonzo Church and Alan Turing in 1936. By the completeness theorem of first-order logic, a statement is universally valid if and only
Entscheidungsproblem
Area of mathematical logic
stability spectrum theorem, which implies that every complete theory T in a countable signature falls in one of the following classes: There are no cardinals
Model_theory
Describes the objects of a given type, up to some equivalence
only classifies every class, but provides a distinguished (canonical) element of each class. There exist many classification theorems in mathematics, as
Classification_theorem
Sufficiency theorem for reconstructing signals from samples
reconstructed exactly from those samples. Strictly speaking, the theorem only applies to a class of mathematical functions having a Fourier transform that is
Nyquist–Shannon sampling theorem
Nyquist–Shannon_sampling_theorem
Gives necessary and sufficient conditions for two braids to have equivalent closures
In mathematics the Markov theorem gives necessary and sufficient conditions for two braids to have closures that are equivalent knots or links. The conditions
Markov_theorem
17th-century conjecture proved by Andrew Wiles in 1994
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that there are no positive integers a
Fermat's_Last_Theorem
Quadratic imaginary number fields with unique factorisation
In number theory, the Heegner theorem or Stark-Heegner theorem establishes the complete list of the quadratic imaginary number fields whose rings of integers
Stark–Heegner_theorem
1995 publication in mathematics
Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the modularity theorem were believed to be
Wiles's proof of Fermat's Last Theorem
Wiles's_proof_of_Fermat's_Last_Theorem
Complexity class used to classify decision problems
would exist for solving NP-complete, and by corollary, all NP problems. The complexity class NP is related to the complexity class co-NP, for which the answer
NP_(complexity)
Concept in mathematical logic
\lor \}} is also functionally complete. (Its functional completeness is also proved by the Disjunctive Normal Form Theorem.) But this is still not minimal
Functional_completeness
Topics referred to by the same term
Gödel's completeness theorem, correspondence between semantic truth and syntactic provability in first-order logic Gödel's incompleteness theorems, limits
Completeness
In mathematics, a statement that has been proven
mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The proof of a theorem is a logical argument that uses
Theorem
Characteristic of some logical systems
called complete with respect to a particular property if every formula having the property can be derived using that system, i.e. is one of its theorems; otherwise
Completeness_(logic)
Statistical property
1214/aos/1176344563. Brown, L. D.; Cohen, Arthur; Strawderman, W. E. (1976). "A Complete Class Theorem for Strict Monotone Likelihood Ratio With Applications". Ann. Statist
Bias_of_an_estimator
Metric geometry
are complete are called geodesic manifolds; completeness follows from the Hopf–Rinow theorem. Every compact metric space is complete, though complete spaces
Complete_metric_space
Subfield of automated reasoning and mathematical logic
Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving
Automated_theorem_proving
Due to the concise yet complete description of many dynamical systems, Conley's theorem is also known as the fundamental theorem of dynamical systems.
Conley's fundamental theorem of dynamical systems
Conley's_fundamental_theorem_of_dynamical_systems
Theorem in geometry
various generalizations of the Riemann–Roch theorem; among the most famous is the Grothendieck–Riemann–Roch theorem, which is further generalized by the formulation
Riemann–Roch-type_theorem
Mathematical proposition equivalent to the axiom of choice
the proofs of several theorems of crucial importance, for instance the Hahn–Banach theorem in functional analysis, the theorem that every vector space
Zorn's_lemma
Mathematical construction
include very elegant proofs of the compactness theorem and the completeness theorem, Keisler's ultrapower theorem, which gives an algebraic characterization
Ultraproduct
Listing all imaginary quadratic fields with a given class number
With the proof of the Gross–Zagier theorem in 1986, a complete list of imaginary quadratic fields with a given class number could be specified by a finite
Class_number_problem
Set of sentences in a formal language
first-order logic, the most important case, it follows from the completeness theorem that the two meanings coincide. In other logics, such as second-order
Theory_(mathematical_logic)
Ability of a computing system to simulate Turing machines
items of an existing array). However, another theorem shows that there are problems solvable by Turing-complete languages that cannot be solved by any language
Turing_completeness
Theorem for proving more complex theorems
also known as a "helping theorem" or an "auxiliary theorem". In many cases, a lemma derives its importance from the theorem it aims to prove; however
Lemma_(mathematics)
Graph that can be embedded in the plane
Kuratowski's theorem: A finite graph is planar if and only if it does not contain a subgraph that is a subdivision of the complete graph K5 or the complete bipartite
Planar_graph
Existence and cardinality of models of logical theories
In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf
Löwenheim–Skolem_theorem
Branch of mathematical combinatorics
dimensions. The Hales–Jewett theorem implies Van der Waerden's theorem. A theorem similar to van der Waerden's theorem is Schur's theorem: for any given c there
Ramsey_theory
On graph coloring and neighborhood size
colors, except for two cases, complete graphs and cycle graphs of odd length, which require Δ + 1 colors. The theorem is named after R. Leonard Brooks
Brooks'_theorem
Concept in mathematical logic
be formulated cannot be complete, as demonstrated by Gödel's first incompleteness theorem. This syntactic sense of complete is distinct from the semantic
Complete_theory
Theorem in stochastic calculus
proved such a theorem, which became known as the Doob-Meyer decomposition. In honor of Doob, Meyer used the term "class D" to refer to the class of supermartingales
Doob–Meyer decomposition theorem
Doob–Meyer_decomposition_theorem
Algorithmic complexity class
space complexity classes in the following way: P ⊆ NP ⊆ PSPACE ⊆ EXPTIME ⊆ NEXPTIME ⊆ EXPSPACE. Furthermore, by the time hierarchy theorem and the space
EXPTIME
Result about when a matrix can be diagonalized
operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators
Spectral_theorem
Class of algebraic structures
abelian groups, the rings, the monoids etc. According to Birkhoff's theorem, a class of algebraic structures of the same signature is a variety if and only
Variety_(universal_algebra)
Proof all ranked voting rules have spoilers
Arrow's impossibility theorem is a key result in social choice theory, proved by American economist Kenneth Arrow. It shows that no procedure for group
Arrow's_impossibility_theorem
Unsolved problem in computer science
Kurt Gödel to John von Neumann, Gödel asked whether theorem-proving (now known to be co-NP-complete) could be solved in quadratic or linear time, and posited
P_versus_NP_problem
Non-contradiction of a theory
and complete. Gödel's incompleteness theorems show that any sufficiently strong recursively enumerable theory of arithmetic cannot be both complete and
Consistency
Theorem in computability theory
In computability theory, Rice's theorem states that all non-trivial semantic properties of programs are undecidable. A semantic property is one about
Rice's_theorem
Partially ordered set in which all subsets have both a supremum and infimum
them as a special class of lattices. Complete lattices must not be confused with complete partial orders (CPOs), a more general class of partially ordered
Complete_lattice
Approach to optimizing robustness to failure
function and prior distribution (this is the statement of the complete class theorems), and thus that non-probabilistic methods such as info-gap are
Info-gap_decision_theory
Type of theory in mathematical logic
categoricity theorem (1965) confirms that these are the only possibilities: Morley's categoricity theorem (Morley 1965)—If T is a consistent and complete theory
Categorical_theory
On coloring the edges of graphs
In graph theory, Vizing's theorem states that every simple undirected graph may be edge colored using a number of colors that is at most one larger than
Vizing's_theorem
computational complexity theory, NL-complete is a complexity class containing the languages that are complete for NL, the class of decision problems that can
NL-complete
Group of mathematical theorems
specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship among quotients
Isomorphism_theorems
Undecidability of equality of real numbers
Richardson of the University of Bath. Specifically, the class of expressions for which the theorem holds is that generated by rational numbers, the number
Richardson's_theorem
structure theorem, introduced by Cohen (1946), describes the structure of complete Noetherian local rings. Some consequences of Cohen's structure theorem include
Cohen_structure_theorem
Category of mathematical proof
In mathematics, an impossibility theorem is a theorem that demonstrates a problem or general set of problems cannot be solved. These are also known as
Proof_of_impossibility
theory, Trakhtenbrot's theorem (due to Boris Trakhtenbrot) states that the problem of validity in first-order logic on the class of all finite models is
Trakhtenbrot's_theorem
In board games that cannot end in a draw, one of the two players has a winning strategy
example game of chess in 1913. Zermelo's theorem can be applied to all finite-stage two-player games with complete information and alternating moves. The
Zermelo's theorem (game theory)
Zermelo's_theorem_(game_theory)
Theorem about metric spaces
Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach–Caccioppoli theorem) is an important
Banach_fixed-point_theorem
Classification of symplectic toric manifolds
of mathematics, Delzant's theorem provides a complete classification of symplectic toric manifolds in term of a special class of polytopes. It was proved
Delzant's_theorem
On collapse of the polynomial hierarchy if NP is in non-uniform polynomial time class
(Adleman's theorem), the theorem is also evidence that the use of randomization does not lead to polynomial time algorithms for NP-complete problems. The
Karp–Lipton_theorem
Relationship between derivatives and integrals
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at every
Fundamental theorem of calculus
Fundamental_theorem_of_calculus
Statistical property
1214/aos/1176344563. Brown, L.D.; Cohen, Arthur; Strawderman, W.E. (1976). "A complete class theorem for strict monotone likelihood ratio with applications". Annals
Monotone_likelihood_ratio
Every finite abelian extension of Q is contained within some cyclotomic field
a fact generalised in class field theory. The theorem was first stated by Kronecker (1853) though his argument was not complete for extensions of degree
Kronecker–Weber_theorem
Index of articles associated with the same name
sign with an inequality sign, form a broad class of such auxiliary relations. One instance of such theorem was used by Aronson and Weinberger to characterize
Comparison_theorem
Concept in model theory
that every model realizes p (at least if the theory is complete). The omitting types theorem says that conversely if p is not isolated then there is
Type_(model_theory)
Logic statement about a formal system proven in a metalanguage
is a class consisting of the sets satisfying the formula. Consistency proofs of systems such as Peano arithmetic. Gödel's completeness theorem states
Metatheorem
On finding a maximal set of solutions of a system of first-order homogeneous linear PDEs
In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system
Frobenius theorem (differential topology)
Frobenius_theorem_(differential_topology)
Model theory concept
theory T. In this section T is a countable complete theory and κ is a cardinal. The Löwenheim–Skolem theorem shows that if I(T,κ) is nonzero for one infinite
Spectrum_of_a_theory
System of mathematical set theory
quantifiers range over classes. NBG is finitely axiomatizable, while ZFC and MK are not. A key theorem of NBG is the class existence theorem, which states that
Von Neumann–Bernays–Gödel set theory
Von_Neumann–Bernays–Gödel_set_theory
Every set is smaller than its power set
elegant and remarkably simple. The complete proof is presented below, with detailed explanations to follow. Theorem (Cantor)—Let f {\displaystyle f} be
Cantor's_theorem
Existence of values making formula true
to consistency for first-order logic, a result known as Gödel's completeness theorem. The negation of satisfiability is unsatisfiability, and the negation
Satisfiability
Type of "good" decision rule in Bayesian statistics
possible to define a generalized Bayes rule. According to the complete class theorems, under mild conditions every admissible rule is a (generalized)
Admissible_decision_rule
Generalizes the Kodaira vanishing theorem
of holomorphic (p,0)-forms taking values on F. The theorem states that, if the first Chern class of F is negative, H q ( M ; Ω p ( F ) ) = 0 when q
Nakano_vanishing_theorem
Theorem in computational complexity theory
theory, the PCP theorem (also known as the PCP characterization theorem) states that every decision problem in the NP complexity class has probabilistically
PCP_theorem
Visual depiction of a partially ordered set
& Tamassia (1995a), Theorem 9, p. 118; Baker, Fishburn & Roberts (1971), theorem 4.1, page 18. Garg & Tamassia (1995a), Theorem 15, p. 125; Bertolazzi
Hasse_diagram
Form of logic that allows quantification over predicates
used in the context of Courcelle's theorem, an algorithmic meta-theorem in graph theory. The MSO theory of the complete infinite binary tree (S2S) is decidable
Second-order_logic
On the structure of complete Riemannian manifolds of non-positive sectional curvature
mathematics, the Cartan–Hadamard theorem is a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non-positive
Cartan–Hadamard_theorem
Branch of applied probability theory
can arise from departures from the probability axioms, and the complete class theorems, which show that all admissible decision rules are equivalent to
Decision_theory
Theorem that arithmetical truth cannot be defined in arithmetic
Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1933, is an important limitative result in mathematical logic, the foundations
Tarski's undefinability theorem
Tarski's_undefinability_theorem
Axiom of set theory
orthonormal basis. The Banach–Alaoglu theorem about compactness of sets of functionals. The Baire category theorem about complete metric spaces, and its consequences
Axiom_of_choice
Ideals in a Boolean algebra can be extended to prime ideals
In mathematics, the Boolean prime ideal theorem states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement
Boolean_prime_ideal_theorem
follows from the proof of the Cook–Levin theorem because the reduction used is parsimonious. ⊕P is a counting class, and can be seen as finding the least
Parity_P
Computation modulo a fixed integer
important theorems relating to modular arithmetic: Carmichael's theorem Chinese remainder theorem Euler's theorem Fermat's little theorem (a special
Modular_arithmetic
Theorem in category theory
theorem, Russell's paradox, Gödel's first incompleteness theorem, Turing's solution to the Entscheidungsproblem, and Tarski's undefinability theorem.
Lawvere's_fixed-point_theorem
Theorem in mathematical logic
compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important
Compactness_theorem
Every symmetric convex set in R^n with volume > 2^n contains a non-zero integer point
is in the class PPP, and it was conjectured to be PPP-complete. Danzer set Pick's theorem Dirichlet's unit theorem Minkowski's second theorem Ehrhart's
Minkowski's_theorem
Class of theorems about Nash equilibrium payoff profiles in repeated games
In game theory, folk theorems are a class of theorems describing an abundance of Nash equilibrium payoff profiles in repeated games (Friedman 1971). The
Folk_theorem_(game_theory)
Existence of group elements of prime order
prove the theorem with the use of strong induction and the class equation, though considerably less machinery is required to prove the theorem in the abelian
Cauchy's theorem (group theory)
Cauchy's_theorem_(group_theory)
Problem in computer science
statement of the incompleteness theorem by asserting that an effective axiomatization of the natural numbers that is both complete and sound is impossible. The
Halting_problem
Finiteness of sets of forbidden graph minors
Wagner's theorem characterizes the planar graphs as being the graphs that do not have the complete graph K 5 {\displaystyle K_{5}} or the complete bipartite
Robertson–Seymour_theorem
complexity classes is infinite. The compression theorem is an important theorem about the complexity of computable functions. The theorem states that
Structural_complexity_theory
On forbidden subgraphs in planar graphs
minors; therefore, these two theorems are equivalent. An extension is the Robertson–Seymour theorem, stating that every class of graphs closed under taking
Kuratowski's_theorem
Counterintuitive result in probability
The infinite monkey theorem states that a monkey hitting keys independently and at random on a typewriter keyboard for an infinite amount of time will
Infinite_monkey_theorem
Theorem classifying finite simple groups
enormous theorem) is a result of group theory stating that every finite simple group is either cyclic, or alternating, or belongs to a broad infinite class called
Classification of finite simple groups
Classification_of_finite_simple_groups
Theorem in mathematics
"nonzero Jacobian determinant". If the function of the theorem belongs to a higher differentiability class, the same is true for the inverse function. There
Inverse_function_theorem
Graph linking pairs of comparable elements in a partial order
is Mirsky's theorem, and the perfection of their complements is Dilworth's theorem; these facts, together with the perfect graph theorem can be used to
Comparability_graph
analysis and the sequential probability ratio test and on Wald's complete class theorem characterizing admissible decision rules as limits of Bayesian procedures
List of publications in statistics
List_of_publications_in_statistics
Basic result in harmonic analysis on compact topological groups
In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are
Peter–Weyl_theorem
Statement that is taken to be true
interpretation". Gödel's completeness theorem establishes the completeness of a certain commonly used type of deductive system. Note that "completeness" has a different
Axiom
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