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Consistency of the axioms of arithmetic
In mathematics, Hilbert's second problem was posed by David Hilbert in 1900 as one of his 23 problems. It asks for a proof that arithmetic is consistent
Hilbert's_second_problem
23 mathematical problems stated in 1900
Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several
Hilbert's_problems
On solvability of Diophantine equations
Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge
Hilbert's_tenth_problem
Mathematical problems related to differential equations
In mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential
Riemann–Hilbert_problem
German mathematician (1862–1943)
_{j}}({\vec {x}})q_{j}({\vec {x}})} . This result is known as the Hilbert root theorem, or "Hilberts Nullstellensatz" in German. He also proved that the correspondence
David_Hilbert
On uniformization of analytic relations
Hilbert's twenty-second problem is the penultimate entry in the celebrated list of 23 Hilbert problems compiled in 1900 by David Hilbert. It entails the
Hilbert's twenty-second problem
Hilbert's_twenty-second_problem
On dissections between polyhedra
The third of Hilbert's problems presented in 1900 was the first to be solved. The problem asks the following: Given any two polyhedra of equal volume,
Hilbert's_third_problem
On lattices and sphere packing in Euclidean space
Hilbert's eighteenth problem is one of the 23 problems set out in a celebrated list compiled in 1900 by mathematician David Hilbert. It asks three separate
Hilbert's_eighteenth_problem
On transcendence of certain numbers
Hilbert's seventh problem is one of David Hilbert's list of open mathematical problems posed in 1900. It concerns the irrationality and transcendence of
Hilbert's_seventh_problem
Limitative results in mathematical logic
he turned to a second problem for his habilitation. His original goal was to obtain a positive solution to Hilbert's second problem. At the time, theories
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
On topology of algebraic curves and surfaces
Hilbert's 16th problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900, as part of his list
Hilbert's_sixteenth_problem
Impossible task in computing
Entscheidungsproblem (German for 'decision problem'; pronounced [ɛntˈʃaɪ̯dʊŋspʁoˌbleːm]) is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. It
Entscheidungsproblem
When are solutions in the calculus of variations analytic
Hilbert's nineteenth problem is one of the 23 Hilbert problems, set out in a list compiled by David Hilbert in 1900. It asks whether the solutions of
Hilbert's_nineteenth_problem
On solutions of 7th-degree equations
Hilbert's thirteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It entails proving whether
Hilbert's_thirteenth_problem
2008 British TV series or programme
had formulated the Incompleteness Theorem based on his study of Hilbert's second problem: This statement cannot be proved Using a code based on prime numbers
The_Story_of_Maths
Classify quadratic forms over algebraic number fields
Hilbert's eleventh problem is one of David Hilbert's list of open mathematical problems posed at the Second International Congress of Mathematicians in
Hilbert's_eleventh_problem
Type of vector space in math
plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates
Hilbert_space
Proposition in mathematical logic
problems in set theory, and establishing its truth or falsehood was the first of Hilbert's 23 problems presented in 1900. The answer to this problem is
Continuum_hypothesis
Logical principle
debate had a profound effect on Hilbert. Reid indicates that Hilbert's second problem (one of Hilbert's problems from the Second International Conference in
Law_of_excluded_middle
Thought experiment of infinite sets
Hilbert's paradox of the Grand Hotel (colloquially the Infinite Hotel Paradox or Hilbert's Hotel) is a thought experiment which illustrates a counterintuitive
Hilbert's paradox of the Grand Hotel
Hilbert's_paradox_of_the_Grand_Hotel
Attempt to formalize all of mathematics, based on a finite set of axioms
In mathematics, Hilbert's program, formulated by German mathematician David Hilbert in the early 1920s, was a proposed solution to the foundational crisis
Hilbert's_program
Mathematical formal infinite series
Hahn embedding theorem and then studied by him in relation to Hilbert's second problem. The field of Hahn series K [ [ T Γ ] ] {\displaystyle K\left[\left[T^{\Gamma
Hahn_series
Problem in computer science
halting problem which emerged in the 1950s. 1900 (1900): David Hilbert poses his "23 questions" (now known as Hilbert's problems) at the Second International
Halting_problem
Foundational controversy in twentieth-century mathematics
solvability of every mathematical problem." This Third Insight is referring to Hilbert's second problem and Hilbert's ongoing attempt to axiomatize all
Brouwer–Hilbert_controversy
Study of mathematics itself
white. The Entscheidungsproblem (German for 'decision problem') is a challenge posed by David Hilbert in 1928. The Entscheidungsproblem asks for an algorithm
Metamathematics
System of formal deduction in logic
a Hilbert system, sometimes called Hilbert calculus, Hilbert-style system, Hilbert-style proof system, Hilbert-style deductive system or Hilbert–Ackermann
Hilbert_system
Conjecture on zeros of the zeta function
make up Hilbert's eighth problem in David Hilbert's list of twenty-three unsolved problems; it is also one of the Millennium Prize Problems of the Clay
Riemann_hypothesis
Problem that can be possibly solved via mathematics
planets in the Solar System, or a problem of a more abstract nature, such as Hilbert's problems. It can also be a problem referring to the nature of mathematics
Mathematical_problem
maximal determinant problem: what is the largest determinant of a matrix with entries all equal to 1 or −1? Hilbert's fifteenth problem: put Schubert calculus
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
3-volume treatise on mathematics, 1910–1913
not been established for Principia's axioms of set theory. (See Hilbert's second problem.) Russell and Whitehead suspected that the system in PM is incomplete:
Principia_Mathematica
Theorem in formal logic
to the strong normalization of the Girard/Reynold's System F. Hilbert's second problem Dag Prawitz, 1968. Hauptsatz for higher order logic. Journal of
Takeuti's_conjecture
Partially unsolved problem in mathematics
In July 2023, a second and independent preprint of Neville appeared on arXiv, claiming the solution of the problem for separable Hilbert spaces.[non-primary
Invariant_subspace_problem
Study of geometries as axiomatic systems
period after David Hilbert's famous address on unsolved problems, remarked that his colleagues had already solved Hilbert's second problem. At the University
Foundations_of_geometry
Non-contradiction of a theory
Equiconsistency – Being equally consistent Hilbert's problems – 23 mathematical problems stated in 1900 Hilbert's second problem – Consistency of the axioms of arithmetic
Consistency
Math theorem about sphere packing
1900, David Hilbert included it in his list of twenty three unsolved problems of mathematics—it forms part of Hilbert's eighteenth problem. The next step
Kepler_conjecture
operators on Hilbert spaces. In the theory of partial differential equations, it is very useful in solving elliptic boundary value problems. Let (H, ⟨
Hilbert–Schmidt_theorem
Basic framework of mathematics
axiom of choice is unprovable in ZF even without urelements. 1970: Hilbert's tenth problem is proven unsolvable: there is no recursive solution to decide
Foundations_of_mathematics
Yes-or-no question that cannot ever be solved by a computer
theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm
Undecidable_problem
Basis for Euclidean geometry
Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book Grundlagen der Geometrie (tr. The Foundations of Geometry) as
Hilbert's_axioms
Question about single-shape aperiodic tiling
problem can be seen as a natural extension of the second part of Hilbert's eighteenth problem, which asks for a single polyhedron that tiles Euclidean 3-space
Einstein_problem
Concept in quantum information theory
In quantum information theory, a set of bases in Hilbert space Cd are said to be mutually unbiased if when a system is prepared in an eigenstate of one
Mutually_unbiased_bases
Unsolved problem in computer science
Unsolved problem in computer science If the solution to a problem can be checked in polynomial time, must the problem be solvable in polynomial time? More
P_versus_NP_problem
Millennium Prize Problem
existence and mass gap problem is an unsolved problem in mathematical physics and mathematics, and one of the seven Millennium Prize Problems defined by the Clay
Yang–Mills existence and mass gap
Yang–Mills_existence_and_mass_gap
Russian mathematician and computer scientist (born 1947)
computer scientist. He is best known for his negative solution of Hilbert's tenth problem (Matiyasevich's theorem), which was presented in his 1972 doctoral
Yuri_Matiyasevich
American mathematician (born 1989)
he placed second in the Intel Science Talent Search competition, with a generalization to rectifiable curves of the carpenter's rule problem for polygons
John_Pardon
Polynomial ideals are finitely generated
was stated and proved by David Hilbert in 1890 in his seminal article on invariant theory, where he solved several problems on invariants. In this article
Hilbert's_basis_theorem
Unsolved problem in mathematics
Unsolved problem in mathematics Is every sufficiently large sequence of square numbers with constant second difference necessarily a sequence of consecutive
Büchi's_problem
Paradox in set theory
incompleteness theorems – Limitative results in mathematical logic Hilbert's first problem – Proposition in mathematical logicPages displaying short descriptions
Russell's_paradox
Class of ordinary differential equations
In mathematics and its applications, a Sturm–Liouville problem is a second-order linear ordinary differential equation of the form d d x [ p ( x ) d y
Sturm–Liouville_theory
Complexity class used to classify decision problems
second phase consists of a deterministic algorithm that verifies whether the guess is a solution to the problem. The complexity class P (all problems
NP_(complexity)
Yes/no problem in computer science
decision problem is a computational problem that can be posed as a yes–no question on a set of input values. An example of a decision problem is deciding
Decision_problem
Abstract mathematics problem
paradox (also known as the balls and vase problem or the ping pong ball problem) is a hypothetical problem in abstract mathematics and logic designed
Ross–Littlewood_paradox
Proof by Alan Turing
to the Entscheidungsproblem". It was the second proof (after Church's theorem) of the negation of Hilbert's Entscheidungsproblem; that is, the conjecture
Turing's_proof
Type of measurement in quantum mechanics
discovered with Hilbert's twelfth problem. Unsolved problem in mathematics Do SIC-POVMs exist in all dimensions? More unsolved problems in mathematics
SIC-POVM
Subfield of mathematics
these problems shaped the direction of mathematical logic, as did the effort to resolve Hilbert's Entscheidungsproblem, posed in 1928. This problem asked
Mathematical_logic
Even integers as sums of two primes
Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural
Goldbach's_conjecture
Swedish mathematician and concert pianist
consider Hilbert's fifth problem in the spirit of functional analysis. In two years, 1969–1970, Enflo published five papers on Hilbert's fifth problem; these
Per_Enflo
German mathematician (1895–1941)
including polygons and tessellations. He solved one of the parts of Hilbert's eighteenth problem, and is the namesake of the Reinhardt domains in several complex
Karl Reinhardt (mathematician)
Karl_Reinhardt_(mathematician)
Mathematical problem
In mathematical logic, Tarski's high school algebra problem was a question posed by Alfred Tarski. It asks whether there are identities involving addition
Tarski's high school algebra problem
Tarski's_high_school_algebra_problem
Hungarian mathematician (1913–1996)
mathematical conjectures of the 20th century. Erdős pursued and proposed problems in discrete mathematics, graph theory, number theory, mathematical analysis
Paul_Erdős
Computational problems no algorithm can solve
homeomorphic, or if a 5-manifold is homeomorphic to S5. Hilbert's tenth problem: the problem of deciding whether a Diophantine equation (multivariable
List_of_undecidable_problems
Problem in physics and celestial mechanics
In physics, the n-body problem is the problem of predicting the individual motions of a group of celestial objects interacting with each other gravitationally
N-body_problem
Existence of values making formula true
validity problem was posed firstly by David Hilbert, as the so-called Entscheidungsproblem. The universal validity of a formula is a semi-decidable problem by
Satisfiability
Relation between algebraic varieties and polynomial ideals
proven by David Hilbert in his second major paper on invariant theory in 1893 (following his seminal 1890 paper in which he proved Hilbert's basis theorem)
Hilbert's_Nullstellensatz
Unsolved problem in mathematics
Unsolved problem in mathematics Is every finite group the Galois group of a Galois extension of the rational numbers? More unsolved problems in mathematics
Inverse_Galois_problem
to child with the most famous example being the Bernoulli family. This second generation phenomenon also holds in physics but in that field the Nobel
List of second-generation mathematicians
List_of_second-generation_mathematicians
American mathematician and Nobel Laureate (1928–2015)
equations resolved Hilbert's nineteenth problem on regularity in the calculus of variations, which had been a well-known open problem for almost 60 years
John_Forbes_Nash_Jr.
Sum of inverse squares of natural numbers
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed
Basel_problem
Problems which attempt to find the most efficient way to pack objects into containers
infinite-dimensional Hilbert space with no restrictions. It is worth describing in detail here, to give a flavor of the general problem. In this case, a configuration
Packing_problems
In functional analysis, a Hilbert space
kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Specifically, a Hilbert space
Reproducing kernel Hilbert space
Reproducing_kernel_Hilbert_space
σ-weak topology, is a topology on B(H), the space of bounded operators on a Hilbert space H. B(H) admits a predual B*(H), the trace class operators on H. The
Ultraweak_topology
acquire the special definition above. Dilation (operator theory) P. Halmos, A Hilbert Space Problem Book, Second Edition, Springer-Verlag, 1982. v t e
Compression (functional analysis)
Compression_(functional_analysis)
Book by Wilhelm Ackermann
Mathematical Logic is the 1950 American translation of the 1938 second edition of David Hilbert's and Wilhelm Ackermann's classic text Grundzüge der theoretischen
Principles of Mathematical Logic
Principles_of_Mathematical_Logic
Study of optimal transportation and allocation of resources
to the study of optimal transportation and allocation of resources. The problem was formalized by the French mathematician Gaspard Monge in 1781. In the
Transportation theory (mathematics)
Transportation_theory_(mathematics)
unsolved problems in mathematics, these include: The sixth problem of 1900 Hilbert's problems about the axiomatization of physics, the problem is either
List of unsolved problems in physics
List_of_unsolved_problems_in_physics
Disproved conjecture
in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits". International
Kalman's_conjecture
Undecidability of equality of real numbers
elementary functions if and only if a = 0 {\displaystyle a=0} ). After Hilbert's tenth problem was solved in 1970, B. F. Caviness observed that the use of e x
Richardson's_theorem
paradox Cantor's theorem – paradox – diagonal argument Compactness Halting problem Lindström's Löwenheim–Skolem Russell's paradox Logics Traditional Classical
Abstract_model_theory
R-tree variant and index for multidimensional objects
utilization is ≈100%; this structure is called a packed Hilbert R-tree. The second index, called a Dynamic Hilbert R-tree, supports insertions and deletions, and
Hilbert_R-tree
Theory of rapid universe expansion
horizon problem are also solved by inflation theory. The flatness problem (also known as the oldness problem) is a cosmological fine-tuning problem within
Cosmic_inflation
Mathematical logic concept
result on Hilbert's plan to prove the consistency of mathematics. It is likely that all mathematicians ultimately would have accepted Hilbert's approach
Gentzen's_consistency_proof
On polynomial rings over fields
mathematics, Hilbert's syzygy theorem is one of the three fundamental theorems about polynomial rings over fields, first proved by David Hilbert in 1890,
Hilbert's_syzygy_theorem
Mathematical-logic system
computable function can decide the question. This was historically the first problem for which undecidability could be proven. As usual for such a proof, computable
Lambda_calculus
American mathematician and educator (1921–2008)
widely varied areas of mathematics, including the solution of Hilbert's fifth problem, and was a leader in reform and innovation in mathematics teaching
Andrew_M._Gleason
Form of logic that allows quantification over predicates
and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in
Second-order_logic
Numerical method for solving physical or engineering problems
differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis
Finite_element_method
Study of computable functions and Turing degrees
Matiyasevich's theorem, which implies that Hilbert's tenth problem has no effective solution; this problem asked whether there is an effective procedure
Computability_theory
Problem of solving a partial differential equation subject to prescribed boundary values
In mathematics, a Dirichlet problem asks for a function which solves a specified partial differential equation (PDE) in the interior of a given region
Dirichlet_problem
Sequence of operations for a task
concept of algorithms began with attempts to solve David Hilbert's Entscheidungsproblem (decision problem). Later formalizations were framed as attempts to define
Algorithm
Type of infinite structure
dynamical systems and algorithms for tame optimization, and multi-objective problems (PhD thesis). Université Montpellier; Universidad técnica Federico Santa
O-minimal_theory
Omission of operations and relations of a structure
paradox Cantor's theorem – paradox – diagonal argument Compactness Halting problem Lindström's Löwenheim–Skolem Russell's paradox Logics Traditional Classical
Reduct
Axiom of set theory proposed by Peter Aczel in 1988
paradox Cantor's theorem – paradox – diagonal argument Compactness Halting problem Lindström's Löwenheim–Skolem Russell's paradox Logics Traditional Classical
Aczel's_anti-foundation_axiom
Differential calculus on function spaces
foundation. The 20th and the 23rd Hilbert problem published in 1900 encouraged further development. In the 20th century David Hilbert, Oskar Bolza, Gilbert Ames
Calculus_of_variations
Partial differential equations with data on two intersecting characteristics
The Goursat problem (also called the Darboux problem) is a boundary value problem for a second-order hyperbolic partial differential equation (PDE) in
Goursat_problem
Category of mathematical proof
an impossibility theorem is a theorem that demonstrates a problem or general set of problems cannot be solved. These are also known as proofs of impossibility
Proof_of_impossibility
This article contains a list of sample Hilbert-style deductive systems for propositional logics. Classical propositional calculus is the standard propositional
List of axiomatic systems in logic
List_of_axiomatic_systems_in_logic
paradox Cantor's theorem – paradox – diagonal argument Compactness Halting problem Lindström's Löwenheim–Skolem Russell's paradox Logics Traditional Classical
Atomic model (mathematical logic)
Atomic_model_(mathematical_logic)
Statistical learning theory
regularized empirical risk functional defined over a reproducing kernel Hilbert space can be represented as a finite linear combination of kernel products
Representer_theorem
Statement that is taken to be true
vectors ('states') in a separable Hilbert space, and physical quantities as linear operators that act in this Hilbert space. This approach is fully falsifiable
Axiom
Variable that stores data about other variables or program structure
paradox Cantor's theorem – paradox – diagonal argument Compactness Halting problem Lindström's Löwenheim–Skolem Russell's paradox Logics Traditional Classical
Metavariable
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