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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
German mathematician (1862–1943)
Hilbert ring Hilbert–Poincaré series Hilbert series and Hilbert polynomial Hilbert space Hilbert spectrum Hilbert system Hilbert transform Hilbert's arithmetic
David_Hilbert
Type of vector space in math
The mathematical concept of a Hilbert space generalizes the notion of Euclidean space. It extends the methods of Euclidean geometry and calculus from
Hilbert_space
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
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
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
Space-filling curve
The Hilbert curve (also known as the Hilbert space-filling curve) is a continuous fractal space-filling curve first described by the German mathematician
Hilbert_curve
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
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
Method of deriving conclusions
logical systems. Natural deduction systems employ many intuitive rules of inference to reflect how people naturally reason, while Hilbert systems provide
Rule_of_inference
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
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
Mathematical model for deduction or proof systems
rules of inference, theorems from axioms. In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in mathematics. However,
Formal_system
Einstein–Hilbert equations Hilbert algebra Hilbert C*-module Hilbert basis (linear programming) Hilbert class field Hilbert cube Hilbert curve Hilbert curve
List of things named after David Hilbert
List_of_things_named_after_David_Hilbert
Axiom set used in first-order logic
axiomizations of Euclidean geometry are Hilbert's axioms (1899) and Birkhoff's axioms (1932). Using his axiom system, Tarski was able to show that the first-order
Tarski's_axioms
Axiomatization of probability and physics
Hilbert's sixth problem is to axiomatize those branches of physics in which mathematics is prevalent. It occurs on the widely cited list of Hilbert's
Hilbert's_sixth_problem
Line-by-line system for natural deduction proofs
modularization and meta-theoretical analysis, such as cut-elimination. Hilbert system proofs rely on axioms and only a few inference rules, making them concise
Fitch_notation
Mathematical term; concerning axioms used to derive theorems
axiomatic systems were developed in the nineteenth century. They included non-Euclidean geometry, Georg Cantor's abstract set theory, and Hilbert's revisionist
Axiomatic_system
Square matrix where a[i,j]=1/(i+j-1)
In linear algebra, a Hilbert matrix, introduced by Hilbert (1894), is a square matrix with entries being the unit fractions H i j = 1 i + j − 1 . {\displaystyle
Hilbert_matrix
Propositional proof system
rules. Frege systems (more often known as Hilbert systems in general proof theory) are named after Gottlob Frege. The name "Frege system" was first defined
Frege_system
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
Style of formal logical argumentation
deduction and sequent calculus systems are particular distinct kinds of Gentzen-style systems. Hilbert-style systems typically have a very small number
Sequent_calculus
Native American tribal elder (1918–2008)
Vi Hilbert (Lushootseed: taqʷšəblu; née Anderson; July 24, 1918 – December 19, 2008) was an Upper Skagit elder and conservationist of her traditional
Vi_Hilbert
Foundational controversy in twentieth-century mathematics
The Brouwer–Hilbert controversy (German: Grundlagenstreit, lit. 'foundational debate') was a debate in twentieth-century mathematics over fundamental
Brouwer–Hilbert_controversy
Kind of proof calculus
related to the "natural" way of reasoning. This contrasts with Hilbert-style systems, which instead use axioms as much as possible to express the logical
Natural_deduction
Formal language used to prove statements
in widespread use: The class of Hilbert systems, of which the most famous example is the 1928 Hilbert–Ackermann system of first-order logic; Gerhard Gentzen's
Proof_calculus
Stream in the American state of Missouri
the area had the surname Hilbert. List of rivers of Missouri U.S. Geological Survey Geographic Names Information System: Hilbert Branch "Lewis County Place
Hilbert_Branch
Debate about credit for general relativity
mathematics. When he met Einstein in the summer of 1915, Hilbert had started working on an axiomatic system for a unified field theory, combining the ideas of
General relativity priority dispute
General_relativity_priority_dispute
Moduli scheme of subschemes of a scheme, represents the flat-family-of-subschemes functor
variety. The Hilbert scheme is a disjoint union of projective subschemes corresponding to Hilbert polynomials. The basic theory of Hilbert schemes was
Hilbert_scheme
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
Statement in plane geometry
the axioms. Hilbert uses Pasch's axiom in his axiomatic treatment of Euclidean geometry. Given the remaining axioms in Hilbert's system, it can be shown
Pasch's_axiom
Integral transform and linear operator
In mathematics and signal processing, the Hilbert transform is a specific singular integral that takes a function, u(t) of a real variable and produces
Hilbert_transform
Limitative results in mathematical logic
philosophy of mathematics. The theorems are interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Algebraic manipulation of "true" and "false"
tautology is a theorem. Propositional calculus is commonly organized as a Hilbert system, whose operations are just those of Boolean algebra and whose theorems
Boolean_algebra
Theorem in the mathematical formulation of quantum mechanics
are represented on the Hilbert space of states. The physical states in a quantum theory are represented by unit vectors in Hilbert space up to a phase factor
Wigner's_theorem
Topics referred to by the same term
High-performance computing HPC Challenge Benchmark Hilbert system, a formal proof system in logic Hilbert–Pólya conjecture, a conjecture in number theory
HPC
Mathematical entity to describe the probability of each possible measurement on a system
a vector in a Hilbert space. Mixed states are statistical mixtures of pure states and cannot be represented as vectors on that Hilbert space, and instead
Quantum_state
Symbolic logic system
the implication calculus, which do not involve negations, the page on Hilbert system presents it through propositional forms of the axioms of law of identity
Minimal_logic
Construct all metric spaces where lines resemble those on a sphere
In mathematics, Hilbert's fourth problem in the 1900 list of Hilbert's problems is a foundational question in geometry. In one statement derived from the
Hilbert's_fourth_problem
Concept in general relativity
The Einstein–Hilbert action in general relativity yields the Einstein field equations through the principle of stationary action. With the ( − , + , +
Einstein–Hilbert_action
Branch of mathematical logic
theory is often seen as being established by David Hilbert, who initiated what is called Hilbert's program in the Foundations of Mathematics. The central
Proof_theory
these is due to Hilbert who created a system in the same style as Euclid. Unfortunately, Hilbert's system requires 21 axioms. Other systems have used fewer
Point–line–plane_postulate
Generalized Euclidean space in mathematics
quantum mechanics, the projective Hilbert space or ray space P ( H ) {\displaystyle \mathbf {P} (H)} of a complex Hilbert space H {\displaystyle H} is the
Projective_Hilbert_space
Tensor product space endowed with a special inner product
product of Hilbert spaces is a way to extend the tensor product construction so that the result of taking a tensor product of two Hilbert spaces is another
Tensor product of Hilbert spaces
Tensor_product_of_Hilbert_spaces
Basic framework of mathematics
and reduced to a purely formal system as envisaged in Hilbert's program. This dealt a final blow to the heart of Hilbert's program, the hope that consistency
Foundations_of_mathematics
Formal language and associated computer program
interactively on the website, in a user-friendly way. Most databases use a Hilbert system of formal deduction though this is not a requirement. The Metamath Proof
Metamath
Subfield of mathematics
arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of
Mathematical_logic
On surjectivity of linear map to anti-dual
analysis and Hilbert space theory, the fundamental theorem of Hilbert spaces gives a necessary and sufficient condition for a Hausdorff pre-Hilbert space to
Fundamental theorem of Hilbert spaces
Fundamental_theorem_of_Hilbert_spaces
Mathematical description of quantum state
and multiplied by complex numbers to form new wave functions and form a Hilbert space. The inner product of two wave functions is a measure of the overlap
Wave_function
Tool in mathematical dimension theory
In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a
Hilbert series and Hilbert polynomial
Hilbert_series_and_Hilbert_polynomial
German politician
Systems, Member of the Board of Trustees Dresden. "Oberbürgermeister Dirk Hilbert". www.dresden.de (in German). Retrieved 2019-12-10. "Dirk Hilbert kandidiert
Dirk_Hilbert
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
Conjecture in topology
In mathematics, the Hilbert–Smith conjecture is concerned with the transformation groups of manifolds; and in particular with the limitations on topological
Hilbert–Smith_conjecture
Description of a quantum-mechanical system
The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of square-integrable
Schrödinger_equation
Distance function
In mathematics, the Hilbert metric, also known as the Hilbert projective metric, is an explicitly defined distance function on a bounded convex subset
Hilbert_metric
Two geometries based on axioms closely related to those specifying Euclidean geometry
several unstated assumptions that should also have been taken as axioms. Hilbert's system consisting of 20 axioms most closely follows the approach of Euclid
Non-Euclidean_geometry
Various systems of symbolic logic
(\chi \to \psi ){\big )}} . A formal proof of the latter using the Hilbert system is given on that page. With ⊥ {\displaystyle \bot } for ψ {\displaystyle
Intuitionistic_logic
American politician (born 1994)
Kyle Hilbert (born March 23, 1994) is a Republican member of the Oklahoma House of Representatives and the current Speaker of the Oklahoma House of Representatives
Kyle_Hilbert
Systematic method for producing proofs
popular are natural deduction, sequent calculi (i.e., Gentzen-type systems), Hilbert systems, and semantic tableaux or trees. A given proof procedure will
Proof_procedure
Space filling fractal curve
of the Hilbert curve. Precisely, it is the loop version of the Hilbert curve, and it may be thought as the union of four copies of the Hilbert curves
Moore_curve
Theory of logic to account for observations from quantum theory
interacting quantum systems. In an example due to Foulis and Randall, there are orthomodular propositions with finite-dimensional Hilbert models whose pairing
Quantum_logic
Programming language designed 1942 to 1945
for a formal system—as in Hilbert-Kalkül, the original name for the Hilbert-style deduction system—so Plankalkül refers to a formal system for planning
Plankalkül
Branch of logic
{\displaystyle \varphi } using the rules of the formal system. A Hilbert-style axiomatic system, or Hilbert system, is a set of axioms or assumptions from which
Propositional_logic
Concept in mathematics
The Riemann–Hilbert correspondence is a correspondence between abstract algebra (specifically group theory) and mathematical analysis (specifically differential
Riemann–Hilbert correspondence
Riemann–Hilbert_correspondence
Concept in quantum information theory
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 of the
Mutually_unbiased_bases
Theorem in quantum mechanics
quantum mechanics, each physical system is associated with a Hilbert space. For the purposes of this overview, the Hilbert space is assumed to be finite-dimensional
Gleason's_theorem
Logical principle
elderly mathematician], Hilbert's proof of the finiteness of the basis of the invariant system was simply not mathematics. Hilbert, on the other hand, throughout
Law_of_excluded_middle
Formulation of quantum mechanics on a Hilbert Space
{H} } is a fixed complex Hilbert space of countably infinite dimension (as a hilbert-basis). The observables of a quantum system are defined to be the (possibly
Dirac–von_Neumann_axioms
Mathematical model of the time dependence of a point in space
(e.g. the state of an atom can be described by a set of functions in an hilbert space and a set of probabilities for these), or a manifold (e.g. the state
Dynamical_system
On linear differential equations with certain properties
The twenty-first problem of the 23 Hilbert problems, from the celebrated list put forth in 1900 by David Hilbert, concerns the existence of a certain
Hilbert's twenty-first problem
Hilbert's_twenty-first_problem
View that mathematics does not necessarily represent reality, but is more akin to a game
no contradictions can be derived from the system). The way that Hilbert tried to show that an axiomatic system was consistent was by formalizing it using
Formalism (philosophy of mathematics)
Formalism_(philosophy_of_mathematics)
Any entity that can be measured
seemingly unintuitive properties. Specifically, if a system is in a state described by a vector in a Hilbert space, the measurement process affects the state
Observable
Logical formalism using combinators instead of variables
signatures. Specifically, a typed combinatory logic corresponds to a Hilbert system in proof theory. The K and S combinators correspond to the axioms AK:
Combinatory_logic
Mathematical problem concerning limit cycles in dynamical systems
problems in mathematics In mathematics, particularly in dynamical systems, the Hilbert–Arnold problem is an unsolved problem concerning the estimation of
Hilbert–Arnold_problem
Social Scientist
examines the role of digital technologies, information systems, and data in society. Hilbert academic work includes studies on the measurement and analysis
Martin_Hilbert
Unincorporated community in West Virginia, United States
Hilbert is an unincorporated community in Wirt County, West Virginia, United States. U.S. Geological Survey Geographic Names Information System: Hilbert
Hilbert,_West_Virginia
formalism as a foundation of mathematics (see Hilbert's program). He believed that mathematics is a system of formal rules and that its truth lies in the
Mathematical_object
Computer-based technologies
such as SOAP, describing "data-in-transit rather than... data-at-rest". Hilbert and Lopez identify the exponential pace of technological change (a kind
Information_technology
Relation between algebraic varieties and polynomial ideals
In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros" or, more literally, "zero-locus-theorem") is a theorem that establishes a fundamental
Hilbert's_Nullstellensatz
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
Expression of polynomials as sum of squares
Hilbert's seventeenth problem is one of the 23 of Hilbert's problems set out in a celebrated list compiled in 1900 by David Hilbert. It concerns the expression
Hilbert's_seventeenth_problem
line space, is the space of operators on Hilbert space. Liouville space is itself a Hilbert space under the Hilbert-Schmidt inner product. Abstractly, Liouville
Liouville_space
Mathematical structures that allow quantum mechanics to be explained
mathematical formalism uses mainly a part of functional analysis, especially Hilbert spaces, which are a kind of linear space. Such are distinguished from mathematical
Mathematical formulation of quantum mechanics
Mathematical_formulation_of_quantum_mechanics
where H is the Hilbert transform. It possesses infinitely many conserved densities and symmetries; thus it is a completely integrable system. Bretherton
Benjamin–Ono_equation
Theorem of quantum information theory
moving from one Hilbert space to another Hilbert space. Since the wave function contains all the relevant information about a physical system, the conservation
No-hiding_theorem
Study of geometries as axiomatic systems
to Hilbert, but also to differing choices made by the two translators. What follows will be based on the Unger translation. Hilbert's axiom system is
Foundations_of_geometry
Subspace of a quantum system's Hilbert space that is invariant to non-unitary dynamics
quantum system's Hilbert space that is invariant to non-unitary dynamics. Alternatively stated, they are a small section of the system Hilbert space where
Decoherence-free_subspaces
Propositional logic theorem
litotes. In Hilbert-style deductive systems for propositional logic, double negation is not always taken as an axiom (see list of Hilbert systems), and is
Double_negation
Positive integer of the form 4n + 1
a Hilbert number is a positive integer of the form 4n + 1 (Flannery & Flannery (2000, p. 35)). The Hilbert numbers were named after David Hilbert. The
Hilbert_number
American environmentalist (1917–1998)
Morton Shelly Hilbert (January 3, 1917 – December 24, 1998) was a professor of public health, environmentalist, and co-founder of Earth Day that was first
Morton_Hilbert
Job scheduling method in computing
the Hilbert curve scheduling method turns a multidimensional task allocation problem into a one-dimensional space filling problem using Hilbert curves
Hilbert_curve_scheduling
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
Formulation of classical mechanics in terms of Hilbert spaces
Koopman observed that the phase space of the classical system can be converted into a Hilbert space. According to this formulation, functions representing
Koopman–von Neumann classical mechanics
Koopman–von_Neumann_classical_mechanics
Rewriting system and type of formal grammar
the real line R: Prouhet-Thue-Morse system Well-known L-systems on a plane R2 are: space-filling curves (Hilbert curve, Peano's curves, Dekking's church
L-system
quadratic fields. Such systems have been studied for their connection with Hilbert's Twelfth Problem. In the case of a Bost–Connes system over Q, the absolute
Bost–Connes_system
Mathematics award
Mathematiche [Mathematical Lives: Protagonists of the Twentieth Century From Hilbert to Wiles] (2011 ed.). Springer. pp. 2013–2014. ISBN 978-3642136054. "Fields
Fields_Medal
Property of certain dynamical systems
quantum integrable systems. In the quantum setting, functions on phase space must be replaced by self-adjoint operators on a Hilbert space, and the notion
Integrable_system
Representation of a quantum mechanical system
system (qubit), named after the physicist Felix Bloch. Mathematically each quantum mechanical system is associated with a separable complex Hilbert space
Bloch_sphere
Paradox in set theory
a contradiction (to Cantor's theorem), as he told Hilbert and Richard Dedekind by letter. Hilbert also formulated his own paradox, which relied on reasoning
Russell's_paradox
Impossible task in computing
problem'; pronounced [ɛntˈʃaɪ̯dʊŋspʁoˌbleːm]) is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. It asks for an algorithm that considers
Entscheidungsproblem
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