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Mathematical logic

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Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. [1]

263 relations: A K Peters, Ltd., Abraham Fraenkel, Abstract algebra, Academic Press, ACM Computing Classification System, Alan Turing, Alfred North Whitehead, Alfred Tarski, Algebraic geometry, Algebraic logic, Algorithmically random sequence, Alonzo Church, Alpha recursion theory, American Mathematical Monthly, Analysis, Arithmetic, Arithmetic function, Arithmetical hierarchy, Arithmetization of analysis, Augustin-Louis Cauchy, Augustus De Morgan, Automated theorem proving, Axiom, Axiom of choice, Axiom schema of replacement, Banach–Tarski paradox, Begriffsschrift, Benedikt Löwe, Bernard Bolzano, Bertrand Russell, Bijection, injection and surjection, Boolean algebra (structure), Brouwer–Heyting–Kolmogorov interpretation, Burali-Forti paradox, Busy beaver, Cambridge University Press, Cantor's diagonal argument, Cantor's first uncountability proof, Cantor's theorem, Cardinal number, Cardinality, Categorical logic, Category theory, Cesare Burali-Forti, Chapman & Hall, Charles Sanders Peirce, Combinatory logic, Compactness theorem, Computability, Computability theory, ..., Computable function, Computable model theory, Computational complexity theory, Constructible universe, Constructivism (mathematics), Continuous function, Continuum hypothesis, Countable set, Cumulative hierarchy, Cut-elimination theorem, Cylindric algebra, D. C. Heath and Company, David Hilbert, Decision problem, Dedekind cut, Deductive reasoning, Definable set, Descriptive complexity theory, Determinacy, Differentiable function, Domain of discourse, Double-negation translation, Elementary class, Elementary equivalence, Elliptic geometry, Elsevier, Emil Leon Post, Entscheidungsproblem, Ernst Schröder, Ernst Zermelo, Euclid, Fagin's theorem, First-order logic, Forcing (mathematics), Formal system, Formal verification, Foundations of mathematics, Fourier series, Function (mathematics), Function problem, Fuzzy logic, Gödel's completeness theorem, Gödel's incompleteness theorems, Geometry, Georg Cantor, George Boole, George Peacock, Gerhard Gentzen, Giuseppe Peano, Gottfried Wilhelm Leibniz, Gottlob Frege, Great circle, Halting problem, Herbert Enderton, Hermann Weyl, Heyting algebra, Higher-order logic, Hilary Putnam, Hilbert system, Hilbert's axioms, Hilbert's problems, Hilbert's program, Hilbert's tenth problem, History of logic, Hyperarithmetical theory, If and only if, Impredicativity, Inaccessible cardinal, Independence (mathematical logic), Indian logic, Infinitary logic, Infinitesimal, Integer, Intuitionism, Intuitionistic logic, Isomorphism, J. L. Austin, Jean van Heijenoort, Johann Heinrich Lambert, Jules Richard, Julia Robinson, Karl Weierstrass, Knowledge representation and reasoning, Kripke semantics, Kripke–Platek set theory, Kurt Gödel, L. E. J. Brouwer, Lambda calculus, Large cardinal, Lattice (order), Law of excluded middle, Löb's theorem, Löwenheim–Skolem theorem, Leopold Kronecker, Leopold Löwenheim, Limitation of size, Lindström's theorem, List of computability and complexity topics, List of first-order theories, List of logic symbols, List of mathematical logic topics, List of set theory topics, Logic, Logic in China, Logic in computer science, Logic in Islamic philosophy, Logic programming, Logical consequence, London Mathematical Society, Martin Davis, Mathematical analysis, Mathematical induction, Mathematical proof, Mathematics, Mathematische Annalen, Metalogic, Metamathematics, Michael D. Morley, Modal logic, Model checking, Model theory, Morley's categoricity theorem, Morse–Kelley set theory, Natural deduction, Natural number, New Foundations, New York City, Nicolas Bourbaki, Nikolai Lobachevsky, Non-classical logic, Non-Euclidean geometry, Non-standard model of arithmetic, NP (complexity), O-minimal theory, On Formally Undecidable Propositions of Principia Mathematica and Related Systems, Ordinal analysis, Ordinal number, Oxford University Press, Parallel postulate, Pasch's axiom, Paul Cohen, Peano axioms, Philosophy, Polish space, Power set, Primitive recursive function, Principia Mathematica, Programming language, Proof mining, Proof theory, Propositional calculus, Pyotr Novikov, Quantifier (logic), Quantifier elimination, RAND Corporation, Real analysis, Real closed field, Real line, Recursive definition, Recursive set, Recursively enumerable set, Reverse mathematics, Rhetoric, Richard Dedekind, Richard's paradox, Robert Lawson Vaught, Russell's paradox, Saunders Mac Lane, Second-order logic, Semantics, Semantics (computer science), Sequent calculus, Set (mathematics), Set theory, Signature (logic), Skolem's paradox, Springer Science+Business Media, Stanford Encyclopedia of Philosophy, Stefan Banach, Stephen Cole Kleene, Stewart Shapiro, Structure (mathematical logic), Successor function, Syllogism, Syntax, Theoretical computer science, Theory (mathematical logic), Thoralf Skolem, Tibor Radó, Topos, Transactions of the American Mathematical Society, Transfinite induction, Transfinite number, Truth value, Turing degree, Turing machine, Type theory, Ulrich Kohlenbach, Universal algebra, Urelement, Vaught conjecture, Von Neumann–Bernays–Gödel set theory, W. Hugh Woodin, Weierstrass function, Well-formed formula, Well-order, Wilfrid Hodges, Word problem for groups, World Scientific, Yuri Matiyasevich, Zermelo set theory, Zermelo–Fraenkel set theory, (ε, δ)-definition of limit. Expand index (213 more) »

A K Peters, Ltd.

A K Peters, Ltd. was a publisher of scientific and technical books, specializing in mathematics and in computer graphics, robotics, and other fields of computer science.

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Abraham Fraenkel

Abraham Halevi (Adolf) Fraenkel (אברהם הלוי (אדולף) פרנקל; February 17, 1891, Munich, Germany – October 15, 1965, Jerusalem, Israel), known as Abraham Fraenkel, was a German-born Israeli mathematician.

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Abstract algebra

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.

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Academic Press

Academic Press is an academic book publisher.

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ACM Computing Classification System

The ACM Computing Classification System is a subject classification system for computing devised by the Association for Computing Machinery.

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Alan Turing

Alan Mathison Turing, OBE, FRS (23 June 1912 – 7 June 1954) was a British pioneering computer scientist, mathematician, logician, cryptanalyst, theoretical biologist, and marathon and ultra distance runner.

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Alfred North Whitehead

Alfred North Whitehead, OM FRS (15 February 1861 – 30 December 1947) was an English mathematician and philosopher.

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Alfred Tarski

Alfred Tarski (January 14, 1901 – October 26, 1983) was a Polish logician, mathematician and philosopher.

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Algebraic geometry

Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.

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Algebraic logic

In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables.

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Algorithmically random sequence

Intuitively, an algorithmically random sequence (or random sequence) is an infinite sequence of binary digits that appears random to any algorithm.

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Alonzo Church

Alonzo Church (June 14, 1903 – August 11, 1995) was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science.

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Alpha recursion theory

In recursion theory, α recursion theory is a generalisation of recursion theory to subsets of admissible ordinals \alpha.

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American Mathematical Monthly

The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894.

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Analysis

Analysis is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it.

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Arithmetic

Arithmetic or arithmetics (from the Greek ἀριθμός arithmos, "number") is the oldest and most elementary branch of mathematics.

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Arithmetic function

In number theory, an arithmetic, arithmetical, or number-theoretic function is a real or complex valued function f(n) defined on the set of natural numbers (i.e. positive integers) that "expresses some arithmetical property of n".

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Arithmetical hierarchy

In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy classifies certain sets based on the complexity of formulas that define them.

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Arithmetization of analysis

The arithmetization of analysis was a research program in the foundations of mathematics carried out in the second half of the 19th century.

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Augustin-Louis Cauchy

Baron Augustin-Louis Cauchy FRS FRSE (21 August 1789 – 23 May 1857) was a French mathematician reputed as a pioneer of analysis.

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Augustus De Morgan

Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician and logician.

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Automated theorem proving

Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs.

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Axiom

An axiom or postulate is a premise or starting point of reasoning.

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Axiom of choice

In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.

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Axiom schema of replacement

In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZFC) that asserts that the image of any set under any definable mapping is also a set.

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Banach–Tarski paradox

The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball.

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Begriffsschrift

Begriffsschrift (German for, roughly, "concept-script") is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book.

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Benedikt Löwe

Benedikt Löwe (born 1972) is a German mathematician and logician, and Professor at the University of Hamburg, known for initiating the interdisciplinary conference "Foundations of the Formal Sciences" (FotFS) in 1999.

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Bernard Bolzano

Bernhard Placidus Johann Nepomuk Bolzano (Bernard Bolzano in English; 5 October 1781 – 18 December 1848) was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his antimilitarist views.

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Bertrand Russell

Bertrand Arthur William Russell, 3rd Earl Russell (18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematician, historian, writer, social critic and political activist.

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Bijection, injection and surjection

In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments (input expressions from the domain) and images (output expressions from the codomain) are related or mapped to each other.

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Boolean algebra (structure)

In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice.

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Brouwer–Heyting–Kolmogorov interpretation

In mathematical logic, the Brouwer–Heyting–Kolmogorov interpretation, or BHK interpretation, of intuitionistic logic was proposed by L. E. J. Brouwer, Arend Heyting and independently by Andrey Kolmogorov.

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Burali-Forti paradox

In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that naïvely constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction.

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Busy beaver

In computability theory, a busy beaver is a Turing machine that attains the maximum number of steps performed, or maximum number of nonblank symbols finally on the tape, among all Turing machines in a certain class.

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Cambridge University Press

Cambridge University Press (CUP) is the publishing business of the University of Cambridge.

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Cantor's diagonal argument

In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers.

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Cantor's first uncountability proof

Georg Cantor's first proof of uncountability demonstrates that the set of all real numbers is uncountably, rather than countably, infinite.

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Cantor's theorem

In elementary set theory, Cantor's theorem states that, for any set A, the set of all subsets of A (the power set of A) has a strictly greater cardinality than A itself.

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Cardinal number

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets.

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Cardinality

In mathematics, the cardinality of a set is a measure of the "number of elements of the set".

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Categorical logic

Categorical logic is a branch of category theory within mathematics, adjacent to mathematical logic but more notable for its connections to theoretical computer science.

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Category theory

Category theory formalizes mathematical structure and its concepts in terms of a collection of objects and of arrows (also called morphisms).

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Cesare Burali-Forti

Cesare Burali-Forti (13 August 1861 – 21 January 1931) was an Italian mathematician.

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Chapman & Hall

Chapman & Hall was a British publishing house in London, founded in the first half of the 19th century by Edward Chapman and William Hall.

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Charles Sanders Peirce

Charles Sanders Peirce (like "purse", September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician, and scientist who is sometimes known as "the father of pragmatism".

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Combinatory logic

Combinatory logic is a notation to eliminate the need for quantified variables in mathematical logic.

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Compactness theorem

In mathematical logic, the 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.

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Computability

Computability is the ability to solve a problem in an effective manner.

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Computability theory

Computability theory, also called recursion theory, is a branch of mathematical logic, of computer science, and of the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees.

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Computable function

Computable functions are the basic objects of study in computability theory.

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Computable model theory

Computable model theory is a branch of model theory which deals with questions of computability as they apply to model-theoretical structures.

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Computational complexity theory

Computational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other.

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Constructible universe

In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted L, is a particular class of sets that can be described entirely in terms of simpler sets.

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Constructivism (mathematics)

In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists.

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Continuous function

In mathematics, a continuous function is, roughly speaking, a function for which small changes in the input result in small changes in the output.

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Continuum hypothesis

In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets.

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Countable set

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.

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Cumulative hierarchy

In mathematical set theory, a cumulative hierarchy is a family of sets Wα indexed by ordinals α such that.

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Cut-elimination theorem

The cut-elimination theorem (or Gentzen's Hauptsatz) is the central result establishing the significance of the sequent calculus.

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Cylindric algebra

The notion of cylindric algebra, invented by Alfred Tarski, arises naturally in the algebraization of first-order logic with equality.

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D. C. Heath and Company

D.C. Heath and Company was an American publishing company located at 125 Spring Street in Lexington, Massachusetts, specializing in textbooks.

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David Hilbert

David Hilbert (23 January 1862 – 14 February 1943) was a German mathematician.

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Decision problem

In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer, depending on the values of some input parameters.

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Dedekind cut

In mathematics, a Dedekind cut, named after Richard Dedekind, is a partition of the rational numbers into two non-empty sets A and B, such that all elements of A are less than all elements of B, and A contains no greatest element.

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Deductive reasoning

Deductive reasoning, also deductive logic or logical deduction or, informally, "top-down" logic, is the process of reasoning from one or more statements (premises) to reach a logically certain conclusion.

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Definable set

In mathematical logic, a definable set is an n-ary relation on the domain of a structure whose elements are precisely those elements satisfying some formula in the language of that structure.

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Descriptive complexity theory

Descriptive complexity is a branch of computational complexity theory and of finite model theory that characterizes complexity classes by the type of logic needed to express the languages in them.

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Determinacy

In set theory, a branch of mathematics, determinacy is the study of under what circumstances one or the other player of a game must have a winning strategy, and the consequences of the existence of such strategies.

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Differentiable function

In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain.

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Domain of discourse

In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range.

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Double-negation translation

In proof theory, a discipline within mathematical logic, double-negation translation, sometimes called negative translation, is a general approach for embedding classical logic into intuitionistic logic, typically by translating formulas to formulas which are classically equivalent but intuitionistically inequivalent.

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Elementary class

In model theory, a branch of mathematical logic, an elementary class (or axiomatizable class) is a class consisting of all structures satisfying a fixed first-order theory.

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Elementary equivalence

In model theory, a branch of mathematical logic, two structures M and N of the same signature σ are called elementarily equivalent if they satisfy the same first-order σ-sentences.

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Elliptic geometry

Elliptic geometry, a special case of Riemannian geometry, is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p, as all lines in elliptic geometry intersect.

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Elsevier

Elsevier B.V. is an academic publishing company that publishes medical and scientific literature.

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Emil Leon Post

Emil Leon Post (February 11, 1897 – April 21, 1954) was a Polish-born American mathematician and logician.

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Entscheidungsproblem

In mathematics and computer science, the Entscheidungsproblem (German for 'decision problem') is a challenge posed by David Hilbert in 1928.

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Ernst Schröder

Friedrich Wilhelm Karl Ernst Schröder (25 November 1841 in Mannheim, Baden, Germany – 16 June 1902 in Karlsruhe, Germany) was a German mathematician mainly known for his work on algebraic logic.

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Ernst Zermelo

Ernst Friedrich Ferdinand Zermelo (27 July 1871 – 21 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics.

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Euclid

Euclid (Εὐκλείδης Eukleidēs; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referred to as the "Father of Geometry".

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Fagin's theorem

Fagin's theorem is a result in descriptive complexity theory that states that the set of all properties expressible in existential second-order logic is precisely the complexity class NP.

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First-order logic

First-order logic is a formal system used in mathematics, philosophy, linguistics, and computer science.

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Forcing (mathematics)

In the mathematical discipline of set theory, forcing is a technique discovered by Paul Cohen for proving consistency and independence results.

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Formal system

A formal system is broadly defined as any well-defined system of abstract thought based on the model of mathematics.

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Formal verification

In the context of hardware and software systems, formal verification is the act of proving or disproving the correctness of intended algorithms underlying a system with respect to a certain formal specification or property, using formal methods of mathematics.

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Foundations of mathematics

Foundations of mathematics is the study of the logical and philosophical basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics.

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Fourier series

In mathematics, a Fourier series is a way to represent a (wave-like) function as the sum of simple sine waves.

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Function (mathematics)

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.

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Function problem

In computational complexity theory, a function problem is a computational problem where a single output (of a total function) is expected for every input, but the output is more complex than that of a decision problem, that is, it isn't just YES or NO.

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Fuzzy logic

Fuzzy logic is a form of many-valued logic in which the truth values of variables may be any real number between 0 and 1.

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Gödel's completeness theorem

Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic.

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Gödel's incompleteness theorems

Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic.

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Geometry

Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

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Georg Cantor

Georg Ferdinand Ludwig Philipp Cantor (– January 6, 1918) was a German mathematician.

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George Boole

George Boole (2 November 1815 – 8 December 1864) was an English mathematician, philosopher and logician.

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George Peacock

George Peacock (9 April 1791 – 8 November 1858) was an English mathematician.

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Gerhard Gentzen

Gerhard Karl Erich Gentzen (November 24, 1909 – August 4, 1945) was a German mathematician and logician.

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Giuseppe Peano

Giuseppe Peano (27 August 1858 – 20 April 1932) was an Italian mathematician.

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Gottfried Wilhelm Leibniz

Gottfried Wilhelm von Leibniz (also Godefroi Guillaume Leibnitz,; or; July 1, 1646 – November 14, 1716) was a German polymath and philosopher, and to this day he occupies a prominent place in the history of mathematics and the history of philosophy.

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Gottlob Frege

Friedrich Ludwig Gottlob Frege (8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician.

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Great circle

A great circle, also known as an orthodrome or Riemannian circle, of a sphere is the intersection of the sphere and a plane which passes through the center point of the sphere.

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Halting problem

In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running or continue to run forever.

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Herbert Enderton

Herbert Bruce Enderton (April 15, 1936 – October 20, 2010) was a Professor Emeritus of Mathematics at UCLA and a former member of the faculties of Mathematics and of Logic and the Methodology of Science at the University of California, Berkeley.

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Hermann Weyl

Hermann Klaus Hugo Weyl, (9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher.

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Heyting algebra

In mathematics, a Heyting algebra is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation a → b of implication such that c ∧ a ≤ b is equivalent to c ≤ a → b. From a logical standpoint, A → B is by this definition the weakest proposition for which modus ponens, the inference rule A → B, A ⊢ B, is sound.

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Higher-order logic

In mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and a stronger semantics.

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Hilary Putnam

Hilary Whitehall Putnam (born July 31, 1926) is an American philosopher, mathematician, and computer scientist who has been a central figure in analytic philosophy since the 1960s, especially in philosophy of mind, philosophy of language, philosophy of mathematics, and philosophy of science.

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Hilbert system

In logic, especially mathematical logic, a Hilbert system, sometimes called Hilbert calculus or Hilbert–Ackermann system, is a type of system of formal deduction attributed to Gottlob FregeMáté & Ruzsa 1997:129 and David Hilbert.

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Hilbert's axioms

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 the foundation for a modern treatment of Euclidean geometry.

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Hilbert's problems

Hilbert's problems are a list of twenty-three problems in mathematics published by German mathematician David Hilbert in 1900.

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Hilbert's program

In mathematics, Hilbert's program, formulated by German mathematician David Hilbert, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies.

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Hilbert's tenth problem

Hilbert's tenth problem is the tenth on the list of Hilbert's problems of 1900.

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History of logic

The history of logic is the study of the development of the science of valid inference (logic).

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Hyperarithmetical theory

In recursion theory, hyperarithmetic theory is a generalization of Turing computability.

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If and only if

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

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Impredicativity

In mathematics and logic, a self-referencing definition is called impredicative.

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Inaccessible cardinal

In set theory, an uncountable regular cardinal number is called weakly inaccessible if it is a weak limit cardinal, and strongly inaccessible, or just inaccessible, if it is a strong limit cardinal.

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Independence (mathematical logic)

In mathematical logic, independence refers to the unprovability of a sentence from other sentences.

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Indian logic

The development of Indian logic dates back to the anviksiki of Medhatithi Gautama (c. 6th century BCE) the Sanskrit grammar rules of Pāṇini (c. 5th century BCE); the Vaisheshika school's analysis of atomism (c. 2nd century BCE); the analysis of inference by Gotama (c. 2nd century), founder of the Nyaya school of Hindu philosophy; and the tetralemma of Nagarjuna (c. 2nd century CE).

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Infinitary logic

An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs.

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Infinitesimal

In mathematics, infinitesimals are things so small that there is no way to measure them.

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Integer

An integer (from the Latin ''integer'' meaning "whole")Integer 's first, literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

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Intuitionism

In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality.

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Intuitionistic logic

Intuitionistic logic, sometimes more generally called constructive logic, is a system of symbolic logic that differs from classical logic by replacing the traditional concept of truth with the concept of constructive provability.

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Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism (or more generally a morphism) that admits an inverse.

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J. L. Austin

John Langshaw "J.

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Jean van Heijenoort

Jean Louis Maxime van Heijenoort (July 23, 1912 – March 29, 1986) was a pioneer historian of mathematical logic.

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Johann Heinrich Lambert

Johann Heinrich Lambert (Jean-Henri Lambert in French; 26 August 1728 – 25 September 1777) was a Swiss polymath who made important contributions to the subjects of mathematics, physics (particularly optics), philosophy, astronomy and map projections.

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Jules Richard

Jules Richard (born 12 August 1862 in Blet, Département Cher, died 14 October 1956 in Châteauroux, Département Indre) was a French mathematician.

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Julia Robinson

Julia Hall Bowman Robinson (December 8, 1919 – July 30, 1985) was an American mathematician best known for her work on decision problems and Hilbert's Tenth Problem.

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Karl Weierstrass

Karl Theodor Wilhelm Weierstrass (Weierstraß; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis".

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Knowledge representation and reasoning

Knowledge representation and reasoning (KR) is the field of artificial intelligence (AI) dedicated to representing information about the world in a form that a computer system can utilize to solve complex tasks such as diagnosing a medical condition or having a dialog in a natural language.

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Kripke semantics

Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André Joyal.

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Kripke–Platek set theory

The Kripke–Platek axioms of set theory (KP), pronounced, are a system of axiomatic set theory based on the ideas of and.

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Kurt Gödel

Kurt Friedrich Gödel (April 28, 1906 – January 14, 1978) was an Austrian, and later American, logician, mathematician, and philosopher.

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L. E. J. Brouwer

Luitzen Egbertus Jan Brouwer ForMemRS (27 February 1881 – 2 December 1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, a graduate of the University of Amsterdam, who worked in topology, set theory, measure theory and complex analysis.

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Lambda calculus

Lambda calculus (also written as λ-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution.

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Large cardinal

In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers.

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Lattice (order)

In mathematics, a lattice is a partially ordered set in which every two elements have a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet).

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Law of excluded middle

In logic, the law of excluded middle (or the principle of excluded middle) is the third of the three classic laws of thought.

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Löb's theorem

In mathematical logic, Löb's theorem states that in a theory with Peano arithmetic, for any formula P, if it is provable that "if P is provable then P is true", then P is provable.

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Löwenheim–Skolem theorem

In mathematical logic, the Löwenheim–Skolem theorem, named for Leopold Löwenheim and Thoralf Skolem, states that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ. The result implies that first-order theories are unable to control the cardinality of their infinite models, and that no first-order theory with an infinite model can have a unique model up to isomorphism.

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Leopold Kronecker

Leopold Kronecker (7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory and algebra.

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Leopold Löwenheim

Leopold Löwenheim (26 June 1878 in Krefeld – 5 May 1957 in Berlin) was a German mathematician, known for his work in mathematical logic.

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Limitation of size

In the philosophy of mathematics, specifically the philosophical foundations of set theory, limitation of size is a concept developed by Philip Jourdain and/or Georg Cantor to avoid Cantor's paradox.

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Lindström's theorem

In mathematical logic, Lindström's theorem (named after Swedish logician Per Lindström, who published it in 1969) states that first-order logic is the strongest logic (satisfying certain conditions, e.g. closure under classical negation) having both the (countable) compactness property and the (downward) Löwenheim–Skolem property.

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List of computability and complexity topics

This is a list of computability and complexity topics, by Wikipedia page.

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List of first-order theories

In mathematical logic, a first-order theory is given by a set of axioms in some language.

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List of logic symbols

In logic, a set of symbols is commonly used to express logical representation.

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List of mathematical logic topics

This is a list of mathematical logic topics, by Wikipedia page.

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List of set theory topics

This page is a list of articles related to set theory.

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Logic

Logic (from the λογική, logike) is the branch of philosophy concerned with the use and study of valid reasoning.

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Logic in China

Logic in China plays a particularly interesting role in the history of logic due to its repression and abandonment compared to the strong ancient adoption and continued development of the study of logic in Europe, India, and the Islamic world.

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Logic in computer science

Logic in computer science covers the overlap between the field of logic and that of computer science.

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Logic in Islamic philosophy

Logic (منطق) plays an important role in Islamic philosophy.

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Logic programming

Logic programming is a programming paradigm based on formal logic.

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Logical consequence

Logical consequence (also entailment) is one of the most fundamental concepts in logic.

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London Mathematical Society

The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS) and the Institute of Mathematics and its Applications (IMA)).

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Martin Davis

Martin David Davis (born 1928) is an American mathematician, known for his work on Hilbert's tenth problem.

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Mathematical analysis

Mathematical analysis is a branch of mathematics that studies continuous change and includes the theories of differentiation, integration, measure, limits, infinite series, and analytic functions.

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Mathematical induction

Mathematical induction is a method of mathematical proof typically used to establish a given statement for all natural numbers.

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Mathematical proof

In mathematics, a proof is a deductive argument for a mathematical statement.

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Mathematics

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change.

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Mathematische Annalen

Mathematische Annalen (abbreviated as Math. Ann. or, formerly, Math. Annal.) is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann.

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Metalogic

Metalogic is the study of the metatheory of logic.

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Metamathematics

Metamathematics is the study of mathematics itself using mathematical methods.

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Michael D. Morley

Michael Darwin Morley (born 1930) is an American mathematician, currently professor emeritus at Cornell University.

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Modal logic

Modal logic is a type of formal logic primarily developed in the 1960s that extends classical propositional and predicate logic to include operators expressing modality.

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Model checking

In computer science, model checking or property checking refers to the following problem: Given a model of a system, exhaustively and automatically check whether this model meets a given specification.

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Model theory

In mathematics, model theory is the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic.

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Morley's categoricity theorem

In model theory, a branch of mathematical logic, a theory is κ-categorical (or categorical in κ) if it has exactly one model of cardinality κ up to isomorphism.

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Morse–Kelley set theory

In the foundation of mathematics, Morse–Kelley set theory (MK) or Kelley–Morse set theory (KM) or Quine–Morse set theory (QM) or the system of Quine and Morse is a first order axiomatic set theory that is closely related to von Neumann–Bernays–Gödel set theory (NBG).

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Natural deduction

In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning.

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Natural number

In mathematics, the natural numbers (sometimes called the whole numbers): "whole number An integer, though sometimes it is taken to mean only non-negative integers, or just the positive integers." give definitions of "whole number" under several headwords: INTEGER … Syn. whole number.

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New Foundations

In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica.

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New York City

New York – often called New York City or the City of New York to distinguish it from the State of New York, of which it is a part – is the most populous city in the United States and the center of the New York metropolitan area, the premier gateway for legal immigration to the United States and one of the most populous urban agglomerations in the world.

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Nicolas Bourbaki

Nicolas Bourbaki is the collective pseudonym under which a group of (mainly French) 20th-century mathematicians, with the aim of reformulating mathematics on an extremely abstract and formal but self-contained basis, wrote a series of books beginning in 1935.

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Nikolai Lobachevsky

Nikolai Ivanovich Lobachevsky (a; &ndash) was a Russian mathematician and geometer, known primarily for his work on hyperbolic geometry, otherwise known as Lobachevskian geometry.

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Non-classical logic

Non-classical logics (and sometimes alternative logics) is the name given to formal systems that differ in a significant way from standard logical systems such as propositional and predicate logic.

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Non-Euclidean geometry

In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry.

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Non-standard model of arithmetic

In mathematical logic, a non-standard model of arithmetic is a model of (first-order) Peano arithmetic that contains non-standard numbers.

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NP (complexity)

In computational complexity theory, NP is one of the most fundamental complexity classes.

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O-minimal theory

In mathematical logic, and more specifically in model theory, an infinite structure (M,<,...) which is totally ordered by Knight, Pillay and Steinhorn (1986), Pillay and Steinhorn (1988).

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On Formally Undecidable Propositions of Principia Mathematica and Related Systems

Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I ("On Formally Undecidable Propositions of Principia Mathematica and Related Systems I") is a paper in mathematical logic by Kurt Gödel.

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Ordinal analysis

In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength.

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Ordinal number

In set theory, an ordinal number, or ordinal, is the order type of a well-ordered set.

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Oxford University Press

Oxford University Press (OUP) is the largest university press in the world, and the second-oldest, after Cambridge University Press.

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Parallel postulate

In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry.

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Pasch's axiom

In geometry, Pasch's axiom is a statement in plane geometry, used implicitly by Euclid, which cannot be derived from the postulates as Euclid gave them.

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Paul Cohen

Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American mathematician.

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Peano axioms

In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano.

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Philosophy

Philosophy is the study of the general and fundamental nature of reality, existence, knowledge, values, reason, mind, and language.

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Polish space

In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset.

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Power set

In mathematics, the power set (or powerset) of any set, written, ℘(),, or 2''S'', is the set of all subsets of, including the empty set and itself.

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Primitive recursive function

In computability theory, primitive recursive functions are a class of functions that are defined using primitive recursion and composition as central operations and are a strict subset of the total µ-recursive functions (µ-recursive functions are also called partial recursive).

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Principia Mathematica

The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913.

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Programming language

A programming language is a formal constructed language designed to communicate instructions to a machine, particularly a computer.

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Proof mining

In proof theory, a branch of mathematical logic, proof mining (or unwinding) is a research program that analyzes formalized proofs, especially in analysis, to obtain explicit bounds or rates of convergence from proofs that, when expressed in natural language, appear to be nonconstructive.

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Proof theory

Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques.

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Propositional calculus

Propositional calculus (also called propositional logic, sentential calculus, or sentential logic) is the branch of mathematical logic concerned with the study of propositions (whether they are true or false) that are formed by other propositions with the use of logical connectives, and how their value depends on the truth value of their components.

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Pyotr Novikov

Pyotr Sergeyevich Novikov (Пётр Серге́евич Но́виков; 15 August 1901, Moscow, Russian Empire – 9 January 1975, Moscow, Soviet Union) was a Soviet mathematician.

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Quantifier (logic)

In logic, quantification is a construct that specifies the quantity of specimens in the domain of discourse that satisfy an open formula.

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Quantifier elimination

Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science.

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RAND Corporation

RAND Corporation (Research ANd Development) is a nonprofit global policy think tank originally formed by Douglas Aircraft Company to offer research and analysis to the United States Armed Forces.

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Real analysis

Real analysis (traditionally, the theory of functions of a real variable) is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable.

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Real closed field

In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers.

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Real line

In mathematics, the real line, or real number line is the line whose points are the real numbers.

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Recursive definition

A recursive definition (or inductive definition) in mathematical logic and computer science is used to define the elements in a set in terms of other elements in the set (Aczel 1978:740ff).

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Recursive set

In computability theory, a set of natural numbers is called recursive, computable or decidable if there is an algorithm which terminates after a finite amount of time and correctly decides whether or not a given number belongs to the set.

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Recursively enumerable set

In computability theory, traditionally called recursion theory, a set S of natural numbers is called recursively enumerable, computably enumerable, semidecidable, provable or Turing-recognizable if.

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Reverse mathematics

Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics.

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Rhetoric

Rhetoric (pronounced) is the art of discourse, an art that aims to improve the capability of writers or speakers to inform, persuade, or motivate particular audiences in specific situations.

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Richard Dedekind

Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory), algebraic number theory and the definition of the real numbers.

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Richard's paradox

In logic, Richard's paradox is a semantical antinomy of set theory and natural language described first by the French mathematician Jules Richard during 1905.

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Robert Lawson Vaught

Robert Lawson Vaught (April 4, 1926 – April 2, 2002) was a mathematical logician, and one of the founders of model theory.

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Russell's paradox

In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that some attempted formalizations of the naive set theory created by Georg Cantor led to a contradiction.

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Saunders Mac Lane

Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg.

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Second-order logic

In logic and mathematics second-order logic is an extension of first-order logic, which itself is an extension of propositional logic.

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Semantics

Semantics (from σημαντικός sēmantikós, "significant") is the study of meaning.

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Semantics (computer science)

In programming language theory, semantics is the field concerned with the rigorous mathematical study of the meaning of programming languages.

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Sequent calculus

Sequent calculus is, in essence, a style of formal logical argumentation where every line of a proof is a conditional tautology (called a sequent by Gerhard Gentzen) instead of an unconditional tautology.

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Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

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Set theory

Set theory is the branch of mathematical logic that studies sets, which informally are collections of objects.

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Signature (logic)

In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language.

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Skolem's paradox

In mathematical logic and philosophy, Skolem's paradox is a seeming contradiction that arises from the downward Löwenheim–Skolem theorem.

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Springer Science+Business Media

Springer Science+Business Media or Springer is a global publishing company that publishes books, e-books and peer-reviewed journals in science, technical and medical (STM) publishing.

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Stanford Encyclopedia of Philosophy

The Stanford Encyclopedia of Philosophy (SEP) combines an online encyclopedia of philosophy with peer reviewed publication of original papers in philosophy, freely-accessible to internet users.

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Stefan Banach

Stefan Banach (30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the world's most important and influential 20th-century mathematicians.

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Stephen Cole Kleene

Stephen Cole Kleene (January 5, 1909 – January 25, 1994) was an American mathematician.

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Stewart Shapiro

Stewart Shapiro (born 1951) is O'Donnell Professor of Philosophy at the Ohio State University and a regular visiting professor at the University of St Andrews in Scotland.

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Structure (mathematical logic)

In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations, and relations that are defined on it.

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Successor function

In mathematics, the successor function or successor operation is a primitive recursive function S such that S(n).

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Syllogism

A syllogism (συλλογισμός syllogismos, "conclusion, inference") is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two or more propositions that are asserted or assumed to be true.

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Syntax

In linguistics, syntax is the set of rules, principles, and processes that govern the structure of sentences in a given language.

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Theoretical computer science

Theoretical computer science is a division or subset of general computer science and mathematics that focuses on more abstract or mathematical aspects of computing and includes the theory of computation.

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Theory (mathematical logic)

In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language.

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Thoralf Skolem

Thoralf Albert Skolem (23 May 1887 – 23 March 1963) was a Norwegian mathematician who worked in mathematical logic and set theory.

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Tibor Radó

Tibor Radó (June 2, 1895 – December 29, 1965) was a Hungarian mathematician who moved to the USA after World War I.

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Topos

In mathematics, a topos (or; plural topoi or, or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site).

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Transactions of the American Mathematical Society

The Transactions of the American Mathematical Society is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society.

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Transfinite induction

Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers.

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Transfinite number

Transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite.

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Truth value

In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth.

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Turing degree

In computer science and mathematical logic the Turing degree (named after Alan Turing) or degree of unsolvability of a set of natural numbers measures the level of algorithmic unsolvability of the set.

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Turing machine

A Turing machine is an abstract "machine" that manipulates symbols on a strip of tape according to a table of rules; to be more exact, it is a mathematical model that defines such a device.

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Type theory

In mathematics, logic, and computer science, a type theory is any of a class of formal systems, some of which can serve as alternatives to set theory as a foundation for all mathematics.

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Ulrich Kohlenbach

Ulrich Wilhelm Kohlenbach (* July 27, 1962 in Frankfurt am Main) is a German professor of mathematics and a researcher in logic.

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Universal algebra

Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.

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Urelement

In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ur-, 'primordial') is an object (concrete or abstract) that is not a set, but that may be an element of a set.

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Vaught conjecture

The Vaught conjecture is a conjecture in the mathematical field of model theory originally proposed by Robert Lawson Vaught in 1961.

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Von Neumann–Bernays–Gödel set theory

In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of the canonical Zermelo–Fraenkel set theory (ZFC).

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W. Hugh Woodin

William Hugh Woodin (born April 23, 1955) is an American mathematician and set theorist at Harvard University.

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Weierstrass function

In mathematics, the Weierstrass function is an example of a pathological real-valued function on the real line.

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Well-formed formula

In mathematical logic, a well-formed formula, shortly wff, often simply formula, is a word (i.e. a finite sequence of symbols from a given alphabet) that is part of a formal language.

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Well-order

In mathematics, a well-order relation (or well-ordering) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering.

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Wilfrid Hodges

Wilfrid Augustine Hodges, FBA (born May 27, 1941) is a British mathematician, known for his work in model theory.

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Word problem for groups

In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group G is the algorithmic problem of deciding whether two words in the generators represent the same element.

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World Scientific

World Scientific Publishing is an academic publisher of scientific, technical, and medical books and journals headquartered in Singapore.

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Yuri Matiyasevich

Yuri Vladimirovich Matiyasevich, (Ю́рий Влади́мирович Матиясе́вич; born March 2, 1947, in Leningrad) is a Russian mathematician and computer scientist.

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Zermelo set theory

Zermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory.

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Zermelo–Fraenkel set theory

In mathematics, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets free of paradoxes such as Russell's paradox.

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(ε, δ)-definition of limit

In calculus, the (ε, δ)-definition of limit ("epsilon-delta definition of limit") is a formalization of the notion of limit.

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Formal Logic, Logic (math), Logic (mathematics), Logic (maths), Logic (symbolic), Logic modeling, Logic modelling, Mathematical Logic, Mathematical formalism, Mathematical logician, Symbolic Logic, Symbolic logic (disambiguation).

References

[1] https://en.wikipedia.org/wiki/Mathematical_logic

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