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# Mathematical logic

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

## A K Peters

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.

## Abraham Fraenkel

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

## Abstract algebra

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

## ACM Computing Classification System

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

## Alan Turing

Alan Mathison Turing (23 June 1912 – 7 June 1954) was an English computer scientist, mathematician, logician, cryptanalyst, philosopher, and theoretical biologist.

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

## Alfred Tarski

Alfred Tarski (January 14, 1901 &ndash; October 26, 1983), born Alfred Teitelbaum,School of Mathematics and Statistics, University of St Andrews,, School of Mathematics and Statistics, University of St Andrews.

## Algebraic geometry

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

## Algebraic logic

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

## Algorithmically random sequence

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

## 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.

## Alpha recursion theory

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

## American Mathematical Monthly

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

## Analysis

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

## Arithmetic

Arithmetic (from the Greek ἀριθμός arithmos, "number") is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations on them—addition, subtraction, multiplication and division.

## Arithmetic function

In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function f(n) whose domain is the positive integers and whose range is a subset of the complex numbers.

## 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.

## Arithmetices principia, nova methodo exposita

The 1889 treatise Arithmetices principia, nova methodo exposita (The principles of arithmetic, presented by a new method; 1889) by Giuseppe Peano is a seminal document in mathematical logic and set theory, introducing what is now the standard axiomatization of the natural numbers, and known as the Peano axioms, as well as some pervasive notations, such as the symbols for the basic set operations ∈, ⊂, ∩, ∪, and ''A''&minus;''B''.

## 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.

## Augustin-Louis Cauchy

Baron Augustin-Louis Cauchy FRS FRSE (21 August 178923 May 1857) was a French mathematician, engineer and physicist who made pioneering contributions to several branches of mathematics, including: mathematical analysis and continuum mechanics.

## Augustus De Morgan

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

## 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.

## Axiom

An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

## 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.

## Axiom schema of replacement

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

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.

## 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.

## 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.

## Benson Mates

Benson Mates (May 19, 1919 in Portland, Oregon – May 14, 2009 in Berkeley, California) was an American philosopher, noted for his work in logic, the history of philosophy, and skepticism.

## Bernard Bolzano

Bernard Bolzano (born Bernardus Placidus Johann Nepomuk Bolzano; 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.

## Bertrand Russell

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

## 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.

## Boolean algebra

In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 respectively.

## Boolean algebra (structure)

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

## 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 and Arend Heyting, and independently by Andrey Kolmogorov.

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

## Busy beaver

The busy beaver game consists of designing a halting, binary-alphabet Turing machine which writes the most 1s on the tape, using only a limited set of states.

## Cambridge University Press

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

## 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.

## Cantor's theorem

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

## 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.

## Cardinality

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

## Carl Gustav Hempel

Carl Gustav "Peter" Hempel (January 8, 1905 – November 9, 1997) was a German writer and philosopher.

## Categorical logic

Categorical logic is the branch of mathematics in which tools and concepts from category theory are applied to the study of mathematical logic.

## Category theory

Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms).

## Cesare Burali-Forti

Cesare Burali-Forti (13 August 1861 – 21 January 1931) was an Italian mathematician, after whom the Burali-Forti paradox is named.

## 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.

## Charles Sanders Peirce

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

## Christmas

Christmas is an annual festival commemorating the birth of Jesus Christ,Martindale, Cyril Charles.

## Christmas and holiday season

The Christmas season, also called the festive season, or the holiday season (mainly in the U.S. and Canada; often simply called the holidays),, is an annually recurring period recognized in many Western and Western-influenced countries that is generally considered to run from late November to early January.

## Christmas Eve

Christmas Eve is the evening or entire day before Christmas Day, the festival commemorating the birth of Jesus.

Christmas traditions vary from country to country.

## Claude Shannon

Claude Elwood Shannon (April 30, 1916 – February 24, 2001) was an American mathematician, electrical engineer, and cryptographer known as "the father of information theory".

## Combinatory logic

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

## 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.

## Computability

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

## Computability theory

Computability theory, also known as 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.

## Computable function

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

## 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.

## Computational complexity theory

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

## Computer science

Computer science deals with the theoretical foundations of information and computation, together with practical techniques for the implementation and application of these foundations.

## 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.

## 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.

## Continuous function

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

## Continuum hypothesis

In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets.

## 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.

## Cumulative hierarchy

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

## Cut-elimination theorem

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

## Cylindric algebra

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

## 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.

## Daniel H. H. Ingalls Sr.

Daniel Henry Holmes Ingalls Sr. (May 4, 1916 – July 17, 1999) was the Wales Professor of Sanskrit at Harvard University.

## David Hilbert

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

## Decision problem

In computability theory and computational complexity theory, a decision problem is a problem that can be posed as a yes-no question of the input values.

## Dedekind cut

In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind, are а method of construction of the real numbers from the rational numbers.

## Deductive reasoning

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

## 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 first-order language of that structure.

## 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.

## Determinacy

Determinacy is a subfield of set theory, a branch of mathematics, that examines the conditions under which one or the other player of a game has a winning strategy, and the consequences of the existence of such strategies.

## 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.

## 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.

## Dordrecht

Dordrecht, colloquially Dordt, historically in English named Dort, is a city and municipality in the Western Netherlands, located in the province of South Holland.

## 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.

## Edmund Berkeley

Edmund Callis Berkeley (February 22, 1909 – March 7, 1988) was an American computer scientist who co-founded the Association for Computing Machinery (ACM) in 1947.

## 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.

## 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.

## Elliptic geometry

Elliptic geometry is a geometry in which Euclid's parallel postulate does not hold.

## Elsevier

Elsevier is an information and analytics company and one of the world's major providers of scientific, technical, and medical information.

## Emil Leon Post

Emil Leon Post (February 11, 1897 – April 21, 1954) was an American mathematician and logician.

## Entscheidungsproblem

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

Ernest Addison Moody (1903–1975) was a noted philosopher, medievalist, and logician as well as a musician and scientist.

## Ernst Schröder

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

## 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.

## Euclid

Euclid (Εὐκλείδης Eukleidēs; fl. 300 BC), sometimes given the name Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry".

## 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.

## First-order logic

First-order logic—also known as first-order predicate calculus and predicate logic—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science.

## Forcing (mathematics)

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

## Formal system

A formal system is the name of a logic system usually defined in the mathematical way.

## 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.

## Foundations of mathematics

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

## Fourier series

In mathematics, a Fourier series is a way to represent a function as the sum of simple sine waves.

## Frederic Fitch

Frederic Brenton Fitch (1908 – September 18, 1987) was an American logician, a Sterling Professor at Yale University.

## Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

## 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.

## 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.

## 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.

## Gödel's incompleteness theorems

Gödel's incompleteness theorems are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system containing basic arithmetic.

## 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.

## Georg Cantor

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

## Georg Cantor's first set theory article

Georg Cantor's first set theory article was published in 1874 and contains the first theorems of transfinite set theory, which studies infinite sets and their properties.

## George Boole

George Boole (2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Ireland.

## George Peacock

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

## Gerhard Gentzen

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

## Giuseppe Peano

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

## Gottfried Wilhelm Leibniz

Gottfried Wilhelm (von) Leibniz (or; Leibnitz; – 14 November 1716) was a German polymath and philosopher who occupies a prominent place in the history of mathematics and the history of philosophy.

## Gottlob Frege

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

## Great circle

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

## Grundlagen der Mathematik

Grundlagen der Mathematik (English: Foundations of Mathematics) is a two-volume work by David Hilbert and Paul Bernays.

## 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 (i.e., halt) or continue to run forever.

## Hans Reichenbach

Hans Reichenbach (September 26, 1891 – April 9, 1953) was a leading philosopher of science, educator, and proponent of logical empiricism.

## Heinrich Scholz

Heinrich Scholz (December 17, 1884 – December 30, 1956) was a German logician, philosopher, and Protestant theologian who was a peer of Alan Turing, who wrote in his memoirs that he on the inclusion of his essay from 1936 "On Computable Numbers, with an Application to the Entscheidungsproblem": " two people could have understood it, and would have responded – Heinrich Scholz and Richard Bevan Braithwaite." Scholz had an extraordinary career but was not considered a brilliant logician, for example on the same level as Gottlob Frege or Rudolf Carnap, but was considered an outstanding scientist of national importance.

## 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.

## Hermann Weyl

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

## 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.

## 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, sometimes, stronger semantics.

## Hilary Putnam

Hilary Whitehall Putnam (July 31, 1926 – March 13, 2016) was an American philosopher, mathematician, and computer scientist, and a major figure in analytic philosophy in the second half of the 20th century.

## Hilbert system

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

## 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.

## Hilbert's problems

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

## Hilbert's program

In mathematics, Hilbert's program, formulated by German mathematician David Hilbert in the early part of the 20th century, 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.

## Hilbert's tenth problem

Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900.

## History of logic

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

## Hyperarithmetical theory

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

## Impredicativity

Something that is impredicative, in mathematics and logic, is a self-referencing definition.

## Inaccessible cardinal

In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic.

## Independence (mathematical logic)

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

## 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. 6th century BCE to 2nd century BCE); the analysis of inference by Gotama (c. 6th century BC to 2nd century CE), founder of the Nyaya school of Hindu philosophy; and the tetralemma of Nagarjuna (c. 2nd century CE).

## Infinitary logic

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

## Infinitesimal

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

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

## 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.

## Intuitionistic logic

Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof.

## Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.

## J. L. Austin

John Langshaw "J.

## Jan Łukasiewicz

Jan Łukasiewicz (21 December 1878 – 13 February 1956) was a Polish logician and philosopher born in Lwów, a city in the Galician kingdom of Austria-Hungary.

## Józef Maria Bocheński

Józef Maria Bocheński (Czuszów, Congress Poland, Russian Empire, 30 August 1902 – 8 February 1995, Fribourg, Switzerland) was a Polish Dominican, logician and philosopher.

## Jean van Heijenoort

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

## 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.

## John von Neumann

John von Neumann (Neumann János Lajos,; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, and polymath.

## Joseph Henry Woodger

Joseph Henry Woodger (2 May 1894 – 8 March 1981) was a British theoretical biologist and philosopher of biology whose attempts to make biological sciences more rigorous and empirical was significantly influential to the philosophy of biology in the twentieth century.

## Jules Richard

Jules Richard (12 August 1862 – 14 October 1956) was a French mathematician.

## Julia Robinson

Julia Hall Bowman Robinson (December 8, 1919 – July 30, 1985) was an American mathematician renowned for her contributions to the fields of computability theory and computational complexity theory–most notably in decision problems.

## Karl Menger

Karl Menger (January 13, 1902 – October 5, 1985) was an Austrian-American mathematician.

## 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".

## Knowledge representation and reasoning

Knowledge representation and reasoning (KR, KR², KR&R) 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.

## 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.

## Kripke–Platek set theory

The Kripke–Platek axioms of set theory (KP), pronounced, are a system of axiomatic set theory developed by Saul Kripke and Richard Platek.

## Kurt Gödel

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

## L. E. J. Brouwer

Luitzen Egbertus Jan Brouwer (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, who worked in topology, set theory, measure theory and complex analysis.

## 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.

## Large cardinal

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

## Lattice (order)

A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.

## Law of excluded middle

In logic, the law of excluded middle (or the principle of excluded middle) states that for any proposition, either that proposition is true or its negation is true.

## Löb's theorem

In mathematical logic, Löb's theorem states that in any formal system F with Peano arithmetic (PA), for any formula P, if it is provable in F that "if P is provable in F then P is true", then P is provable in F. More formally, if Bew(#P) means that the formula P with Gödel number #P is provable (from the German "beweisbar"), then or An immediate corollary of Löb's theorem is that, if P is not provable in PA, then "if P is provable in PA, then P is true" is not provable in PA.

## 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.

## Leopold Kronecker

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

## 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.

## 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.

## 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.

## List of computability and complexity topics

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

## List of first-order theories

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

## List of logic symbols

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

## Logic

Logic (from the logikḗ), originally meaning "the word" or "what is spoken", but coming to mean "thought" or "reason", is a subject concerned with the most general laws of truth, and is now generally held to consist of the systematic study of the form of valid inference.

## Logic in China

Formal logic in China has a special place in the history of logic due to its repression and abandonment—in contrast to the strong ancient adoption and continued development of the study of logic in Europe, India, and the Islamic world.

## Logic in computer science

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

## Logic in Islamic philosophy

Early Islamic law placed importance on formulating standards of argument, which gave rise to a "novel approach to logic" (منطق manṭiq "speech, eloquence") in Kalam (Islamic scholasticism) However, with the rise of the Mu'tazili philosophers, who highly valued Aristotle's Organon, this approach was displaced by the older ideas from Hellenistic philosophy, The works of al-Farabi, Avicenna, al-Ghazali and other Persian Muslim logicians who often criticized and corrected Aristotelian logic and introduced their own forms of logic, also played a central role in the subsequent development of European logic during the Renaissance.

## Logic programming

Logic programming is a type of programming paradigm which is largely based on formal logic.

## Logical consequence

Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically follows from one or more statements.

## 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)).

## Martin Davis

Martin David Davis (born March 8, 1928) is an American mathematician, known for his work on Hilbert's tenth problem.

## Mathematical analysis

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

## Mathematical induction

Mathematical induction is a mathematical proof technique.

## Mathematical proof

In mathematics, a proof is an inferential argument for a mathematical statement.

## Mathematics

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

## 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.

## Metamathematics

Metamathematics is the study of mathematics itself using mathematical methods.

## Michael D. Morley

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

## 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.

## 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.

## 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.

## 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.

## Morse–Kelley set theory

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

## 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.

## Natural number

In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").

## 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.

## New Year

New Year is the time or day at which a new calendar year begins and the calendar's year count increments by one.

## New Year's Day

New Year's Day, also called simply New Year's or New Year, is observed on January 1, the first day of the year on the modern Gregorian calendar as well as the Julian calendar.

## New Year's Eve

In the Gregorian calendar, New Year's Eve (also known as Old Year's Day or Saint Sylvester's Day in many countries), the last day of the year, is on 31 December which is the seventh day of Christmastide.

## New York City

The City of New York, often called New York City (NYC) or simply New York, is the most populous city in the United States.

## 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.

## Nikolai Lobachevsky

Nikolai Ivanovich Lobachevsky (a; –) was a Russian mathematician and geometer, known primarily for his work on hyperbolic geometry, otherwise known as Lobachevskian geometry and also his fundamental study on Dirichlet integrals known as Lobachevsky integral formula.

## Non-classical logic

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

## Non-Euclidean geometry

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

## 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.

## NP (complexity)

In computational complexity theory, NP (for nondeterministic polynomial time) is a complexity class used to describe certain types of decision problems.

## O-minimal theory

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

## 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.

## Ordinal analysis

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

## Ordinal number

In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another.

## Oskar Morgenstern

Oskar Morgenstern (January 24, 1902 – July 26, 1977) was a German-born economist.

## Oxford University Press

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

## 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.

## 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.

## Paul Bernays

Paul Isaac Bernays (17 October 1888 – 18 September 1977) was a Swiss mathematician, who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics.

## Paul Cohen

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

## Peano axioms

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

## Philosophy

Philosophy (from Greek φιλοσοφία, philosophia, literally "love of wisdom") is the study of general and fundamental problems concerning matters such as existence, knowledge, values, reason, mind, and language.

## Philotheus Boehner

Philotheus Boehner (born Heinrich Boehner; February 17, 1901 &ndash; May 22, 1955) was a member of the Franciscan order known for medieval scholarship.

## 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.

## Power set

In mathematics, the power set (or powerset) of any set is the set of all subsets of, including the empty set and itself, variously denoted as, 𝒫(), ℘() (using the "Weierstrass p"),,, or, identifying the powerset of with the set of all functions from to a given set of two elements,.

## 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).

## Principia Mathematica

The Principia Mathematica (often abbreviated PM) 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.

## Programming language

A programming language is a formal language that specifies a set of instructions that can be used to produce various kinds of output.

## 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.

## Proof theory

Proof theory is a major branchAccording to Wang (1981), pp.

## Propositional calculus

Propositional calculus is a branch of logic.

## Pyotr Novikov

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

## Quantifier (logic)

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

## Quantifier elimination

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

## RAND Corporation

RAND Corporation ("Research ANd Development") is an American nonprofit global policy think tank created in 1948 by Douglas Aircraft Company to offer research and analysis to the United States Armed Forces.

## Rózsa Péter

Rózsa Péter, born Politzer, (17 February 1905 &ndash; 16 February 1977) was a Hungarian mathematician and logician.

## Real analysis

In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real-valued functions.

## 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.

## Real line

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

## 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).

## Recursive set

In computability theory, a set of natural numbers is called recursive, computable or decidable if there is an algorithm which takes a number as input, terminates after a finite amount of time (possibly depending on the given number) and correctly decides whether the number belongs to the set.

## 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.

## Reverse mathematics

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

## Rhetoric

Rhetoric is the art of discourse, wherein a writer or speaker strives to inform, persuade, or motivate particular audiences in specific situations.

## 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), axiomatic foundation for the natural numbers, algebraic number theory and the definition of the real numbers.

Richard Swineshead (also Suisset, Suiseth, etc.; fl. c. 1340 – 1354) was an English mathematician, logician, and natural philosopher.

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

## Robert Lawson Vaught

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

## Rudolf Carnap

Rudolf Carnap (May 18, 1891 – September 14, 1970) was a German-born philosopher who was active in Europe before 1935 and in the United States thereafter.

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 naïve set theory created by Georg Cantor led to a contradiction.

## Saunders Mac Lane

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

## 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.

## Semantics

Semantics (from σημαντικός sēmantikós, "significant") is the linguistic and philosophical study of meaning, in language, programming languages, formal logics, and semiotics.

## Semantics (computer science)

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

## 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.

## Set (mathematics)

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

## Set theory

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

## Signature (logic)

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

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

## Solomon Feferman

Solomon Feferman (December 13, 1928 – July 26, 2016) was an American philosopher and mathematician with works in mathematical logic.

Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

## 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.

## Stanisław Leśniewski

Stanisław Leśniewski (March 30, 1886 – May 13, 1939) was a Polish mathematician, philosopher and logician.

## 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.

## Stephen Cole Kleene

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

## Stewart Shapiro

Stewart Shapiro (born 1951) is O'Donnell Professor of Philosophy at the Ohio State University.

## 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.

## Successor function

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

## 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.

## Syntax

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

## Theoretical computer science

Theoretical computer science, or TCS, is a subset of general computer science and mathematics that focuses on more mathematical topics of computing and includes the theory of computation.

## Theory (mathematical logic)

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

## Thoralf Skolem

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

Tibor Radó (June 2, 1895 &ndash; December 29, 1965) was a Hungarian mathematician who moved to the United States after World War I.

## Topos

In mathematics, a topos (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).

## 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.

## Transfinite induction

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

## 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.

## 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.

## 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.

## Turing machine

A Turing machine is a mathematical model of computation that defines an abstract machine, which manipulates symbols on a strip of tape according to a table of rules.

## 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.

## Ulrich Kohlenbach

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

## Universal algebra

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

## 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.

## Vaught conjecture

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

## 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 Zermelo–Fraenkel set theory (ZFC).

## W. Hugh Woodin

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

## Weierstrass function

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

## Well-order

In mathematics, a well-order (or well-ordering or well-order relation) 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.

## Wilfrid Hodges

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

## 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.

## World Scientific

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

## Yuri Matiyasevich

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

## 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.

## Zermelo–Fraenkel set theory

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

## (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.

## 2018

2018 has been designated as the third International Year of the Reef by the International Coral Reef Initiative.

## 2019

2019 (MMXIX) will be a common year starting on Tuesday of the Gregorian calendar, the 2019th year of the Common Era (CE) and Anno Domini (AD) designations, the 19th year of the 3rd millennium, the 19th year of the 21st century, and the 10th and last year of the 2010s decade.

## References

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