NationMaster - Encyclopedia: Alfred Tarski

Alfred Tarski (January 14, 1902, Warsaw, Russian-ruled Poland – October 26, 1983, Berkeley, California) was a logician and mathematician who spent four decades as a professor of mathematics at the University of California, Berkeley. A member of the interwar Warsaw School of Mathematics, and active in the USA after 1939, he wrote on topology, geometry, measure theory, mathematical logic, set theory, metamathematics, and above all, model theory, abstract algebra, and algebraic logic. His biographers Anita Feferman and Solomon Feferman state (2004) that, "Along with his contemporary, Kurt Gödel, he changed the face of logic in the twentieth century, especially through his work on the concept of truth and the theory of models." January 14 is the 14th day of the year in the Gregorian calendar. ... 1902 (MCMII) was a common year starting on Wednesday (see link for calendar). ... Warsaw (Polish: , , in full The Capital City of Warsaw, Polish: ) is the capital of Poland, its largest city, and a gamma world city. ... October 26 is the 299th day of the year (300th in leap years) in the Gregorian Calendar, with 66 days remaining. ... 1983 (MCMLXXXIII) was a common year starting on Saturday of the Gregorian calendar. ... Berkeley is a city on the east shore of San Francisco Bay in northern California, in the United States. ... The University of California, Berkeley (also known as UC Berkeley, Berkeley, Cal, and by other names, see below) is the oldest and flagship campus of the ten-campus University of California system. ... The Warsaw School of Mathematics describes a group of mathematicians working in logic, set theory, point-set topology and real analysis in the 1920s and 1930s. ... A MÃ¶bius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ... Table of Geometry, from the 1728 Cyclopaedia. ... In mathematics, a measure is a function that assigns a number, e. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In general, metamathematics or meta-mathematics is reflection about mathematics seen as an entity/object in human consciousness and culture. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... Algebraic logic has at least two meanings: the early study of Boolean algebra; and abstract algebraic logic, a branch of contemporary mathematical logic. ... Solomon Feferman is a mathematician and philosopher at Stanford University. ... Kurt GÃ¶del (IPA: ) (April 28, 1906 Brno, then Austria-Hungary, now Czech Republic â€“ January 14, 1978 Princeton, New Jersey) was an Austrian logician, mathematician, and philosopher of mathematics One of the most significant logicians of all time, GÃ¶dels work has had immense impact upon scientific and philosophical... Logic, from Classical Greek Î»ÏŒÎ³Î¿Ï‚ logos (the word), is the study of patterns found in reasoning. ... For other uses, see Truth (disambiguation). ... Look up model in Wiktionary, the free dictionary. ...

Life

Alfred Tarski was born Alfred Teitelbaum (Polish spelling: "Tajtelbaum"), to parents who were Polish Jews in comfortable circumstances. Some of his brilliance is attributed to his mother, Rosa Prussak. Tarski first revealed his mathematical abilities while in secondary school, at Warsaw's Szkoła Mazowiecka. Nevertheless, when he entered the University of Warsaw in 1918, he intended to study biology. From the Middle Ages until the Holocaust, Jews were a significant part of the Polish population. ... Warsaw University (Polish Uniwersytet Warszawski) - the biggest and one of the most prestigious universities in Poland. ... This article or section does not cite its references or sources. ...

In 1919, Poland became a nation again for the first time since 1795, and Warsaw University became a Polish university for the first time in generations. Under the leadership of Jan Łukasiewicz, Stanisław Leśniewski and Wacław Sierpiński, the university immediately became a world leader in logic, foundational mathematics, the philosophy of mathematics, and analytic and linguistic philosophy. At Warsaw University, Tarski had a fateful encounter with Leśniewski, who discovered Tarski's genius and persuaded him to abandon biology for mathematics. Henceforth Tarski attended courses taught by Łukasiewicz, Sierpiński, Stefan Mazurkiewicz and Tadeusz Kotarbiński, and became the only person ever to complete a doctorate under Leśniewski's supervision. Tarski and Leśniewski soon grew cool to each other. In private correspondence, Leśniewski sometimes employed anti-semitic language when talking about Tarski. In later life, Tarski reserved his warmest praise for Kotarbiński. Year 1919 (MCMXIX) was a common year starting on Wednesday (link will display the full calendar). ... 1795 was a common year starting on Thursday (see link for calendar). ... Warsaw University (Polish: ) is one of the largest universities in Poland. ... // Jan Åukasiewicz (21 December 1878 - 13 February 1956) was a Polish mathematician born in Lemberg, Galicia, Austria-Hungary (now Lviv, Ukraine). ... Stanislaw Lesniewski (March 30, 1886â€“May 13, 1939) was a Polish mathematician, philosopher and logician. ... WacÅ‚aw Franciszek SierpiÅ„ski (March 14, 1882 â€” October 21, 1969), a Polish mathematician, was born and died in Warsaw. ... Warsaw University (Polish: ) is one of the largest universities in Poland. ... Stefan Mazurkiewicz (born September 25, 1888 in Warsaw, Poland - died June 19, 1945, Grodzisk Mazowiecki) was a Polish mathematician who worked in mathematical analysis, topology, and probability. ... Tadeusz KotarbiÅ„ski (b. ... Tadeusz KotarbiÅ„ski (b. ...

In 1923, Alfred Teitelbaum and his brother Wacław changed their surname to Tarski, a name they invented because it sounded Polish, was simple to spell and pronounce, and seemed unused. (Years later, Alfred met another Alfred Tarski in northern California.) The Tarski brothers also converted to Roman Catholicism, Poland's dominant religion. Alfred did so even though he was an avowed atheist, because he was about to complete his doctorate and correctly anticipated that it would be difficult for a Jew to obtain a serious position in the new Polish university system. Tarski was a Polish nationalist and wished to be fully accepted as a Pole, which was how he saw himself. He married a Pole of Catholic ancestry, and spoke Polish at home throughout his later American life. Official language(s) English Capital Sacramento Largest city Los Angeles Area Ranked 3rd - Total 158,302 sq mi (410,000 kmÂ²) - Width 250 miles (400 km) - Length 770 miles (1,240 km) - % water 4. ... The Roman Catholic Church, most often spoken of simply as the Catholic Church, is the largest Christian church, with over one billion members. ... The 18th-century French author Baron dHolbach was one of the first self-described atheists. ...

After becoming the youngest person ever to complete a Ph. D. at Warsaw University, Tarski taught logic at the Polish Pedagogical Institute, mathematics and logic at the University, and served as Łukasiewicz's assistant. Because these positions were poorly paid, Tarski also taught mathematics at a Warsaw secondary school; before World War II, it was not uncommon for European intellectuals of research caliber to teach high school. Hence between 1923 and his departure for the United States in 1939, Tarski not only wrote several textbooks and many papers, a number of them ground-breaking, but did so while supporting himself primarily by teaching high-school mathematics.

In 1929 Tarski married a fellow teacher, Maria Witkowska. She had worked as a courier for the army during Poland's fight for independence. They had two children.

Tarski applied for a chair of philosophy at Lwów University, but on Bertrand Russell's recommendation it was awarded to Leon Chwistek. In 1937 Tarski applied for a chair at Poznań University; but rather than award a chair to one of Jewish ancestry, the chair was abolished.^{[citation needed]} The building of the University. ... Bertrand Arthur William Russell, 3rd Earl Russell OM FRS (18 May 1872 â€“ 2 February 1970), was a British philosopher, logician, and mathematician. ... Leon Chwistek Leon Chwistek (b. ... The University of PoznaÅ„ (Polish: Uniwersytet im. ...

In 1930, Tarski visited the University of Vienna, lectured to Menger's colloquium, and met Kurt Gödel. Thanks to a fellowship, Tarski was able to return to Vienna during the first half of 1935 to work with Menger's research group. From Vienna he traveled to Paris to present his ideas on truth at the first meeting of the Unity of Science movement, an outgrowth of the Vienna Circle. Kurt GÃ¶del (IPA: ) (April 28, 1906 Brno, then Austria-Hungary, now Czech Republic â€“ January 14, 1978 Princeton, New Jersey) was an Austrian logician, mathematician, and philosopher of mathematics One of the most significant logicians of all time, GÃ¶dels work has had immense impact upon scientific and philosophical... It has been suggested that this article or section be merged with Unified Science. ... Moritz Schlick around 1930 The Vienna Circle (in German: der Wiener Kreis) was a group of philosophers who gathered around Moritz Schlick when he was called to the Vienna University in 1922, organized in a philosophical association named Verein Ernst Mach (Ernst Mach Society). ...

Tarski's ties to this movement saved his life, because they resulted in his being invited to address the Unity of Science Congress held in September 1939 at Harvard University. Thus he left Poland in August 1939, on the last ship to sail from Poland for the United States before the German invasion of Poland and the outbreak of World War II. Tarski left reluctantly, because Leśniewski had died a few months before, creating a vacancy which Tarski hoped to fill. He was so oblivious to the Nazi threat that he left his wife and children in Warsaw; he did not see them again until 1946. During the war, nearly all his extended family died at the hands of the Nazis. Harvard redirects here. ... Combatants Poland Germany, Soviet Union, Slovakia Commanders Edward Rydz-ÅšmigÅ‚y Fedor von Bock (Army Group North), Gerd von Rundstedt (Army Group South), Mikhail Kovalov (Belorussian Front), Semyon Timoshenko (Ukrainian Front), Ferdinand ÄŒatloÅ¡ (Field Army Bernolak) Strength 39 divisions, 16 brigades, 4,300 guns, 880 tanks, 400 aircraft, Total: 950... Combatants Allied Powers Axis Powers Casualties Military dead: 17,000,000 Civilian dead: 33,000,000 Total dead: 50,000,000 Military dead: 8,000,000 Civilian dead: 4,000,000 Total dead 12,000,000 World War II (abbreviated WWII), or the Second World War, was a worldwide conflict...

Once in the United States, Tarski held a number of temporary teaching and research positions: Harvard University (1939), City College of New York (1940), and thanks to a Guggenheim Fellowship, the Institute for Advanced Study at Princeton (1942), where he again met Gödel. Tarski became an American citizen in 1945. Harvard redirects here. ... The City College of The City University of New York (known more commonly as City College of New York or simply City College, CCNY, or colloquially as City)[1] is a senior college of the City University of New York, in New York City. ... Fuld Hall The Institute for Advanced Study is a private institution in Princeton Township, New Jersey, U.S.A. (although it is not part of Princeton University), designed to foster pure cutting-edge research by scientists in a variety of fields without the complications of teaching or funding, or the...

In 1942, Tarski joined the Mathematics Department at the University of California, Berkeley, where he spent the rest of his career. Although emeritus from 1968, he taught until 1973 and supervised Ph. D. candidates until his death. At Berkeley, Tarski acquired a reputation as an awesome and demanding teacher: The University of California, Berkeley (also known as UC Berkeley, Berkeley, Cal, and by other names, see below) is the oldest and flagship campus of the ten-campus University of California system. ...

Tarski supervised 24 Ph. D. dissertations, 5 by women, and strongly influenced the dissertations of Alfred Lindenbaum, Dana Scott, and Steven Givant. His students include Andrzej Mostowski, Julia Robinson, Robert Vaught, Solomon Feferman, Richard Montague, J. Donald Monk, Donald Pigozzi, Roger Maddux, and the authors of Chang and Jerome Keisler (1973), the classic text on model theory.. Dana Stewart Scott (born 1932) is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, California. ... Andrzej Mostowski (1 November 1913 - 22 August 1975) was a Polish mathematician. ... Julia Hall Bowman Robinson (December 8, 1919 - July 30, 1985) was an American mathematician, born in Saint Louis, Missouri. ... Robert Lawson Vaught (April 4, 1926 Alhambra, California - April 2, 2002) was a mathematical logician, and one of the founders of model theory. ... Solomon Feferman is a mathematician and philosopher at Stanford University. ... Richard Merett Montague (1930â€“1971) was an American mathematician and philosopher. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ...

Tarski lectured at University College, London (1950, 1966), the Institut Henri Poincaré in Paris (1955), the Miller Institute for Basic Research in Science in Berkeley (1958-1960), the University of California at Los Angeles (1967), and the Pontifical Catholic University of Chile (1974-75). He was elected to the National Academy of Sciences and the British Academy, and presided over the Association for Symbolic Logic, 1944-46, and the International Union for the History and Philosophy of Science, 1956-57. The Front Quad University College London, commonly known as UCL, is one of the colleges that make up the University of London. ... The University of California, Los Angeles, popularly known as UCLA, is a public, coeducational university situated in the neighborhood of Westwood within the city of Los Angeles. ... PUC from Cerro Santa LucÃa Inside PUC San Joaquin Campus Pontificia Universidad CatÃ³lica de Chile (PUC) (Spanish Pontifical Catholic University of Chile) is one of Chiles oldest and most prestigious universities. ... President Harding and the National Academy of Sciences at the White House, Washington, DC, April 1921 The National Academy of Sciences (NAS) is a corporation in the United States whose members serve pro bono as advisers to the nation on science, engineering, and medicine. ... The British Academy is the United Kingdoms national academy for the humanities and the social sciences. ... The Association for Symbolic Logic (ASL) is an international organization of specialists in symbolic logicâ€”the largest such organization in the world. ...

Mathematician

Tarski's mathematical interests were exceptionally broad for a mathematical logician. His collected papers run to about 2500 pages, most of them on mathematics, not logic. For a concise survey of Tarski's mathematical and logical accomplishments by his former student Solomon Feferman, see "Interludes I-VI" in Feferman and Feferman (2004). To meet Wikipedias quality standards, this article or section may require cleanup. ... Solomon Feferman is a mathematician and philosopher at Stanford University. ...

Tarski's first paper, published when he was only 19 years old, was on set theory, a subject to which he returned throughout his life. In 1924, he and Stefan Banach proved that a sphere can be cut into a finite number of pieces, and then reassembled into a sphere of larger size, or alternatively it can be reassembled into two spheres whose sizes each equal that of the original one. This result is now called the Banach-Tarski paradox. "Paradox" here means "counterintuitive result." Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Stefan Banach Stefan Banach (March 30, 1892 in KrakÃ³w, Austria-Hungary now Polandâ€“ August 31, 1945 in LwÃ³w, Soviet Union - occupied Poland), was an eminent Polish mathematician, one of the moving spirits of the LwÃ³w School of Mathematics in pre-war Poland. ... A sphere is a perfectly symmetrical geometrical object. ... The Banachâ€“Tarski paradox: A ball can be decomposed and reassembled into two balls the same size as the original. ...

In A decision method for elementary algebra and geometry, Tarski showed, by the method of quantifier elimination, that the first-order theory of the real numbers under addition and multiplication is decidable. This is a very curious result, because Alonzo Church proved in 1936 that Peano arithmetic (effectively the theory Tarski proved decidable, except that the natural numbers replace the reals) is not decidable. Peano arithmetic is also incompletable by Gödel's incompleteness theorem. In his 1953 Undecidable theories, Tarski et al. showed that many mathematical systems, including lattice theory, abstract projective geometry, and closure algebras, are all undecidable. The theory of Abelian groups is decidable, but that of non-Abelian groups is not. Quantifier elimination is a technique in logic, model theory, and theoretical computer science. ... First-order predicate calculus or first-order logic (FOL) is a theory in symbolic logic that permits the formulation of quantified statements such as there is at least one X such that. ... In mathematics, the real numbers may be described informally in several different ways. ... A logical system is decidable iff there exists an algorithm such that for every well-formed formula in that system there exists a maximum finite number N of steps such that the algorithm is capable of deciding in less than or equal to N algorithmic steps whether the formula is... Alonzo Church (June 14, 1903 â€“ August 11, 1995) was an American mathematician and logician who was responsible for some of the foundations of theoretical computer science. ... In mathematics, the Peano axioms (or Peano postulates) are a set of first-order axioms proposed by Giuseppe Peano which determine the theory of Peano arithmetic (also known as first-order arithmetic). ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematical logic, GÃ¶dels incompleteness theorems are two celebrated theorems proven by Kurt GÃ¶del in 1931. ... See lattice for other mathematical as well as non-mathematical meanings of the term. ... Projective geometry is a non-metrical form of geometry that emerged in the early 19th century. ... In abstract algebra, an interior algebra is an algebraic structure of the signature <A, ·, +, , 0, 1, I> where <A, ·, +, , 0, 1> is a Boolean algebra and I is a unary operator, the interior operator, satisfying the identities: xI ≤ x xII = xI (xy)I = xIyI 1I = 1 xI is called... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... Group theory is that branch of mathematics concerned with the study of groups. ...

In the 1920s and 30s, Tarski often taught high school geometry. In 1929, he showed that much of Euclidian solid geometry could be recast as a first order theory whose individuals are spheres, a primitive notion, a single primitive binary relation "is contained in," and two axioms that, among other thing, imply that containment partially orders the spheres. Relaxing the requirement that all individuals be spheres yields a formalization of mereology far easier to exposit that Lesniewski's variant. Starting in 1926, Tarski devised an original axiomatization for plane Euclidian geometry, one considerably more concise than Hilbert's. Tarski's axiomatization is a first-order theory devoid of set theory, whose individuals are points, and having only two primitive relations. In 1930, he proved this theory decidable because it can be mapped into another theory he had already proved decidable, namely his first-order theory of the real numbers. Near the end of his life, Tarski wrote a very long letter, published as Tarski and Givant (1999), summarizing his work on geometry. Table of Geometry, from the 1728 Cyclopaedia. ... In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... Mereology is a collection of axiomatic formal systems dealing with parts and their respective wholes. ... Year 1926 (MCMXXVI) was a common year starting on Friday (link will display the full calendar). ... Tarskis axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry, called elementary, that is formulable in first order logic with identity, and requiring no set theory. ... In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ... Hilberts axioms are a set of 20 assumptions (originally 21), David Hilbert proposed in 1899 as the foundation for a modern treatment of Euclidean geometry. ... First-order predicate calculus or first-order logic (FOL) is a theory in symbolic logic that permits the formulation of quantified statements such as there is at least one X such that. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Point can refer to: Look up Point in Wiktionary, the free dictionary // Mathematics In mathematics: Point (geometry), an entity that has a location in space but no extent Fixed point (mathematics), a point that is mapped to itself by a mathematical function Point at infinity Point group Point charge, an... In mathematics, an n-ary relation (or often simply relation) is a generalization of binary relations such as = and < which occur in statements such as 5 < 6 or 2 + 2 = 4. It is the fundamental notion in the relational model for databases. ... The word decidable has formal meaning in computability theory, the theory of formal languages, and mathematical logic. ...

Cardinal Algebras studied algebras whose models include the arithmetic of cardinal numbers. Ordinal Algebras sets out an algebra for the additive theory of order types. Cardinal, but not ordinal, addition commutes. Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set. ... In mathematics, especially in set theory, ordinals may be used to label the elements of any given well-ordered set (the smallest element being labeled 0, the one after that 1, the next one 2, and so on) and to measure the length of the whole set by the least...

In 1941, Tarski published an important paper on binary relations, which began the work on relation algebra and its metamathematics that occupied Tarski and his students for much of the balance of his life. While that exploration (and the closely related work of Roger Lyndon) uncovered some important limitations of relation algebra, Tarski also showed (Tarski and Givant 1987) that relation algebra can express most axiomatic set theory and Peano arithmetic. For an introduction to relation algebra, see Maddux (2006). In the late 1940s, Tarski and his students devised cylindric algebras, which are to first-order logic what the two-element Boolean algebra is to classical sentential logic. This work culminated in the two monographs by Tarski, Henkin, and Monk (1971, 1985). In mathematics, a binary relation (or a dyadic relation) is an arbitrary association of elements of one set with elements of another (perhaps the same) set. ... In mathematics, relation algebra (RA) is an algebraic structure, a proper extension of the two-element Boolean algebra 2 intended to capture the mathematical properties of binary relations. ... In general, metamathematics or meta-mathematics is reflection about mathematics seen as an entity/object in human consciousness and culture. ... This article or section is in need of attention from an expert on the subject. ... In mathematics, the Peano axioms (or Peano postulates) are a set of first-order axioms proposed by Giuseppe Peano which determine the theory of Peano arithmetic (also known as first-order arithmetic). ... In mathematics, relation algebra (RA) is an algebraic structure, a proper extension of the two-element Boolean algebra 2 intended to capture the mathematical properties of binary relations. ... The notion of cylindric algebra, invented by Alfred Tarski, arises naturally in the algebraization of first-order logic. ... First-order logic (FOL) is a universal language in symbolic science, and is in use everyday by mathematicians, philosophers, linguists, computer scientists and practitioners of artificial intelligence. ... -1... A propositional calculus is a formal, deduction system, or proof theory for reasoning with propositional formulas as symbolic logic. ...

Logician

Along with Aristotle, Gottlob Frege, and Kurt Gödel, Tarski is generally considered one of the four greatest logicians of all time (Vaught 1986). Of these four, he was the best mathematician and the most prolific author. Neither Frege nor Gödel ever supervised a single Ph. D. or coauthored a paper; Tarski, on the other hand, supervised 24 Ph. Ds and coauthored over 100 books and papers. Frege was sternly aloof in person and often bitingly sarcastic in print, and Gödel was a notorious recluse. Meanwhile, Tarski loved to interact with people intellectually and socially. Aristotle (Greece: AristotÃ©lÄ“s) (384 BC â€“ March 7, 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great. ... Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar â€“ 26 July 1925, Bad Kleinen, IPA: ) was a German mathematician who became a logician and philosopher. ... Kurt GÃ¶del (IPA: ) (April 28, 1906 Brno, then Austria-Hungary, now Czech Republic â€“ January 14, 1978 Princeton, New Jersey) was an Austrian logician, mathematician, and philosopher of mathematics One of the most significant logicians of all time, GÃ¶dels work has had immense impact upon scientific and philosophical...

Tarski produced axioms for logical consequence, and worked on deductive systems, the algebra of logic, and the theory of definability. His semantic methods, which culminated in the model theory he and a number of his Berkeley students developed in the 1950s and 60s, radically transformed Hilbert's proof-theoretic metamathematics. In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ...

All formal scientific languages can be studied by model theory and related semantic methods. In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ...

Tarski's 1936 article "On the concept of logical consequence" argued that the conclusion of an argument will follow logically from its premises if and only if every model of the premises is a model of the conclusion. In 1937, he published a paper presenting clearly his views on the nature and purpose of the deductive method, and the role of logic in scientific studies. His high school and undergraduate teaching on logic and axiomatics culminated in a classic short text, published first in Polish, then in German translation, and finally in a 1941 English translation as Introduction to Logic and to the Methodology of Deductive Sciences.

Tarski's 1969 "Truth and proof" considered both Gödel's incompleteness theorems and Tarski's indefinability theorem, and mulled over their consequences for the axiomatic method in mathematics. In mathematical logic, GÃ¶dels incompleteness theorems, proved by Kurt GÃ¶del in 1931, are two celebrated theorems stating inherent limitations of all but the most trivial formal systems for arithmetic of mathematical interest. ... In mathematical logic, Tarskis Indefinability Theorem is a theorem due to Alfred Tarski concerning the foundations of mathematics. ...

Truth in formalized languages

In 1933, Tarski published a very long (more than 100pp) paper in Polish, titled "Pojęcie prawdy w językach nauk dedukcyjnych," setting out a mathematical definition of truth for formal languages. The 1936 German translation was titled "Der Wahrheitsbegriff in den Sprachen der deduktiven Disziplinen," sometimes shortened to "Wahrheitsbegriff." An English translation had to await the 1956 first edition of the volume Logic, Semantics, Metamathematics. This enormously cited paper is a landmark event in 20th century analytic philosophy, an important contribution to symbolic logic, semantics, and the philosophy of language. For a brief discussion of its content, see Truth for a brief description of the "Convention T" (see also T-schema) standard in Tarski's "inductive definition of truth". Analytic philosophy is a generic term for a style of philosophy that came to prominence during the 20th Century. ... Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ... Semantics (Greek semantikos, giving signs, significant, symptomatic, from sema, sign) refers to the aspects of meaning that are expressed in a language, code, or other form of representation. ... Philosophy of language is the reasoned inquiry into the nature, origins, and usage of language. ... For other uses, see Truth (disambiguation). ... Convention T is the inductive definition that lies at the heart of any realisation of Alfred Tarskis semantic theory of truth, expressing the commutation of truth over logical operators. ...

Some recent philosophical debate examines the extent to which Tarski's theory of truth for formalized languages can be seen as a correspondence theory of truth. The debate centres on how to read Tarski's condition of material adequacy for a truth definition. That condition requires that the truth theory have the following as theorems for all sentences P of the language for which truth is being defined: The correspondence theory of truth states that something (for example, a proposition or statement or sentence) is rendered true by the existence of a fact with corresponding elements and a similar structure. ...

as expressing merely a deflationary theory of truth or as embodying truth as a more substantial property (see Kirkham 1992). The deflationary theory of truth is a family of theories which all have in common the belief that assertions that predicate truth of a statement do not provide any substantive information or insight into the nature of truth. ... For other uses, see Truth (disambiguation). ...

Logical consequence

In 1936, Tarski published Polish and German versions of a lecture he had given the preceding year at the International Congress of Scientific Philosophy in Paris. A new English translation of this paper, Tarski (2002), highlights the many differences between the German and Polish versions of the paper, and corrects a number of mistranslations in Tarski (1983).

This publication set out the modern model-theoretic definition of (semantic) logical consequence, or at least the basis for it. Whether Tarski's notion was entirely the modern one turns on whether he intended to admit models with varying domains (and in particular, models with domains of different cardinalities). This question is a matter of some debate in the current philosophical literature. John Etchemendy (1999) stimulated much of the recent discussion about Tarski's treatment of varying domains. In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set. ... John W. Etchemendy is Stanford Universitys twelfth and current Provost. ...

Tarski ends by pointing out that his definition of logical consequence depends upon a division of terms into the logical and the extra-logical and he expresses some skepticism that any such objective division will be forthcoming. "What are Logical Notions?" can thus be viewed as continuing "On the Concept of Logical Consequence."

What are logical notions?

Another theory of Tarski's attracting attention in the recent philosophical literature is that outlined in his "What are Logical Notions?" (Tarski 1986). This is the published version of a talk that he gave in 1966; it was edited without his direct involvement.

In the talk, Tarski proposed a demarcation of the logical operations (which he calls "notions") from the non-logical. The suggested criteria was derived from the Erlangen programme of the German 19th century Mathematician, Felix Klein. (Mautner 1946, and possibly an article by the Italian mathematician Silva, anticipated Tarski in applying the Erlangen Program to logic.) An influential research programme and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen. ... Felix Christian Klein (April 25, 1849, DÃ¼sseldorf, Germany â€“ June 22, 1925, GÃ¶ttingen) was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory. ...

That program classified the various types of geometry (Euclidean geometry, affine geometry, topology, etc.) by the type of one-one transformation of space onto itself that left the objects of that geometrical theory invariant. (A one-to-one transformation is a functional map of the space onto itself so that every point of the space is associated with or mapped to one other point of the space. So, "rotate 30 degrees" and "magnify by a factor of 2" are intuitive descriptions of simple uniform one-one transformations.) Continuous transformations give rise to the objects of topology, similarity transformations to those of Euclidean geometry, and so on. Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. ... In geometry, affine geometry occupies a place intermediate between Euclidean geometry and projective geometry. ... A MÃ¶bius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ...

As the range of permissible transformations becomes broader, the range of objects one is able to distinguish as preserved by the application of the transformations becomes narrower. Similarity transformations are fairly narrow (they preserve the relative distance between points) and thus allow us to distinguish relatively many things (e.g., equilateral triangles from non-equilateral triangles). Continuous transformations (which can intuitively be thought of as transformations which allow non-uniform stretching, compression, bending, and twisting, but no ripping or glueing) allow us to distinguish a polygon from an annulus (ring with a hole in the centre), but does not allow us to distinguish two polygons from each other. Look up polygon in Wiktionary, the free dictionary. ... An annulus In mathematics, an annulus (the Latin word for little ring, with plural annuli) is a ring-shaped geometric figure, or more generally, a term used to name a ring-shaped object. ...

Tarski's proposal was to demarcate the logical notions by considering all possible one-to-one transformations (automorphisms) of a domain onto itself. By domain is meant the universe of discourse of a model for the semantic theory of a logic. If one identifies the truth value True with the domain set and the truth-value False with the empty set, then the following operations are counted as logical under the proposal: In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... The term universe of discourse generally refers to the entire set of terms used in a specific discourse, i. ...

1. Truth-functions: All truth-functions are admitted by the proposal. This includes, but is not limited to, all n-ary truth-functions for finite n. (It also admits of truth-functions with any infinite number of places.) In logic a truth function is a function generated from sentences of the language. ...

4. Quantifiers: Tarski explicitly discusses only monadic quantifiers and points out that all such numerical quantifiers are admitted under his proposal. These include the standard universal and existential quantifiers as well as numerical quantifiers such as "Exactly four", "Finitely many", "Uncountably many", and "Between four and 9 million", for example. While Tarski does not enter into the issue, it is also clear that polyadic quantifiers are admitted under the proposal. These are quantifiers like, given two predicates Fx and Gy, "More(x, y)", which says "More things have F than have G." Logical disjunction (usual symbol or) is a logical operator that results in true if either of the operands is true. ... Negation (i. ... In language and logic, quantification is a construct that specifies the extent of validity of a predicate, that is the extent to which a predicate holds over a range of things. ...

5. Set-Theoretic relations: Relations such as inclusion, intersection and union applied to subsets of the domain are logical in the present sense. In mathematics, inclusion is a partial order on sets. ... The term intersection can mean: a road junction, where two roads intersect each other, such as a roundabout intersection; in mathematics, the set in which two or more other sets intersect each other; see intersection (set theory); a movie; see Intersection (movie). ... Union generally refers to two or more things joined into one, such as an organization of multiple people or organizations, multiple objects combined into one, and so on. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ...

6. Set membership: Tarski ended his lecture with a discussion of whether the set membership relation counted as logical in his sense. (Given the reduction of (most of) mathematics to set theory, this was, in effect, the question of whether most or all of mathematics is a part of logic.) He pointed out that set membership is logical if set theory is developed along the lines of type theory, but is extralogical if set theory is set out axiomatically, as in the canonical Zermelo-Fraenkel set theory. At the broadest level, type theory is the branch of mathematics and logic that concerns itself with classifying entities into collections called types. ... Zermelo-Fraenkel set theory, commonly abbreviated ZFC, is the most common form of axiomatic set theory, and as such is the most common foundation of mathematics. ...

7. Logical notions of higher order: While Tarski confined his discussion to operations of first-order logic, there is nothing about his proposal that necessarily restricts it to first-order logic. (Tarski likely restricted his attention to first-order notions as the talk was given to a non-technical audience.) So, higher-order quantifiers and predicates are admitted as well.

In some ways the present proposal is the obverse of that of Lindenbaum and Tarski (1936), who proved that all the logical operations of Russell and Whitehead's Principia Mathematica are invariant under one-to-one transformations of the domain onto itself. The present proposal is also employed in Tarski and Givant (1987). Bertrand Arthur William Russell, 3rd Earl Russell OM FRS (18 May 1872 â€“ 2 February 1970), was a British philosopher, logician, and mathematician. ... Alfred North Whitehead, OM (February 15, 1861 Ramsgate, Kent, England â€“ December 30, 1947 Cambridge, Massachusetts, USA) was an English-born mathematician who became a philosopher. ... The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910-1913. ...

Solomon Feferman and Vann McGee further discussed Tarski's proposal in work published after his death. Feferman (1999) raises problems for the proposal and suggests a cure: replacing Tarski's preservation by automorphisms with preservation by arbitrary homomorphisms. In essence, this suggestion circumvents the difficulty Tarski's proposal has in dealing with sameness of logical operation across distinct domains of a given cardinality and across domains of distinct cardinalities. Feferman's proposal results in a radical restriction of logical terms as compared to Tarski's original proposal. In particular, it ends up counting as logical only those operators of standard first-order logic without identity. Solomon Feferman is a mathematician and philosopher at Stanford University. ... In abstract algebra, a homomorphism is a structure-preserving map. ...

McGee (1996) provides a precise account of what operations are logical in the sense of Tarski's proposal in terms of expressibility in a language that extends first-order logic by allowing arbitrarily long conjunctions and disjunctions, and quantification over arbitrarily many variables. "Arbitrarily" includes a countable infinity.

Bibliography

Many of Tarski's more important papers written during his Polish years in languages other than English, including "The Concept of Truth in Formalized Languages" and "On the Concept of Logical Consequence" discussed above, are translated in the important collection:

Biographical: Leon Henkin is a logician, currently Emeritus Professor at the University of California at Berkeley. ... Leon Henkin is a logician, currently Emeritus Professor at the University of California at Berkeley. ...

Secondary: Solomon Feferman is a mathematician and philosopher at Stanford University. ...

The December 1986 issue of the Journal of Symbolic Logic surveys Tarski's work on model theory (Robert Vaught), algebra (Jonsson), undecidable theories (McNulty), algebraic logic (Donald Monk), and geometry (Szczerba). The March 1988 issue of the same journal surveys his work on axiomatic set theory (Azriel Levy), real closed fields (Lou Van Den Dries), decidable theory (Doner and Wilfrid Hodges), metamathematics (Blok and Pigozzi), truth and logical consequence (John Etchemendy), and general philosophy (Patrick Suppes). In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ... Robert Lawson Vaught (April 4, 1926 Alhambra, California - April 2, 2002) was a mathematical logician, and one of the founders of model theory. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... A logical system is decidable iff there exists an algorithm such that for every well-formed formula in that system there exists a maximum finite number N of steps such that the algorithm is capable of deciding in less than or equal to N algorithmic steps whether the formula is... Algebraic logic has at least two meanings: the early study of Boolean algebra; and abstract algebraic logic, a branch of contemporary mathematical logic. ... Table of Geometry, from the 1728 Cyclopaedia. ... This article or section is in need of attention from an expert on the subject. ... In mathematics, a real closed field is an ordered field F in which any of the following equivalent conditions are true: Every non-negative element of F has a square root in F, and any polynomial of odd degree with coefficients in F has at least one root in F... A logical system is decidable iff there exists an algorithm such that for every well-formed formula in that system there exists a maximum finite number N of steps such that the algorithm is capable of deciding in less than or equal to N algorithmic steps whether the formula is... Wilfrid Hodges (born 1941) is a British mathematician, known for his work in model theory. ... In general, metamathematics or meta-mathematics is reflection about mathematics seen as an entity/object in human consciousness and culture. ... For other uses, see Truth (disambiguation). ... Logical consequence is the relation that holds between a set of sentences and a sentence when the latter follows from the former. ... John W. Etchemendy is Stanford Universitys twelfth and current Provost. ... For other uses, see Philosophy (disambiguation). ...

Other references: Ivor Grattan-Guiness is a prolific contemporary historian of mathematics and logic. ... Sir Karl Raimund Popper, CH, MA, Ph. ...

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