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List Of Things Named After Alfred Tarski
{{Short description, none In the history of mathematics, Alfred Tarski (1901–1983) is one of the most important logicians. His name is now associated with a number of theorems and concepts in that field. Theorems * Łoś–Tarski preservation theorem * Knaster–Tarski theorem (sometimes referred to as Tarski's fixed point theorem) * Tarski's undefinability theorem * Tarski–Seidenberg theorem * Some fixed point theorems, usually variants of the Kleene fixed-point theorem, are referred to the Tarski–Kantorovitch fixed–point principle or the Tarski–Kantorovitch theorem although the use of this terminology is limited. * Tarski's theorem Other mathematics-related work * Bernays-Tarski axiom system * Banach–Tarski paradox * Lindenbaum–Tarski algebra * Łukasiewicz-Tarski logic * Jónsson–Tarski duality * Jónsson–Tarski algebra * Gödel–McKinsey–Tarski translation * The semantic theory of truth is sometimes referred to as Tarski's definition of truth or Ta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
History Of Mathematics
The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer, Akkad and Assyria, followed closely by Ancient Egypt and the Levantine state of Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the patterns in nature, the field of astronomy and to record time and formulate calendars. The earliest mathematical texts available are from Mesopotamia and Egypt – '' Plimpton 322'' ( Babylonian c. 2000 – 1900 BC), the ''Rhind Mathematical Papyrus'' ( Egyptian c. 1800 BC) and the '' Moscow Mathematical Papyrus'' (Egyptian c. 1890 BC). All of these texts mention the so-called Pythagorean triples, so, by inference, the Pythagorean theorem seems to be the mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gödel–McKinsey–Tarski Translation
In logic, a modal companion of a superintuitionistic (intermediate) logic ''L'' is a normal modal logic that interprets ''L'' by a certain canonical translation, described below. Modal companions share various properties of the original intermediate logic, which enables to study intermediate logics using tools developed for modal logic. Gödel–McKinsey–Tarski translation Let ''A'' be a propositional intuitionistic formula. A modal formula ''T''(''A'') is defined by induction on the complexity of ''A'': :T(p)=\Box p, for any propositional variable p, :T(\bot)=\bot, :T(A\land B)=T(A)\land T(B), :T(A\lor B)=T(A)\lor T(B), :T(A\to B)=\Box(T(A)\to T(B)). As negation is in intuitionistic logic defined by A\to\bot, we also have :T(\neg A)=\Box\neg T(A). ''T'' is called the Gödel translation or Gödel–McKinsey– Tarski translation. The translation is sometimes presented in slightly different ways: for example, one may insert \Box before every subformula. All such variants are pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tarski's World
Tarski's World is a computer-based introduction to first-order logic written by Jon Barwise and John Etchemendy. It is named after the mathematical logician Alfred Tarski. The package includes a book, which serves as a textbook and manual, and a computer program which together serve as an introduction to the semantics of logic through games in which simple, three-dimensional worlds are populated with various geometric figures and these are used to test the truth or falsehood of first-order logic sentences. The program is also included in Language, Proof and Logic package. The programme was later extended into Hyperproof. The programme * Barwise, J., & Etchemendy, J. (1993). ''Tarski's world''. Stanford, Calif: CSLI Publ. * Barker-Plummer, D., Barwise, J., & Etchemendy, J. (2008). ''Tarski's world''. Stanford, Calif: CSLI Publications. The Openproof Project at CSLI:home page of the Tarski's World courseware package, Dave Barker-Plummer, Jon Barwise and John Etchemendy in co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tarski–Vaught Test
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. If ''N'' is a substructure of ''M'', one often needs a stronger condition. In this case ''N'' is called an elementary substructure of ''M'' if every first-order ''σ''-formula ''φ''(''a''1, …, ''a''''n'') with parameters ''a''1, …, ''a''''n'' from ''N'' is true in ''N'' if and only if it is true in ''M''. If ''N'' is an elementary substructure of ''M'', then ''M'' is called an elementary extension of ''N''. An embedding ''h'': ''N'' → ''M'' is called an elementary embedding of ''N'' into ''M'' if ''h''(''N'') is an elementary substructure of ''M''. A substructure ''N'' of ''M'' is elementary if and only if it passes the Tarski–Vaught test: every first-order formula ''φ''(''x'', ''b''1, …, ''b''''n'') with pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Group
In mathematics, the free group ''F''''S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1''t'', but ''s'' ≠ ''t''−1 for ''s'',''t'',''u'' ∈ ''S''). The members of ''S'' are called generators of ''F''''S'', and the number of generators is the rank of the free group. An arbitrary group ''G'' is called free if it is isomorphic to ''F''''S'' for some subset ''S'' of ''G'', that is, if there is a subset ''S'' of ''G'' such that every element of ''G'' can be written in exactly one way as a product of finitely many elements of ''S'' and their inverses (disregarding trivial variations such as ''st'' = ''suu''−1''t''). A related but different notion is a free abelian group; both notions are particular instances of a free object from universal algebra. As such, free groups are defined by their universal property. History ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tarski's Plank Problem
In mathematics, Tarski's plank problem is a question about coverings of convex regions in ''n''-dimensional Euclidean space by "planks": regions between two hyperplanes. Tarski asked if the sum of the widths of the planks must be at least the minimum width of the convex region. The question was answered affirmatively by . Statement Given a convex body ''C'' in R''n'' and a hyperplane ''H'', the width of ''C'' parallel to ''H'', ''w''(''C'',''H''), is the distance between the two supporting hyperplanes of ''C'' that are parallel to ''H''. The smallest such distance (i.e. the infimum over all possible hyperplanes) is called the minimal width of ''C'', ''w''(''C''). The (closed) set of points ''P'' between two distinct, parallel hyperplanes in R''n'' is called a plank, and the distance between the two hyperplanes is called the width of the plank, ''w''(''P''). Tarski conjectured that if a convex body ''C'' of minimal width ''w''(''C'') was covered by a collection of planks, then ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tarski Monster Group
In the area of modern algebra known as group theory, a Tarski monster group, named for Alfred Tarski, is an infinite group ''G'', such that every proper subgroup ''H'' of ''G'', other than the identity subgroup, is a cyclic group of order a fixed prime number ''p''. A Tarski monster group is necessarily simple. It was shown by Alexander Yu. Olshanskii in 1979 that Tarski groups exist, and that there is a Tarski ''p''-group for every prime ''p'' > 1075. They are a source of counterexamples to conjectures in group theory, most importantly to Burnside's problem The Burnside problem asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. It was posed by William Burnside in 1902, making it one of the oldest questions in group theory and was infl ... and the von Neumann conjecture. Definition Let p be a fixed prime number. An infinite group G is called a Tarski monster group for p if every nontrivial subgroup ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tarski–Kuratowski Algorithm
In computability theory and mathematical logic the Tarski–Kuratowski algorithm is a non-deterministic algorithm that produces an upper bound for the complexity of a given formula in the arithmetical hierarchy and analytical hierarchy. The algorithm is named after Alfred Tarski and Kazimierz Kuratowski. Algorithm The Tarski–Kuratowski algorithm for the arithmetical hierarchy consists of the following steps: # Convert the formula to prenex normal form. (This is the non-deterministic part of the algorithm, as there may be more than one valid prenex normal form for the given formula.) # If the formula is quantifier-free, it is in \Sigma^0_0 and \Pi^0_0. # Otherwise, count the number of alternations of quantifiers; call this ''k''. # If the first quantifier is ∃, the formula is in \Sigma^0_. # If the first quantifier is ∀ A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tarski's High School Algebra Problem
In mathematical logic, Tarski's high school algebra problem was a question posed by Alfred Tarski. It asks whether there are identities involving addition, multiplication, and exponentiation over the positive integers that cannot be proved using eleven axioms about these operations that are taught in high-school-level mathematics. The question was solved in 1980 by Alex Wilkie, who showed that such unprovable identities do exist. Statement of the problem Tarski considered the following eleven axioms about addition ('+'), multiplication ('·'), and exponentiation to be standard axioms taught in high school: # ''x'' + ''y'' = ''y'' + ''x'' # (''x'' + ''y'') + ''z'' = ''x'' + (''y'' + ''z'') # ''x'' · 1 = ''x'' # ''x'' · ''y'' = ''y'' · ''x'' # (''x'' · ''y'') · ''z'' = ''x'' · (''y'' · ''z'') # ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tarski–Grothendieck Set Theory
Tarski–Grothendieck set theory (TG, named after mathematicians Alfred Tarski and Alexander Grothendieck) is an axiomatic set theory. It is a non-conservative extension of Zermelo–Fraenkel set theory (ZFC) and is distinguished from other axiomatic set theories by the inclusion of Tarski's axiom, which states that for each set there is a Grothendieck universe it belongs to (see below). Tarski's axiom implies the existence of inaccessible cardinals, providing a richer ontology than ZFC. For example, adding this axiom supports category theory. The Mizar system and Metamath use Tarski–Grothendieck set theory for formal verification of proofs. Axioms Tarski–Grothendieck set theory starts with conventional Zermelo–Fraenkel set theory and then adds “Tarski's axiom”. We will use the axioms, definitions, and notation of Mizar to describe it. Mizar's basic objects and processes are fully formal; they are described informally below. First, let us assume that: * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tarski's Exponential Function Problem
In model theory, Tarski's exponential function problem asks whether the theory of the real numbers together with the exponential function is decidable. Alfred Tarski had previously shown that the theory of the real numbers (without the exponential function) is decidable. The problem The ordered real field R is a structure over the language of ordered rings ''L''or = (+,·,−, ''η''−1. Workaround Recently there are attempts at handling the theory of the real numbers with functions such as the exponential function or sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opp ... by relaxing decidability to the weaker notion of quasi-decidability. A theory is quasi-decidable if there is a procedure that decides satisfiability but that may run forever for inputs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tarski's Circle-squaring Problem
Tarski's circle-squaring problem is the challenge, posed by Alfred Tarski in 1925, to take a disc in the plane, cut it into finitely many pieces, and reassemble the pieces so as to get a square of equal area. This was proven to be possible by Miklós Laczkovich in 1990; the decomposition makes heavy use of the axiom of choice and is therefore non-constructive. Laczkovich estimated the number of pieces in his decomposition at roughly 1050. A constructive solution was given by Łukasz Grabowski, András Máthé and Oleg Pikhurko in 2016 which worked everywhere except for a set of measure zero. More recently, gave a completely constructive solution using about 10^ Borel pieces. In 2021 Máthé, Noel and Pikhurko improved the properties of the pieces. In particular, Lester Dubins, Morris W. Hirsch & Jack Karush proved it is impossible to dissect a circle and make a square using pieces that could be cut with an idealized pair of scissors (that is, having Jordan curve boundary). Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
