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| '''Tarski–Grothendieck set theory''' ('''TG''', named after mathematicians [[Alfred Tarski]] and [[Alexander Grothendieck]]) is an [[axiomatic set theory]] that was introduced as part of the [[Mizar system]] for formal verification of proofs.
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| Tarski–Grothendieck set theory 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 cardinal]]s, providing a richer [[ontology]] than that of conventional set theories such as ZFC.
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| ==Axioms==
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| While the [[axiom]]s and [[definition]]s defining Mizar's basic objects and processes are fully [[Formal system|formal]], they are described informally below.
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| * Given any set <math>A</math>, the singleton <math>\{A\}</math> exists.
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| * Given any two sets, their unordered and ordered pairs exist.
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| * Given any family of sets, its union exists.
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| '''TG''' includes the following axioms, which are conventional because also part of [[ZFC]]:
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| * Set axiom: Quantified variables range over sets alone; everything is a set (the same [[ontology]] as [[ZFC]]).
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| * [[Extensionality]] axiom: Two sets are identical if they have the same members.
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| * [[Axiom of regularity]]: No set is a member of itself, and circular chains of membership are impossible.
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| * [[Axiom schema of replacement]]: Let the [[domain (mathematics)|domain]] of the [[function (mathematics)|function]] <math>F</math> be the set <math>A</math>. Then the [[range (mathematics)|range]] of <math>F</math> (the values of <math>F(x)</math> for all members <math>x</math> of <math>A</math>) is also a set.
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| It is Tarski's axiom that distinguishes '''TG''' from other axiomatic set theories. Tarski's axiom also implies the axioms of [[axiom of infinity|infinity]], [[axiom of choice|choice]],<ref>Tarski (1938)</ref><ref>http://mmlquery.mizar.org/mml/current/wellord2.html#T26</ref> and [[axiom of power set|power set]].<ref>Robert Solovay, [http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html Re: AC and strongly inaccessible cardinals].</ref><ref>[http://us.metamath.org/mpegif/grothpw.html Metamath '''grothpw'''.]</ref> It also implies the existence of [[inaccessible cardinal]]s, thanks to which the [[ontology]] of '''TG''' is much richer than that of conventional set theories such as [[ZFC]].
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| * '''Tarski's axiom''' (adapted from Tarski 1939<ref>Tarski (1939)</ref>). For every set <math>x</math>, there exists a set <math>y</math> whose members include:
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| - <math>x</math> itself;
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| - every subset of every member of <math>y</math>;
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| - the power set of every member of <math>y</math>;
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| - every subset of <math>y</math> of [[cardinality]] less than that of <math>y</math>.
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| More formally:
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| :<math>\exists y [x\in y \wedge \forall z\in y(\mathcal P(z)\subseteq y\wedge\mathcal P(z)\in y) \wedge \forall z\in\mathcal P(y)(\neg z\approx y\to z\in y)]</math>
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| where "<math>\mathcal P(x)</math>" denotes the power class of ''x'' and "<math>\approx</math>" denotes [[equinumerosity]]. What Tarski's axiom states (in the vernacular) for each set <math>x</math> there is a [[Grothendieck universe]] it belongs to.
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| ==Implementation in the Mizar system==
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| The Mizar language, underlying the implementation of '''TG''' and providing its logical syntax, is typed and the types are assumed to be non-empty. Hence, the theory is implicitly taken to be [[Axiom of empty set|non-empty]]. The existence axioms, e.g. the existence of the unordered pair, is also implemented indirectly by the definition of term constructors.
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| The system includes equality, the membership predicate and the following standard definitions:
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| * [[Singleton (mathematics)|Singleton]]: A set with one member;
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| * [[Unordered pair]]: A set with two distinct members. <math>\{a,b\} = \{b,a\}</math>;
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| * [[Ordered pair]]: The set <math>\{\{a,b\},\{a\}\} = (a,b) \neq (b,a)</math>;
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| * [[Subset]]: A set all of whose members are members of another given set;
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| * The [[union (set theory)|union]] of a family of sets <math>Y</math>: The set of all members of every member of <math>Y</math>.
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| ==See also==
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| *[[Mizar system]]
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| *[[Grothendieck universe]]
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| *[[Axiom of limitation of size]]
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| ==Notes==
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| <references/>
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| ==References==
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| *Andreas Blass, I.M. Dimitriou, and [[Benedikt Löwe]] (2007) "[http://dare.uva.nl/document/25381 Inaccessible Cardinals without the Axiom of Choice,]" ''Fundamenta Mathematicae'' 194: 179-89.
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| * {{cite conference
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| | first = Nicolas
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| | last = Bourbaki
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| | authorlink = Nicolas Bourbaki
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| | year = 1972
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| | title = Univers
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| | booktitle = Séminaire de Géométrie Algébrique du Bois Marie – 1963-64 – Théorie des topos et cohomologie étale des schémas – (SGA 4) – vol. 1 (Lecture notes in mathematics '''269''')
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| | editor = [[Michael Artin]], [[Alexandre Grothendieck]], [[Jean-Louis Verdier]], eds.
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| | publisher = [[Springer Science+Business Media|Springer-Verlag]]
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| | location = Berlin; New York
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| | language = French
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| | pages = 185–217
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| | url = http://modular.fas.harvard.edu/sga/sga/4-1/4-1t_185.html
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| }}
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| * [[Patrick Suppes]] (1960) ''Axiomatic Set Theory''. Van Nostrand. Dover reprint, 1972.
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| * {{cite journal
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| | last = Tarski
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| | first = Alfred
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| | authorlink = Alfred Tarski
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| | year = 1938
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| | title = Über unerreichbare Kardinalzahlen
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| | journal = Fundamenta Mathematicae
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| | volume = 30
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| | pages = 68–89
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| | url = http://matwbn.icm.edu.pl/ksiazki/fm/fm30/fm30113.pdf
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| }}
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| * {{cite journal
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| | last = Tarski
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| | first = Alfred
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| | authorlink = Alfred Tarski
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| | year = 1939
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| | title = On the well-ordered subsets of any set
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| | journal = Fundamenta Mathematicae
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| | volume = 32
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| | pages = 176–183
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| | url = http://matwbn.icm.edu.pl/ksiazki/fm/fm32/fm32115.pdf
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| }}
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| ==External links==
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| *Trybulec, Andrzej, 1989, "[http://mizar.uwb.edu.pl/JFM/Axiomatics/tarski.html Tarski–Grothendieck Set Theory]", ''Journal of Formalized Mathematics''.
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| *[[Metamath]]: "[http://us.metamath.org/mpegif/mmset.html Proof Explorer Home Page.]" Scroll down to "Grothendieck's Axiom."
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| * [[PlanetMath]]: "[http://planetmath.org/encyclopedia/TarskisAxiom.html Tarski's Axiom]"
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| {{DEFAULTSORT:Tarski-Grothendieck set theory}}
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| [[Category:Systems of set theory]]
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Marvella is what you can contact her but it's not the most female name out there. For many years he's been operating as a receptionist. California is our birth place. What I love performing is performing ceramics but I haven't made a dime with it.
Here is my web site - std testing at home, click the following document,