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| In [[mathematical logic]], a '''definable set''' is an ''n''-ary [[relation (mathematics)|relation]] on the [[domain (mathematics)|domain]] of a [[structure (mathematical logic)|structure]] whose elements are precisely those elements satisfying some [[formula (mathematical logic)|formula]] in the language of that structure. A set can be defined with or without '''parameters''', which are elements of the domain that can be referenced in the formula defining the relation.
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| == Definition ==
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| Let <math>\mathcal{L}</math> be a first-order language, <math>\mathcal{M}</math> an <math>\mathcal{L}</math>-structure with domain <math>M</math>, ''X'' a fixed subset of <math>M</math>, and ''m'' a natural number. Then:
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| * A set <math>A\subseteq M^m</math> is ''definable in <math>\mathcal{M}</math> with parameters from <math>X</math>'' if and only if there exists a formula <math>\varphi[x_1,\ldots,x_m,y_1,\ldots,y_n]</math> and elements <math>b_1,\ldots,b_n\in X</math> such that for all <math>a_1,\ldots,a_m\in M</math>,
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| :<math>(a_1,\ldots,a_m)\in A</math> if and only if <math>\mathcal{M}\models\varphi[a_1,\ldots,a_m,b_1,\ldots,b_n]</math>
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| :The bracket notation here indicates the semantic evaluation of the free variables in the formula.
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| * A set ''<math>A</math> is definable in <math>\mathcal{M}</math> without parameters'' if it is definable in <math>\mathcal{M}</math> with parameters from the empty set (that is, with no parameters in the defining formula).
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| * A function is definable in <math>\mathcal{M}</math> (with parameters) if its graph is definable (with those parameters) in <math>\mathcal{M}</math>.
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| * An element ''a'' is definable in <math>\mathcal{M}</math> (with parameters) if the singleton set {''a''} is definable in <math>\mathcal{M}</math> (with those parameters).
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| == Examples ==
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| === The natural numbers with only the order relation ===
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| Let <math>\mathcal{N}=(\mathbb{N},<)</math> be the structure consisting of the [[natural number]]s with the usual ordering. Then every natural number is definable in <math>\mathcal{N}</math> without parameters. The number <math>0</math> is defined by the formula <math>\varphi(x)</math> stating that there exist no elements less than ''x'':
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| <math>\varphi=\neg\exists y(y<x),</math>
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| and a natural <math>n>0</math> is defined by the formula <math>\varphi(x)</math> stating there exist exactly <math>n</math> elements less than ''x'':
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| <math>\varphi=\exists x_0\cdots\exists x_{n-1}(x_0<x_1\land\cdots\land x_{n-1}<x\land\forall y(y<x\rightarrow(y\equiv x_0\lor\cdots\lor y\equiv x_{n-1})))</math>
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| In contrast, one cannot define any specific [[integer]] without parameters in the structure <math>\mathcal{Z}=(\mathbb{Z},<)</math> consisting of the integers with the usual ordering (see the section on automorphisms below).
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| === The natural numbers with their arithmetical operations ===
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| Let <math>\mathcal{N}=(\mathbb{N},+, \cdot, <)</math> be the first-order structure consisting of the natural numbers and their usual arithmetic operations and order relation. The sets definable in this structure are known as the [[arithmetical set]]s, and are classified in the [[arithmetical hierarchy]]. If the structure is considered in [[second-order logic]] instead of first-order logic, the definable sets of natural numbers in the resulting structure are classified in the [[analytical hierarchy]]. These hierarchies reveal many relationships between definability in this structure and [[computability theory]], and are also of interest in [[descriptive set theory]].
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| === The field of real numbers ===
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| Let <math>\mathcal{R}=(\mathbb{R},0,1,+,\cdot)</math> be the structure consisting of the [[field (mathematics)|field]] of [[real number]]s. Although the usual ordering relation is not directly included in the structure, there is a formula that defines the set of nonnegative reals, since these are the only reals that possess square roots:
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| <math>\varphi=\exists y(y\cdot y\equiv x).</math>
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| Thus any <math>a\in\R</math> is nonnegative if and only if <math>\mathcal{R}\models\varphi[a]</math>. In conjunction with a formula that defines the additive inverse of a real number in <math>\mathcal{R}</math>, one can use <math>\varphi</math> to define the usual ordering in <math>\mathcal{R}</math>: for <math>a,b\in\R</math>, set <math>a\le b</math> if and only if <math>b-a</math> is nonnegative. The enlarged structure <math>\mathcal{R}^{\le}=(\mathbb{R},0,1,+,\cdot,\le)</math>s is called a [[extension by definitions|definitional extension]] of the original structure. It has the same expressive power as the original structure, in the sense that a set is definable over the enlarged structure from a set of parameters if and only if it is definable over the original structure from that same set of parameters.
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| The [[Theory (mathematical logic)|theory]] of <math>\mathcal{R}^{\le}</math> has [[quantifier elimination]]. Thus the definable sets are Boolean combinations of solutions to polynomial equalities and inequalities; these are called [[semi-algebraic sets]]. Generalizing this property of the real line leads to the study of [[o-minimality]].
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| == Invariance under automorphisms ==
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| An important result about definable sets is that they are preserved under [[automorphism]]s.
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| :Let <math>\mathcal{M}</math> be an <math>\mathcal{L}</math>-structure with domain <math>M</math>, <math>X\subseteq M</math>, and <math>A\subseteq M^m</math> definable in <math>\mathcal{M}</math> with parameters from <math>X</math>. Let <math>\pi:M\to M</math> be an automorphism of <math>\mathcal{M}</math> which is the identity on <math>X</math>. Then for all <math>a_1,\ldots,a_m\in M</math>, | |
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| ::<math>(a_1,\ldots,a_m)\in A</math> if and only if <math>(\pi(a_1),\ldots,\pi(a_m))\in A</math>
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| This result can sometimes be used to classify the definable subsets of a given structure. For example, in the case of <math>\mathcal{Z}=(\mathbb{Z},<)</math> above, any translation of <math>\mathcal{Z}</math> is an automorphism preserving the empty set of parameters, and thus it is impossible to define any particular integer in this structure without parameters in <math>\mathcal{Z}</math>. In fact, since any two integers are carried to each other by a translation and its inverse, the only sets of integers definable in <math>\mathcal{Z}</math> without parameters are the empty set and <math>\mathbb{Z}</math> itself. In contrast, there are infinitely many definable sets of pairs (or indeed ''n''-tuples for any fixed ''n''>1) of elements of <math>\mathcal{Z}</math>, since any automorphism (translation) preserves the "distance" between two elements.
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| == Additional results ==
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| The [[Tarski–Vaught test]] is used to characterize the [[elementary substructure]]s of a given structure.
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| == References ==
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| *Hinman, Peter. ''Fundamentals of Mathematical Logic'', A. K. Peters, 2005.
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| *Marker, David. ''Model Theory: An Introduction'', Springer, 2002.
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| *Rudin, Walter. ''Principles of Mathematical Analysis'', 3rd. ed. McGraw-Hill, 1976.
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| *Slaman, Theodore A. and W. Hugh Woodin. ''Mathematical Logic: The Berkeley Undergraduate Course''. Spring 2006.
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| [[Category:Model theory]]
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| [[Category:Logic]]
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| [[Category:Mathematical logic]]
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