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| {{otheruses4|bounded quantification in mathematical logic|bounded quantification in type theory|Bounded quantification}}
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| In the study of formal theories in [[mathematical logic]], '''bounded quantifiers''' are often added to a language in addition to the standard quantifiers "∀" and "∃". Bounded quantifiers differ from "∀" and "∃" in that bounded quantifiers restrict the range of the quantified variable. The study of bounded quantifiers is motivated by the fact that determining whether a [[Sentence (mathematical logic)|sentence]] with only bounded quantifiers is true is often not as difficult as determining whether an arbitrary sentence is true.
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| Examples of bounded quantifiers in the context of real analysis include "∀''x''>0", "∃''y''<0", and "∀''x'' ∊ ℝ". Informally "∀''x''>0" says "for all ''x'' where ''x'' is larger than 0", "∃''y''<0" says "there exists a ''y'' where ''y'' is less than 0" and "∀''x'' ∊ ℝ" says "for all ''x'' where ''x'' is a real number". For example, {{nowrap|1="∀''x''>0 ∃''y''<0 (''x'' = ''y''<sup>2</sup>)"}} says "every positive number is the square of a negative number".
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| == Bounded quantifiers in arithmetic == | |
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| Suppose that ''L'' is the language of [[Peano arithmetic]] (the language of [[second-order arithmetic]] or arithmetic in all finite types would work as well). There are two types of bounded quantifiers: <math>\forall n < t</math> and <math>\exists n < t</math>.
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| These quantifiers bind the number variable ''n'' and contain a numeric term ''t'' which may not mention ''n'' but which may have other free variables. (By "numeric terms" here we mean terms such as "1 + 1", "2", "2 × 3", "''m'' + 3", etc.)
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| These quantifiers are defined by the following rules (<math>\phi</math> denotes formulas):
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| :<math>\exists n < t\, \phi \Leftrightarrow \exists n ( n < t \land \phi)</math>
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| :<math>\forall n < t\, \phi \Leftrightarrow \forall n ( n < t \rightarrow \phi)</math>
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| There are several motivations for these quantifiers.
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| * In applications of the language to [[recursion theory]], such as the [[arithmetical hierarchy]], bounded quantifiers add no complexity. If <math>\phi</math> is a decidable predicate then <math>\exists n < t \, \phi</math> and <math>\forall n < t\, \phi</math> are decidable as well.
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| * In applications to the study of [[Peano Arithmetic]], formulas are sometimes provable with bounded quantifiers but unprovable with unbounded quantifiers.
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| For example, there is a definition of primality using only bounded quantifiers. A number ''n'' is prime if and only if there are not two numbers strictly less than ''n'' whose product is ''n''. There is no quantifier-free definition of primality in the language <math>\langle 0,1,+,\times, <, =\rangle</math>, however. The fact that there is a bounded quantifier formula defining primality shows that the primality of each number can be computably decided.
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| In general, a relation on natural numbers is definable by a bounded formula if and only if it is computable in the linear-time hierarchy, which is defined similarly to the [[polynomial hierarchy]], but with linear time bounds instead of polynomial. Consequently, all predicates definable by a bounded formula are [[ELEMENTARY|Kalmár elementary]], [[context-sensitive grammar|context-sensitive]], and [[primitive recursive]].
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| In the [[arithmetical hierarchy]], an arithmetical formula which contains only bounded quantifiers is called <math>\Sigma^0_0</math>, <math>\Delta^0_0</math>, and <math>\Pi^0_0</math>. The superscript 0 is sometimes omitted.
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| == Bounded quantifiers in set theory ==
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| Suppose that ''L'' is the language <math>\langle \in, \ldots, =\rangle</math> of the [[Zermelo–Fraenkel set theory]], where the ellipsis may be replaced by term-forming operations such as a symbol for the powerset operation. There are two bounded quantifiers: <math>\forall x \in t</math> and <math>\exists x \in t</math>. These quantifiers bind the set variable ''x'' and contain a term ''t'' which may not mention ''x'' but which may have other free variables.
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| The semantics of these quantifiers is determined by the following rules:
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| :<math>\exists x \in t\ (\phi) \Leftrightarrow \exists x ( x \in t \land \phi)</math>
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| :<math>\forall x \in t\ (\phi) \Leftrightarrow \forall x ( x \in t \rightarrow \phi)</math>
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| A ZF formula which contains only bounded quantifiers is called <math>\Sigma_0</math>, <math>\Delta_0</math>, and <math>\Pi_0</math>. This forms the basis of the [[Levy hierarchy]], which is defined analogously with the arithmetical hierarchy.
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| Bounded quantifiers are important in [[Kripke-Platek set theory]] and [[constructive set theory]], where only [[axiom schema of predicative separation|Δ<sub>0</sub> separation]] is included. That is, it includes separation for formulas with only bounded quantifiers, but not separation for other formulas. In KP the motivation is the fact that whether a set ''x'' satisfies a bounded quantifier formula only depends on the collection of sets that are close in rank to ''x'' (as the powerset operation can only be applied finitely many times to form a term). In constructive set theory, it is motivated on [[impredicativity|predicative]] grounds.
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| == See also ==
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| * [[Subtyping]] — bounded quantification in [[type theory]]
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| * [[System F-sub|System F<sub><:</sub>]] — a [[System F|polymorphic]] [[typed lambda calculus]] with bounded quantification
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| == References ==
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| * {{cite book | author = Hinman, P. | title = Fundamentals of Mathematical Logic | publisher = A K Peters | year = 2005 | isbn = 1-56881-262-0}}
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| * {{cite book | author= Kunen, K. | title = Set theory: An introduction to independence proofs | publisher = Elsevier | year = 1980 | isbn = 0-444-86839-9}}
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| [[Category:Quantification]]
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| [[Category:Proof theory]]
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| [[Category:Computability theory]]
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