Angular eccentricity - Revision history

114.70.9.203 at 16:15, 7 May 2014

2014-05-07T16:15:27Z

← Older revision		Revision as of 18:15, 7 May 2014
Line 1:		Line 1:
	~~Hi there~~. My title ~~is Garland although it is not the name on my beginning certificate~~. ~~To perform croquet~~ is ~~the hobby I will never stop performing. Bookkeeping has~~ been ~~his day job~~ for a whilst. ~~Kansas is~~ our ~~beginning place and my parents reside close by~~.<br><br>~~Have a look at~~ my ~~web~~ blog :: [http://~~exitus~~.~~bplaced~~.~~net~~/~~index.php?mod=users&action=view&id=139 http:~~//~~exitus.bplaced.net/index.php?mod=users&action=view&id=139~~]		The individual who wrote the article accurate psychic predictions ([http://clothingcarearchworth.com/index.php?document_srl=441551&mid=customer_review http://clothingcarearchworth.com/index.php?document_srl=441551&mid=customer_review]) is known as Jayson Hirano and he completely digs [http://cspl.postech.ac.kr/zboard/Membersonly/144571 accurate psychic predictions] that title. As a woman what she really likes is style and she's been doing it for quite a whilst. Since I was 18 I've been operating as a bookkeeper but soon my wife and I will start our own company. For years she's been living in Kentucky but her spouse wants them to move.<br><br>Here is my blog post ... [http://www.skullrocker.com/blogs/post/10991 clairvoyant psychic]

en>JamesBWatson: Correcting editing slip

2014-02-28T16:21:24Z

Correcting editing slip

← Older revision		Revision as of 18:21, 28 February 2014
Line 1:		Line 1:
	~~{{otheruses4\|bounded quantification in mathematical logic\|bounded quantification in type theory\|Bounded quantification}}~~		Hi there. My title is Garland although it is not the name on my beginning certificate. To perform croquet is the hobby I will never stop performing. Bookkeeping has been his day job for a whilst. Kansas is our beginning place and my parents reside close by.<br><br>Have a look at my web blog :: [http://exitus.bplaced.net/index.php?mod=users&action=view&id=139 http://exitus.bplaced.net/index.php?mod=users&action=view&id=139]
	~~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.~~

	~~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".~~

	~~== Bounded quantifiers in arithmetic ==~~

	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>.
	~~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.)~~

	~~These quantifiers are defined by the following rules (<math>\phi</math> denotes formulas):~~
	~~:<math>\exists n < t\, \phi \Leftrightarrow \exists n ( n < t \land \phi)</math>~~
	~~:<math>\forall n < t\, \phi \Leftrightarrow \forall n ( n < t \rightarrow \phi)</math>~~

	~~There are several motivations~~ for ~~these quantifiers.~~
	* 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~~.
	* In applications to the study of [[Peano Arithmetic]], formulas are sometimes provable with bounded quantifiers but unprovable with unbounded quantifiers.

	~~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.

	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]].

	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.

	~~== Bounded quantifiers in set theory ==~~
	~~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.~~

	~~The semantics of these quantifiers is determined by the following rules~~:
	:~~<math>\exists x \in t\ (\phi) \Leftrightarrow \exists x ( x \in t \land \phi)</math>~~
	:~~<math>\forall x \in t\ (\phi) \Leftrightarrow \forall x ( x \in t \rightarrow \phi)</math>~~

	~~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~~.

	~~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.

	== ~~See also~~ ==
	* [[Subtyping]] — bounded quantification in [[type theory]]
	* [[System F-sub\|System F<sub><:</~~sub>]] — a [[System F\|polymorphic]] [[typed lambda calculus]] with bounded quantification~~

	~~== References ==~~
	* {{cite book \| author = Hinman, P. ~~\| title = Fundamentals of Mathematical Logic \| publisher = A K Peters \| year = 2005 \| isbn = 1-56881-262-0}}~~
	* {{cite book \| author= Kunen, K. ~~\| title~~ = ~~Set theory: An introduction to independence proofs \| publisher~~ = ~~Elsevier \| year~~ = ~~1980 \| isbn = 0-444-86839-9}}~~

	~~[[Category:Quantification]]~~
	~~[[Category:Proof theory]]~~
	~~[[Category:Computability theory]~~]

en>ChrisGualtieri: General Fixes using AWB

2013-10-20T03:09:34Z

General Fixes using AWB

← Older revision		Revision as of 05:09, 20 October 2013
Line 1:		Line 1:
	~~Jayson Berryhill~~ is ~~how I~~'~~m called~~ and ~~my spouse doesn~~'~~t like it at~~ all~~. Alaska~~ is ~~exactly~~ where I'~~ve usually been residing~~. ~~What me~~ and ~~my family members love~~ is to ~~climb~~ but I'~~m considering~~ on ~~beginning something new~~. ~~Office supervising~~ is ~~exactly~~ where ~~my primary earnings comes from~~ but ~~I've usually needed my own company~~.<br><br>~~Also visit my homepage~~: ~~online psychic chat~~ (~~[http:~~//~~www~~.~~sirudang~~.~~com~~/~~siroo_Notice~~/~~2110 www~~.~~sirudang~~.~~com~~])		{{otheruses4\|bounded quantification in mathematical logic\|bounded quantification in type theory\|Bounded quantification}}
			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.

			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".

			== Bounded quantifiers in arithmetic ==

			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>.
			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.)

			These quantifiers are defined by the following rules (<math>\phi</math> denotes formulas):
			:<math>\exists n < t\, \phi \Leftrightarrow \exists n ( n < t \land \phi)</math>
			:<math>\forall n < t\, \phi \Leftrightarrow \forall n ( n < t \rightarrow \phi)</math>

			There are several motivations for these quantifiers.
			* 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.
			* In applications to the study of [[Peano Arithmetic]], formulas are sometimes provable with bounded quantifiers but unprovable with unbounded quantifiers.

			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.

			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]].

			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.

			== Bounded quantifiers in set theory ==
			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.

			The semantics of these quantifiers is determined by the following rules:
			:<math>\exists x \in t\ (\phi) \Leftrightarrow \exists x ( x \in t \land \phi)</math>
			:<math>\forall x \in t\ (\phi) \Leftrightarrow \forall x ( x \in t \rightarrow \phi)</math>

			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.

			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.

			== See also ==
			* [[Subtyping]] — bounded quantification in [[type theory]]
			* [[System F-sub\|System F<sub><:</sub>]] — a [[System F\|polymorphic]] [[typed lambda calculus]] with bounded quantification

			== References ==
			* {{cite book \| author = Hinman, P. \| title = Fundamentals of Mathematical Logic \| publisher = A K Peters \| year = 2005 \| isbn = 1-56881-262-0}}
			* {{cite book \| author= Kunen, K. \| title = Set theory: An introduction to independence proofs \| publisher = Elsevier \| year = 1980 \| isbn = 0-444-86839-9}}

			[[Category:Quantification]]
			[[Category:Proof theory]]
			[[Category:Computability theory]]

en>Agricolae: redo withbetter coding

2012-06-10T14:10:58Z

redo withbetter coding

New page

Jayson Berryhill is how I'm called and my spouse doesn't like it at all. Alaska is exactly where I've usually been residing. What me and my family members love is to climb but I'm considering on beginning something new. Office supervising is exactly where my primary earnings comes from but I've usually needed my own company.<br><br>Also visit my homepage: online psychic chat ([http://www.sirudang.com/siroo_Notice/2110 www.sirudang.com])