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en>Dadu: /* Basic concepts */

2014-12-29T09:33:47Z

Basic concepts

← Older revision		Revision as of 11:33, 29 December 2014
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	~~Her title~~ is ~~Felicidad Ahmad. Some time~~ in ~~the past I selected to live in Arizona~~ but ~~I require to move for my family~~. ~~To play badminton~~ is ~~some thing he really enjoys doing~~. ~~She is presently~~ a ~~cashier but quickly she'll be on her own~~.<br><br>~~Feel free to visit~~ my ~~homepage;~~ extended car warranty, [http://~~gshappy~~.~~marubaram.wo.tc~~/xe/~~?document_srl=256061 Read the Full Piece of writing~~],		Friends call him Royal. One of his favorite hobbies is taking part in crochet but he hasn't produced a dime with it. Arizona is her birth location and she will never move. Bookkeeping has been his day occupation for a whilst.<br><br>my webpage ... extended car warranty, [http://Ladysmithadventures.com/ActivityFeed/MyProfile/tabid/60/userId/14680/Default.aspx linked here],

200.29.18.116: /* eMule mods */

2014-02-19T00:47:19Z

eMule mods

← Older revision		Revision as of 02:47, 19 February 2014
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	~~In [[constructive mathematics]], a [[Set (mathematics)\|set]] ''A''~~ is ~~'''inhabited''' if there exists an element <math>a\in A</math>~~. ~~In classical mathematics, this is the same as the set being nonempty; however, this equivalence is not valid~~ in ~~[[intuitionistic logic]].~~		Her title is Felicidad Ahmad. Some time in the past I selected to live in Arizona but I require to move for my family. To play badminton is some thing he really enjoys doing. She is presently a cashier but quickly she'll be on her own.<br><br>Feel free to visit my homepage; extended car warranty, [http://gshappy.marubaram.wo.tc/xe/?document_srl=256061 Read the Full Piece of writing],

	~~== Comparison with nonempty sets ==~~
	~~In [[classical mathematics]], a set is inhabited if and only if it is not~~ the ~~[[empty set]]. These definitions diverge~~ in ~~[[constructive mathematics]], however~~. ~~A set ''A''~~ is ~~''nonempty'' if it is not empty, that is, if~~
	~~:<math>\lnot [\forall z (z \not \in A)]~~.~~</math>~~
	It is ~~''inhabited'' if~~
	~~:<math>\exists z (z \in A).</math>~~
	~~In intuitionistic logic, the negation of~~ a ~~universal quantifier is weaker than an existential quantifier, not equivalent to it as in [[classical logic]].~~

	~~== Example ==~~
	Because inhabited sets are the same as nonempty sets in classical logic, it is not possible to produce a [[structure (mathematical logic)\|model]] in the classical sense that contains a nonempty set ''X'' but ~~does not satisfy "~~''X'' is inhabited". But it is possible to construct a [[Kripke model]] ''M'' that satisfies "''X'' is nonempty" without satisfying "''X'' is inhabited". Because an implication is provable in intuitionistic logic if and only if it is true in every Kripke model, this means that one cannot prove in this logic that "''X'' is nonempty" implies "''X'' is inhabited".

	~~The possibility of this construction relies~~ on ~~the intuitionistic interpretation of the existential quantifier~~. ~~In an intuitionistic setting, in order for~~ <~~math~~>~~\exists z \phi(z)~~<~~/math~~> to ~~hold~~, ~~for some formula <math>\phi<~~/~~math>, it is necessary for a specific value of ''z'' satisfying <math>\phi<~~/~~math> to be known.~~

	For example, consider a subset ''X'' of {0,1} [[axiom of specification\|specified]] by the following rule: 0 belongs to ''X'' if and only if the [[Riemann hypothesis]] is true, and 1 belongs to ''X'' if and only if the Riemann hypothesis is false. ~~If we assume that Riemann hypothesis is either true or false, then ''X'' is not empty, but any constructive proof that ''X'' is inhabited would either prove that 0 is in ''X'' or that 1 is in ''X''~~. Thus a constructive proof that ''X'' is inhabited would determine the truth value of the Riemann hypothesis, which is not known, By replacing the Riemann hypothesis in this example by a generic proposition, one can construct a Kripke model with a set that is neither empty nor inhabited (even if the Riemann hypothesis itself is ever proved or refuted).

	=~~= References ==~~
	*D. Bridges and F. Richman. 1987. ''Varieties of ~~Constructive Mathematics''. Oxford University Press. ISBN 978-0-521-31802-0~~

	~~{{PlanetMath attribution\|id=5931\|title=Inhabited set}}~~

	~~[[Category:Constructivism (mathematics)]~~]

en>Glenn: /* External links */ catcorr

2014-01-25T09:24:35Z

External links: catcorr

← Older revision		Revision as of 11:24, 25 January 2014
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	Hi there, ~~I am Alyson Boon although~~ it is not the ~~title on my beginning certificate~~. I'~~ve always cherished living~~ in ~~Alaska~~. ~~I am currently~~ a ~~travel agent~~. ~~One of~~ the ~~issues she enjoys most~~ is ~~canoeing~~ and ~~she~~'~~s been performing~~ it for ~~fairly~~ a ~~while.~~<br><br>~~Feel free~~ to ~~visit my web site~~ :~~: online reader;~~ [~~http://~~1.~~234~~.36.~~240/fxac/m001_2/7330 1~~.~~234~~.36.~~240~~],		In [[constructive mathematics]], a [[Set (mathematics)\|set]] ''A'' is '''inhabited''' if there exists an element <math>a\in A</math>. In classical mathematics, this is the same as the set being nonempty; however, this equivalence is not valid in [[intuitionistic logic]].

			== Comparison with nonempty sets ==
			In [[classical mathematics]], a set is inhabited if and only if it is not the [[empty set]]. These definitions diverge in [[constructive mathematics]], however. A set ''A'' is ''nonempty'' if it is not empty, that is, if
			:<math>\lnot [\forall z (z \not \in A)].</math>
			It is ''inhabited'' if
			:<math>\exists z (z \in A).</math>
			In intuitionistic logic, the negation of a universal quantifier is weaker than an existential quantifier, not equivalent to it as in [[classical logic]].

			== Example ==
			Because inhabited sets are the same as nonempty sets in classical logic, it is not possible to produce a [[structure (mathematical logic)\|model]] in the classical sense that contains a nonempty set ''X'' but does not satisfy "''X'' is inhabited". But it is possible to construct a [[Kripke model]] ''M'' that satisfies "''X'' is nonempty" without satisfying "''X'' is inhabited". Because an implication is provable in intuitionistic logic if and only if it is true in every Kripke model, this means that one cannot prove in this logic that "''X'' is nonempty" implies "''X'' is inhabited".

			The possibility of this construction relies on the intuitionistic interpretation of the existential quantifier. In an intuitionistic setting, in order for <math>\exists z \phi(z)</math> to hold, for some formula <math>\phi</math>, it is necessary for a specific value of ''z'' satisfying <math>\phi</math> to be known.

			For example, consider a subset ''X'' of {0,1} [[axiom of specification\|specified]] by the following rule: 0 belongs to ''X'' if and only if the [[Riemann hypothesis]] is true, and 1 belongs to ''X'' if and only if the Riemann hypothesis is false. If we assume that Riemann hypothesis is either true or false, then ''X'' is not empty, but any constructive proof that ''X'' is inhabited would either prove that 0 is in ''X'' or that 1 is in ''X''. Thus a constructive proof that ''X'' is inhabited would determine the truth value of the Riemann hypothesis, which is not known, By replacing the Riemann hypothesis in this example by a generic proposition, one can construct a Kripke model with a set that is neither empty nor inhabited (even if the Riemann hypothesis itself is ever proved or refuted).

			== References ==
			*D. Bridges and F. Richman. 1987. ''Varieties of Constructive Mathematics''. Oxford University Press. ISBN 978-0-521-31802-0

			{{PlanetMath attribution\|id=5931\|title=Inhabited set}}

			[[Category:Constructivism (mathematics)]]

en>TheChampionMan1234: Added {{merge from}} tag to article (TW)

2012-09-01T09:45:42Z

Added {{merge from}} tag to article (TW)

New page

Hi there, I am Alyson Boon although it is not the title on my beginning certificate. I've always cherished living in Alaska. I am currently a travel agent. One of the issues she enjoys most is canoeing and she's been performing it for fairly a while.<br><br>Feel free to visit my web site :: online reader; [http://1.234.36.240/fxac/m001_2/7330 1.234.36.240],