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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]].
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| == Comparison with nonempty sets ==
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| 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
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| :<math>\lnot [\forall z (z \not \in A)].</math>
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| It is ''inhabited'' if
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| :<math>\exists z (z \in A).</math>
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| In intuitionistic logic, the negation of a universal quantifier is weaker than an existential quantifier, not equivalent to it as in [[classical logic]].
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| == Example ==
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| 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".
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| 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.
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| 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).
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| == References ==
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| *D. Bridges and F. Richman. 1987. ''Varieties of Constructive Mathematics''. Oxford University Press. ISBN 978-0-521-31802-0
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| {{PlanetMath attribution|id=5931|title=Inhabited set}}
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| [[Category:Constructivism (mathematics)]]
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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.
my webpage ... extended car warranty, linked here,