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# Inhabited set

In constructive mathematics, a set A is inhabited if there exists an element $ain A$. 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

$lnot \left[forall z \left(z not in A\right)\right]$.
It is inhabited if
$exists z \left(z in A\right)$.
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 model in the classical sense that contains a nonempty, noninhabited set. But it is possible to construct a Kripke model that does. It is also possible to use the BHK interpretation of intuitionism to produce an example of a nonempty, noninhabited set.

In each of these settings, the construction of a nonempty, noninhabited set relies on the intuitionistic interpretation of the existential quantifier. In an intutionistic setting, in order for $exists z phi\left(z\right)$ to hold, for some formula $phi$, it is necessary for a specific value of z satisfying $phi$ to be known.

For example, consider a subset X of {0,1} 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. Then X is not empty (since assuming that it is empty leads to a contradiction), but X is it not currently known to be occupied either. Indeed, if X is occupied, then either 0 belongs to it or 1 belongs to it, so the Riemann hypothesis is either true or false, but neither of these possibilities has yet been proved. 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 978-0521318020

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