In constructive mathematics, a set is inhabited if there exists an element In classical mathematics, this is the same as the set being nonempty; however, this equivalence is not valid in intuitionistic logic (or constructive 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 is empty if while is nonempty if it is not empty, that is, if

It is inhabited if

Every inhabited set is a nonempty set (because if is an inhabitant of then is false and consequently so is ). In intuitionistic logic, the negation of a universal quantifier is weaker than an existential quantifier, not equivalent to it as in classical logic so a nonempty set is not automatically guaranteed to be inhabited.

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 set but does not satisfy " is inhabited". But it is possible to construct a Kripke model that satisfies " is nonempty" without satisfying " 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 " is nonempty" implies " is inhabited".

The possibility of this construction relies on the intuitionistic interpretation of the existential quantifier. In an intuitionistic setting, in order for to hold, for some formula , it is necessary for a specific value of satisfying to be known.

For example, consider a subset of specified by the following rule: belongs to if and only if the Riemann hypothesis is true, and belongs to if and only if the Riemann hypothesis is false. If we assume that Riemann hypothesis is either true or false, then is not empty, but any constructive proof that is inhabited would either prove that is in or that is in Thus a constructive proof that 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).

See also

References

This article incorporates material from Inhabited set on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.