Inhabited set

In constructive mathematics, a set $A$ is inhabited if there exists an element $a\in A.$ 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 $A$ is empty if $\forall z(z\not \in A)$ while $A$ is nonempty if it is not empty, that is, if $$\lnot [\forall z(z\not \in A)].$$ It is inhabited if $$\exists z(z\in A).$$

Every inhabited set is a nonempty set (because if $a\in A$ is an inhabitant of $A$ then $a\not \in A$ is false and consequently so is $\forall z(z\not \in A)$). 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 $X$ 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 $\exists z\phi (z)$ 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. 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).

See also

• Intersection (set theory) – Set of elements common to all of some sets
• Nothing – Complete absence of anything; the opposite of everything

References

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

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