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## Summary

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).