# Inhabited set

In mathematics, a set is **inhabited** if there exists an element .

In classical mathematics, the property of being inhabited is equivalent to being non-empty. However, this equivalence is not valid in constructive or intuitionistic logic, and so this separate terminology is mostly used in the set theory of constructive mathematics.

## Definition[edit]

In the formal language of first-order logic, set has the property of being *inhabited* if

### Related definitions[edit]

A set has the property of being *empty* if , or equivalently . Here stands for the negation .

A set is *non-empty* if it is not empty, that is, if , or equivalently .

## Theorems[edit]

Modus ponens implies , and taking any a false proposition for establishes that is always valid. Hence, any inhabited set is provably also non-empty.

## Discussion[edit]

In constructive mathematics, the double-negation elimination principle is not automatically valid. In particular, an existence statement is generally stronger than its double-negated form. The latter merely expresses that the existence cannot be ruled out, in the strong sense that it cannot consistently be negated. In a constructive reading, in order for to hold for some formula , it is necessary for a specific value of satisfying to be constructed or known. Likewise, the negation of a universal quantified statement is in general weaker than an existential quantification of a negated statement. In turn, a set may be proven to be non-empty without one being able to prove it is inhabited.

### Examples[edit]

Sets such as or are inhabited, as e.g. witnessed by . The set is empty and thus not inhabited. Naturally, the example section thus focuses on non-empty sets that are not provably inhabited.

It is easy to give examples for any simple set theoretical property, because logical statements can always be expressed as set theoretical ones, using an axiom of separation. For example, with a subset defined as , the proposition may always equivalently be stated as . The double-negated existence claim of an entity with a certain property can be expressed by stating that the set of entities with that property is non-empty.

#### Example relating to excluded middle[edit]

Define a subset via

Clearly and , and from the principle of non-contradiction one concludes . Further, and in turn

Already minimal logic proves , the double-negation for any excluded middle statement, which here is equivalent to . So by performing two contrapositions on the previous implication, one establishes . In words: It *cannot consistently be ruled out* that exactly one of the numbers and inhabits . In particular, the latter can be weakened to , saying is proven non-empty.

As example statements for , consider the consistency of the theory at hand, an infamous provenly theory-independent statement such as the continuum hypothesis, or, informally, an unknowable claim about the past or future. By design, these are chosen to be unprovable. A variant of this is to consider mathematiacal propositions that are merely not yet established - see also Brouwerian counterexamples. Knowledge of the validity of either or is equivalent to knowledge about as above, and cannot be obtained. As neither nor can be proven in the theory, it will also not prove to be inhabited by some particular number. Further, a constructive framework with the disjunction property then cannot prove either. There is no evidence for , nor for , and constructive unprovability of their disjunction reflects this. Nonetheless, since ruling out excluded middle is provenly always inconsistent, it is also established that is not empty. Classical logic adopts axiomatically, spoiling a constructive reading.

#### Example relating to choice[edit]

The full axiom of choice is independent of and provenly not constructively compatible with other common set theory axioms. So a theory including these axioms that does not permit excluded middle does also not validate that function existence principle.

In , the choice axiom is equivalent to the statement that for every vector space there exists basis. So more concretely, consider the question of existence of a Hamel bases of the real numbers over the rational numbers. Such an object is elusive in the sense that are different models that either negate and validate its existence. So it is also consistent to just postulate that existence cannot be ruled out here, in the sense that it cannot be negated. Again, that postulate may be expressed as saying that the set of such Hamel bases is non-empty.

### Model theory[edit]

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".

However, it is possible to construct a Kripke model that differentiates between the two notions. Since an implication is true in every Kripke model if and only if it is provable in intuitionistic logic, this indeed establishes that one cannot intuitionistically prove that " is nonempty" implies " is inhabited".

## See also[edit]

- Intersection (set theory) – Set of elements common to all of some sets
- Nothing – Complete absence of anything; the opposite of everything
- Type inhabitation in type theory.

## References[edit]

- 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.*