Inhabited set

Short description: Property of sets used in constructive mathematics

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

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.

## Contents

* 1 Definition
* 1.1 Related definitions
* 2 Theorems
* 3 Discussion
* 3.1 Examples
* 3.1.1 Example relating to excluded middle
* 3.1.2 Example relating to choice
* 3.2 Model theory
* 4 See also
* 5 References

## Definition

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

∃z.(z∈A).

### Related definitions

A set A has the property of being empty if ∀z.(z∉A), or equivalently ¬∃z.(z∈A). Here z∉A stands for the negation ¬(z∈A).

A set A is non-empty if it is not empty, that is, if ¬∀z.(z∉A), or equivalently ¬¬∃z.(z∈A).

## Theorems

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

## Discussion

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 ∃z.ϕ(z) to hold for some formula ϕ, it is necessary for a specific value of z satisfying ϕ 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

Sets such as {2,3,4,7} or ℚ are inhabited, as e.g. witnessed by 3∈{2,3,4,7}. 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 such examples by using the axiom of separation, as with it logical statements can always be translated to set theoretical ones. For example, with a subset S⊂{0} defined as S:={n∈{0}∣P}, the proposition P may always equivalently be stated as 0∈S. 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

Define a subset A⊂{0,1} via

A:={n∈{0,1}∣(P∧n=0)∨(¬P∧n=1)}

Clearly P↔0∈A and (¬P)↔1∈A, and from the principle of non-contradiction one concludes ¬(0∈A∧1∈A). Further, (P∨¬P)↔(0∈A∨1∈A) and in turn

(P∨¬P)->∃!(n∈{0,1}).n∈A

Already minimal logic proves ¬¬(P∨¬P), the double-negation for any excluded middle statement, which here is equivalent to ¬(0∉A∧1∉A). So by performing two contrapositions on the previous implication, one establishes ¬¬∃!(n∈{0,1}).n∈A. In words: It cannot consistently be ruled out that exactly one of the numbers 0 and 1 inhabits A. In particular, the latter can be weakened to ¬¬∃n.n∈A, saying A is proven non-empty.

As example statements for P, consider the infamous provenly theory-independent statement such as the continuum hypothesis, consistency of the sound theory at hand, 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 mathematical propositions that are merely not yet established - see also Brouwerian counterexamples. Knowledge of the validity of either 0∈A or 1∈A is equivalent to knowledge about P as above, and cannot be obtained. Given neither P nor ¬P can be proven in the theory, it will also not prove A to be inhabited by some particular number. Further, a constructive framework with the disjunction property then cannot prove P∨¬P either. There is no evidence for 0∈A, nor for 1∈A, and constructive unprovability of their disjunction reflects this. Nonetheless, since ruling out excluded middle is provenly always inconsistent, it is also established that A is not empty. Classical logic adopts P∨¬P axiomatically, spoiling a constructive reading.

#### Example relating to choice

There are various easily characterized sets the existence of which is not provable in ZF, but which are implied to exist by the full axiom of choice AC. As such, that axiom is itself independent of ZF. It in fact contradicts other potential axioms for a set theory. Further, it indeed also contradicts constructive principles, in a set theory context. A theory that does not permit excluded middle does also not validate the function existence principle AC.

In ZF, the AC 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. This object is elusive in the sense that there are different ZF 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 consistently be negated. Again, that postulate may be expressed as saying that the set of such Hamel bases is non-empty. Over a constructive theory, such a postulate is weaker than the plain existence postulate, but (by design) is still strong enough to then negate all propositions that would imply the non-existence of a Hamel basis.

### Model theory

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".

However, it is possible to construct a Kripke model M 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 "X is nonempty" implies "X is inhabited".

## See also

* Intersection (set theory) - Set of elements common to all of some sets
* Company:Nothing
* Type inhabitation in type theory.

## References

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

* v
* t
* e

Mathematical logic

General|

* Formal language
* Formation rule
* Formal proof
* Formal semantics
* Well-formed formula
* Set
* Element
* Class
* Classical logic
* Axiom
* Rule of inference
* Relation
* Theorem
* Logical consequence
* Type theory
* Symbol
* Syntax
* Theory

Systems|

* Formal system
* Deductive system
* Axiomatic system
* Hilbert style systems
* Natural deduction
* Sequent calculus

Traditional logic|

* Proposition
* Inference
* Argument
* Validity
* Cogency
* Syllogism
* Square of opposition
* Venn diagram

Propositional calculus
and Boolean logic

* Boolean functions
* Propositional calculus
* Propositional formula
* Logical connectives
* Truth tables
* Many-valued logic

Predicate logic|

* First-order
* Quantifiers
* Predicate
* Second-order
* Monadic predicate calculus

Naive set theory|

* Set
* Empty set
* Element
* Enumeration
* Extensionality
* Finite set
* Infinite set
* Subset
* Power set
* Countable set
* Uncountable set
* Recursive set
* Domain
* Codomain
* Image
* Map
* Function
* Relation
* Ordered pair

Set theory|

* Foundations of mathematics
* Zermelo–Fraenkel set theory
* Axiom of choice
* General set theory
* Kripke–Platek set theory
* Von Neumann–Bernays–Gödel set theory
* Morse–Kelley set theory
* Tarski–Grothendieck set theory

Model theory|

* Model
* Interpretation
* Non-standard model
* Finite model theory
* Truth value
* Validity

Proof theory|

* Formal proof
* Deductive system
* Formal system
* Theorem
* Logical consequence
* Rule of inference
* Syntax

Computability
theory|

* Recursion
* Recursive set
* Recursively enumerable set
* Decision problem
* Church–Turing thesis
* Computable function
* Primitive recursive function

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

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