In
constructive mathematics
In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove th ...
, a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
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
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...
(or constructive logic).
Comparison with nonempty sets
In
classical mathematics In the foundations of mathematics, classical mathematics refers generally to the mainstream approach to mathematics, which is based on classical logic and ZFC set theory. It stands in contrast to other types of mathematics such as constructive m ...
, a set is inhabited if and only if it is not the
empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
. These definitions diverge in
constructive mathematics
In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove th ...
, however.
A set
is if
while
is if it is not empty, that is, if
It is
if
Every inhabited set is a nonempty set (because if
is an
inhabitant
Domicile is relevant to an individual's "personal law," which includes the law that governs a person's status and their property. It is independent of a person's nationality. Although a domicile may change from time to time, a person has only one ...
of
then
is false and consequently so is
).
In intuitionistic logic, the negation of a universal quantifier is weaker than an
existential quantifier
In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, w ...
, not equivalent to it as in
classical logic
Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy.
Characteristics
Each logical system in this class ...
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
A model is an informative representation of an object, person or system. The term originally denoted the Plan_(drawing), plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a mea ...
in the classical sense that contains a nonempty set
but does not satisfy "
is inhabited". But it is possible to construct a
Kripke model
Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André Jo ...
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
In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwee ...
, it is necessary for a specific value of
satisfying
to be known.
For example, consider a
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of
specified by the following rule:
belongs to
if and only if the
Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
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
* D. Bridges and F. Richman. 1987. ''Varieties of Constructive Mathematics''. Oxford University Press.
{{PlanetMath attribution, id=5931, title=Inhabited set
Basic concepts in set theory
Concepts in logic
Constructivism (mathematics)
Mathematical objects
Set theory