Completeness axiom
Encyclopedia
In mathematics
the completeness axiom, also called Dedekind completeness of the real numbers, is a fundamental property of the set R of real numbers. It is the property that distinguishes R from other ordered field
s, especially from the set of rational number
s.
The axiom states that every non-empty subset S of R that has an upper bound in R has a least upper bound, or supremum
, in R. See the article on construction of the real numbers for a full explanation.
The completeness axiom should not be confused with the topological property of completeness of a metric space. The two properties are related, since R, as a metric space with the standard absolute-value metric (where the distance between x and y is |x−y|), does have the latter property as a consequence of its Dedekind completeness. Indeed, R is the completion, in the sense of metric spaces, of the set Q of rational numbers under the absolute-value metric. Thus, the completeness property of metric spaces is one generalization of the completeness axiom itself.
Another generalization focuses on the ordering of the real numbers. In any partially ordered set
, the analog of Dedekind completeness is the property that every non-empty subset that is bounded above has a least upper bound; in other words, the same axiom interpreted in greater generality. A partially ordered set with this property is a lattice
, specifically a conditionally complete lattice. In practice a stronger property is usually employed: that every subset, whether or not it is empty or bounded above, has a least upper bound. Such a partially ordered set is called a complete lattice
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
the completeness axiom, also called Dedekind completeness of the real numbers, is a fundamental property of the set R of real numbers. It is the property that distinguishes R from other ordered field
Ordered field
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and...
s, especially from the set of rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s.
The axiom states that every non-empty subset S of R that has an upper bound in R has a least upper bound, or supremum
Supremum
In mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...
, in R. See the article on construction of the real numbers for a full explanation.
The completeness axiom should not be confused with the topological property of completeness of a metric space. The two properties are related, since R, as a metric space with the standard absolute-value metric (where the distance between x and y is |x−y|), does have the latter property as a consequence of its Dedekind completeness. Indeed, R is the completion, in the sense of metric spaces, of the set Q of rational numbers under the absolute-value metric. Thus, the completeness property of metric spaces is one generalization of the completeness axiom itself.
Another generalization focuses on the ordering of the real numbers. In any partially ordered set
Partially ordered set
In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...
, the analog of Dedekind completeness is the property that every non-empty subset that is bounded above has a least upper bound; in other words, the same axiom interpreted in greater generality. A partially ordered set with this property is a lattice
Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
, specifically a conditionally complete lattice. In practice a stronger property is usually employed: that every subset, whether or not it is empty or bounded above, has a least upper bound. Such a partially ordered set is called a complete lattice
Complete lattice
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum . Complete lattices appear in many applications in mathematics and computer science...
.