Dedekind cut
E634836
A Dedekind cut is a method of constructing the real numbers from the rational numbers by partitioning them into two nonempty sets that capture the idea of a "cut" point on the number line.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
construction of real numbers
ⓘ
mathematical concept ⓘ order-theoretic structure ⓘ |
| alternativeView | pair (A,B) of subsets of Q with A union B = Q and A less than B ⓘ |
| appearsIn |
axiomatic construction of the real numbers
ⓘ
foundations of analysis ⓘ foundations of mathematics ⓘ |
| basedOn | rational numbers ⓘ |
| captures | idea of a cut point on the number line ⓘ |
| constructionType | completion of an ordered field ⓘ |
| contrastsWith |
decimal expansion representation of real numbers
ⓘ
geometric construction of real numbers as points on a line ⓘ |
| correspondsTo |
a point on the real number line
ⓘ
a unique real number ⓘ |
| definition |
a partition of the rational numbers into two nonempty sets A and B such that every element of A is less than every element of B
ⓘ
a subset A of the rationals that is nonempty, not all of Q, downward closed, and has no greatest element ⓘ |
| distinguishes | rational and irrational real numbers ⓘ |
| ensures |
least upper bound property of the real numbers
ⓘ
order completeness of the real numbers ⓘ |
| equivalentTo | Cauchy sequence construction up to isomorphism of ordered fields ⓘ |
| field |
number theory
ⓘ
order theory ⓘ real analysis ⓘ set theory ⓘ |
| formalizedAs | lower set of Q with no maximum ⓘ |
| generalizationOf | Dedekind completion of a linearly ordered set ⓘ |
| hasOperation |
addition of cuts
ⓘ
multiplication of cuts ⓘ order relation induced by set inclusion ⓘ |
| introducedBy | Richard Dedekind NERFINISHED ⓘ |
| introducedIn | 19th century ⓘ |
| namedAfter | Richard Dedekind NERFINISHED ⓘ |
| property |
downward closed in the usual order on the rationals
ⓘ
has no greatest rational element ⓘ nonempty lower set of the rationals ⓘ proper subset of the rationals ⓘ |
| relatedTo | Cauchy sequence construction of the reals ⓘ |
| represents |
irrational real numbers by cuts whose complement has no least element
ⓘ
rational real numbers by cuts with a greatest element in the upper complement ⓘ |
| requires |
Archimedean property of the rationals
ⓘ
total order on the rational numbers ⓘ |
| usedFor | constructing the real numbers from the rational numbers ⓘ |
| usedIn |
constructive analysis
ⓘ
order-completion of ordered sets ⓘ |
| usedToDefine | complete ordered field of real numbers ⓘ |
| yieldsStructure |
complete densely ordered set without endpoints
ⓘ
complete ordered field isomorphic to the real numbers ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.