Difference between revisions of "Dedekind theorem"
From Encyclopedia of Mathematics
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| − | + | ''on the continuity of the real axis; Dedekind principle, Dedekind axiom'' | |
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| + | A form of the [[continuity axiom]] for the [[real number]] system in terms of [[Dedekind cut]]s. It states that for any cut $A|B$ of the set of real numbers there exists a real number $\alpha$ which is either the largest in the class $A$ or the smallest in the class $B$. The number $\alpha$ is the [[least upper bound]] of $A$ and the [[greatest lower bound]] of $B$. | ||
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| + | ====References==== | ||
| + | * Richard Dedekind, "Essays on the Theory of Numbers" (tr. W.W.Beman) Dover (1963) [1901] {{ISBN|0-486-21010-3}} {{ZBL|32.0185.01}} {{ZBL|0112.28101}} | ||
Latest revision as of 20:44, 23 November 2023
on the continuity of the real axis; Dedekind principle, Dedekind axiom
A form of the continuity axiom for the real number system in terms of Dedekind cuts. It states that for any cut $A|B$ of the set of real numbers there exists a real number $\alpha$ which is either the largest in the class $A$ or the smallest in the class $B$. The number $\alpha$ is the least upper bound of $A$ and the greatest lower bound of $B$.
References
- Richard Dedekind, "Essays on the Theory of Numbers" (tr. W.W.Beman) Dover (1963) [1901] ISBN 0-486-21010-3 Zbl 32.0185.01 Zbl 0112.28101
How to Cite This Entry:
Dedekind theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dedekind_theorem&oldid=17055
Dedekind theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dedekind_theorem&oldid=17055
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article