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Difference between revisions of "Dedekind theorem"

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''on the continuity of the real axis''
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For any cut <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030560/d0305601.png" /> of the set of real numbers (see [[Dedekind cut|Dedekind cut]]) there exists a real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030560/d0305602.png" /> which is either the largest in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030560/d0305603.png" /> or the smallest in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030560/d0305604.png" />. This statement is also known as the Dedekind principle (axiom) of continuity of the real axis (cf. [[Real number|Real number]]). The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030560/d0305605.png" /> is the least upper bound of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030560/d0305606.png" /> and the greatest lower bound of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030560/d0305607.png" />.
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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====
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* 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

How to Cite This Entry:
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