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read some more on Dedekind cuts.
The first serious consideration of the real numbers was by [[Archimedes]] and followed by other [[Greek]]s such as [[Euclid]], [[Pappus]], and [[Zeno]]. To honor Archimedes' contribution, real analysts have named a property of the real numbers the [[Archimedean|Archimedean property]]. Real analysis remained in [[geometry]]'s shadow until the development of the subfield of [[calculus]]. This subject [[coordinatization|coordinatized]] all geometry known at the time, subsuming it into its scope.
The first serious consideration of the real numbers was by [[Archimedes]] and followed by other [[Greek]]s such as [[Euclid]], [[Pappus]], and [[Zeno]]. To honor Archimedes' contribution, real analysts have named a property of the real numbers the [[Archimedean|Archimedean property]]. Real analysis remained in [[geometry]]'s shadow until the development of the subfield of [[calculus]]. This subject [[coordinatization|coordinatized]] all geometry known at the time, subsuming it into its scope.
The foundations of real analysis were shaken at the end of the 19th century with the work of [[Richard Dedekind]], who relaid the Archimedean-style groundwork with his own radical concepts. His [[Dedekind cut]]s undercut the assumption of the continuity of the real line, by ''cutting'' at ''gaps'' between points. Mathematicians were worried that his techniques used the dubious [[Axiom of Choice]] and seemingly [[elementary_proof|non-elementary]] methods. However, the unifying ideas of [[Cauchy]], specifically that of the [[Cauchy sequence]] and [[complete metric space|completeness]], helped eliminate doubts and gain acceptance for Dedekind's ideas among real analysts. Dedekind cuts are now viewed as a solid foundation for real analysis, more than Archimedes' ideas ever were.<ref>http://plato.stanford.edu/entries/dedekind-foundations/#FouAna</ref>
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[[category:mathematics]]
[[category:mathematics]]