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When did the constructions of Reals take place? What is the latest construction the one due to Cantor (by Cauchy Sequences) or the Dedekind? I ask because the trustful reference (baby Rudin) that I looked, tells they were published in the same year 1872.

That seems to me, Cantor's idea was more fruitful (however was quite similar) because it was used to complete any Metric space. Nevertheless the Dedekind construction may rest as an Historical thing when the Cantor idea should be the one taught obligatorily in the Analysis courses.

Personal commentary: I don't think the Dedekind's idea can be used in any other construction.

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    Did you check the [wikipedia article on real numbers](http://en.wikipedia.org/wiki/Real_numbers#History) before asking?2012-02-19
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    The article is wrong Cantor and Dedekind gave the construction of the Reals in 1872. See Rudin, Principles of Real Analysis.2012-02-19
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    @chessmath: To avoid further downvotes, closing votes, and to attract better answers you might want to consider editing the question to add your previous comment; what you have read on the topic and so forth. Regarding the latest construction, I can tell you that our freshman year just finished constructing it last semester. The model they built hasn't been used yet!2012-02-20

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