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It's sort of a meaningless and subjective question to ask how valuable a proof or construction is among others if they are all equivalent and they all lead to the same theory. I think, however, it is worth asking whether some proofs and constructions have more pedagogical value than others. (For example, the topological definition of continuity, while true in the limited domain of AB calculus, has almost no pedagogical value until the student is ready to tackle college level analysis.)

My main question is: What are the pedagogical advantages of using Dedekind cuts to construct the real numbers as opposed to the classic approach of metric space completion of the rationals through Cauchy Sequences, or Rudin's construction of the real field by assuming the least upper bound property? I have noticed that some people have strong opinions on the value of Dedekind cuts ("...the stupidest thing ever..."), so I guess what I'm asking for is a defense of the approach at least in analysis pedagogy, or even what new perspectives/theories Dedekind cuts raise that Cauchy sequences or L.U.B. do not elucidate. (This is also for my own education, as I have been taught using Cauchy sequences and L.U.B.)

I apologize if this question is not suited for this site.

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    $\begingroup$ I like the Dedekind cuts, because they make it intuitively obvious that the real line has the LUB property. After all, that's what the cuts are: LUBs of bounded sets. $\endgroup$ – Arthur Feb 6 '18 at 9:31
  • $\begingroup$ I like that way of thinking about it. To play devil’s advocate, why is that more intuitively obvious than the classic example of a set in an ambient space that does not have an LUB, $\{x \in \mathbb{Q} | x^2 <2 \}$? $\endgroup$ – Marcus Aurelius Feb 6 '18 at 9:47
  • $\begingroup$ The construction of reals via Dedekind cuts is easiest to understand because it requires only the knowledge of arithmetic / order relations of rationals (available to 12 year old 7th graders). It's a pity that this wonderful thing is purposely hidden from students until they have done a course in calculus and are starting with a course in real analysis. $\endgroup$ – Paramanand Singh Feb 6 '18 at 10:28
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    $\begingroup$ You might get better answers over at matheducators.stackexchange.com $\endgroup$ – Stig Hemmer Feb 6 '18 at 10:52
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    $\begingroup$ @MarcusAurelius : the definitions for $<, >, +, - $ using Dedekind cuts are a cakewalk, but some care is needed while defining $\times, /$. You can find nice explanations for these in Dedekind's original paper Stetigkeit und irrationale zahlen (English translation available). $\endgroup$ – Paramanand Singh Feb 7 '18 at 0:52

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