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Reading Dedekind original manuscript (https://archive.org/details/essaysintheoryof00dedeuoft) it seems that he was after formalizing the completeness of the real numbers, which was known to him as the continuity of real numbers. Formalizing the completeness of real numbers then easily led him to construction of real numbers using cuts of rational numbers, which also provided a model for the real numbers(https://en.wikipedia.org/wiki/Dedekind_cut).

However, I have seen some sources talking about Eudoxus ratios (https://en.wikipedia.org/wiki/Eudoxus_of_Cnidus) being the predecessor of the Dedekind cuts, which sort of means that Dedekind knew how real numbers could be constructed using rational numbers. This in turn means that the Dedekind construction of real numbers suggested the completeness of real numbers to Dedekind.

So the question is that how did Dedekind conclude the real numbers are complete? Was he convinced that the reals were complete because they could be constructed using cuts of rational numbers (was this an acceptable (logical) argument to him)? or he followed a different logic?

In other words, what made Dedekind accept real numbers are complete, was it the construction of the real numbers or was it something else?

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    $\begingroup$ In a certain sense, completeness has been "postulated" from the start... We have the well known "continuity" of the geometric line: the continuum. The ancient Greek math discovered that $\sqrt 2$ is not a "number" and thus that there are more points than numbers. So, the idea was to find an "enlarged" definition of number to "match" the continuum. $\endgroup$
    – Mauro ALLEGRANZA
    May 25, 2017 at 6:03
  • $\begingroup$ Dedekind " seems to have been the first to recognize that the property of density, possessed by the ordered set of rational numbers, is insufficient to guarantee continuity. In Continuity (1872) he remarks that when the rational numbers are associated to points on a straight line, “there are infinitely many points [on the line] to which no rational number corresponds” so that the rational numbers manifest “a gappiness, incompleteness, discontinuity”, in contrast with the straight line's “absence of gaps, completeness, continuity.” $\endgroup$
    – Mauro ALLEGRANZA
    May 25, 2017 at 6:04
  • $\begingroup$ @Mauro ALLEGRANZA: I disagree with your second remark. That there are infinitely many irrational points was perfectly understood by the Greeks, and Nicola Oresme even tried to prove that the "majority of numbers are irrational". $\endgroup$
    – Alexandre Eremenko
    May 25, 2017 at 7:18
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    $\begingroup$ @AlexandreEremenko - it is a quote from SEP; but I agree: points on the geometrical lines and numbers were not the same. To cut a segment of lenght $\sqrt 2$ is quite easy, but they know that it was impossible to express that lenght as a ration between (natural) numbers. To assume that we have some sort of "number" to measure that lenght required a "stretching" of the concept of number that was of course "imagined" somewhere by someone, but that was not defined in a mathematical correct way until Dedekind. This is my "reading" of Bell's entry. $\endgroup$
    – Mauro ALLEGRANZA
    May 25, 2017 at 7:22

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That the set of "real numbers" must complete was intuitively understood already by the Greeks, though they had neither a formal concept of real number nor the formal definition of completeness. For them, the "numbers" were rational numbers, and the "irrationality of $\sqrt{n}$" meant that there are no enough numbers to measure the lengths of all segments in geometry. But the intuitive notion of numbers which allow to measure any length was desirable and it slowly evolved. By the time of Dedekind, it was clear what mathematicians want of real numbers, and the problem was to "construct" them from rationals. In particular it was known that what we call "completeness" is a desirable property. Dedekind found a way to formally construct from rational numbers the system of numbers with desired properties, including completeness.

This was not the first such construction. Eudoxus's theory of proportions also achieves this purpose. Its disadvantage is that it is more cumbersome and awkward to use.

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  • $\begingroup$ Thanks I have seen in some sources that Dedekind's cut was inspired by Eudoxus's theory of proportions. I think he also rigorously defined completeness (continuity) was this also inspired by Eudoxus's theory of proportions or this was just a property that was desired but was not formalized before him. $\endgroup$
    – abk
    May 25, 2017 at 12:28

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