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So in this videos at 45:46, the third property says that a Dedekind cut should not have a maximum element. So the way I understand the definition is that the real numbers can be thought of as the set of rational numbers less than it. But in the case that a real number is a rational number, then what could go wrong when saying that it is the set of rational numbers less than or equals to it?

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    $\begingroup$ Cf. baby Rudin chapter 1 exercise 20 $\endgroup$ – J. W. Tanner Jul 30 '19 at 15:16
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There really isn't any huge reason to do it one way or the other. The most important point is that one must make a choice, and everyone ought to agree on what choice has been made. This is the convention that has been chosen.

There are other arbitrary things that have been chosen as well. For instance, why is a cut the set of rationals below a given real, rather than the set of rationals above?

One choice may have advantages and disadvantages over another, but the most important thing in this case is to choose so that everyone agrees (competing conventions benefit almost no one). And that is what has been done here with Dedekind cuts.

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