As far as I understand Dedekind cuts, they are really about ranges, not sets. In other words, they are about all of the values greater than or less than some number. But by using sets, I'm led to thinking about odd sets of numbers like "all the numbers with a 3 as the third decimal digit", or "all of the numbers that are the cosine of an angle that is a power of two". Is this generality necessary in defining Dedekind cuts? And if so, why?
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