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The Dedekind cut formulation of the real numbers is that a real number is defined to be a nonempty "downwards closed" subset of rational numbers. Then the min o

• thaumasiotes 7y ago
  The somewhat-standard definition of the naturals (0 = {}, 1 = {0}, 2 = {0, 1}, 3 = {0, 1, 2}, ...) also maps min to set intersection. GCD, which someone else mentions in another comment, is very straightforwardly a set intersection problem. The intuition here is that an operation throws away information that isn't relevant while keeping the relevant information. If you can't lose relevant information, you can't fail to be associative. Probably all such operations can be straightforwardly mapped to set intersection or set union. (Under the definition of the naturals given above, note that set union gives you the max operator.)
• thaumasiotes 7y ago
  > The Dedekind cut formulation of the real numbers is that a real number is defined to be a nonempty "downwards closed" subset of rational numbers. Then the min operation corresponds to taking intersections They didn't figure that out exactly. They figured out something much less significant: that a real number is defined by a nonempty upwards closed subset of real numbers: >> My second epiphany was that equality of real numbers can also be characterized by having the same “downsets”, i.e. two real numbers are equal if and only if the sets of real numbers less than or equal to them are the same. In other words, we choose to represent the real number r as the closed interval (-∞, r]. The point of Dedekind cuts is that you can define real numbers without using real numbers in the definition. They do then set up min as set intersection as suggested by the interval definition. Writing out the choice of real number representation makes it clear that we could also represent a real number as the interval [r, ∞), in which case minimum would be set union rather than set intersection. It doesn't really matter whether we use a closed or open interval.