How many real numbers exist? New proof moves closer to an answer

Re:How many real numbers exist? New proof moves closer to an... by CamperBob2 on Thursday July 15, 2021 @07:02PM (#27851120)

Not a mathematician, so this probably has an obvious answer, but who says that "size" is a necessary universal property of a set? It doesn't seem any more rational to speak of size as a required property of a set than it would be to treat color or weight that way. Some sets have those attributes, but not all.

It seems perfectly reasonable to say that the set of real numbers is one of those sets that doesn't have a "size" property.

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Re:How many real numbers exist? New proof moves closer to an... by dwohnitmok on Thursday July 15, 2021 @07:55PM (#27851546)

Size is a hand-wavy way of saying it. What we're really asking is given two sets A and B, is there a function from A to B such that no two elements from A are mapped to a single element in B?

It turns out if you use ZFC (in particular choice) then one direction must hold. Either such a function exists from A to B or from B to A. This means that we can order all sets using the existence of these functions, and so in that sense size is a "universal property."

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