I have read that the real numbers can be defined in several ways, but that these definitions are all 'equivalent'. What does this actually mean? Is it that there is a unique set of numbers satisfying each definition, and that these sets are all the same?
2026-05-15 16:01:29.1778860889
What does it mean for two definitions (of the reals) to be equivalent?
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Usually equivalent means, there is an isomorphism between these objects preserving the structure.
No matter which definition of real numbers you choose, it will always be a totally ordered archimedean field verifying, that every nonempty bounded subset has a supremum, which makes the real numbers essentially unique. There are just different ways of constructing such an object.