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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?

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  • $\begingroup$ It's not going to be a unique set for any definition, but any set satisfying one definition will be isomorph in their structure to any set satisfying another definition. $\endgroup$
    – Bram28
    Jul 27 '17 at 16:32
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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.

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