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?

  • $\begingroup$ 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

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.


Not the answer you're looking for? Browse other questions tagged or ask your own question.