# What does it mean for two definitions (of the reals) to be equivalent? [duplicate]

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?

• 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. Jul 27 '17 at 16:32