Completeness of the real numbers
Forms of completeness

The real numbers can be defined synthetically as an ordered field satisfying some version of the completeness axiom. Different versions of this axiom are all equivalent in the sense that any ordered field that satisfies one form of completeness satisfies all of them, apart from Cauchy completeness and nested intervals theorem, which are strictly weaker in that there are non Archimedean fields that are ordered and Cauchy complete. When the real numbers are instead constructed using a model, completeness becomes a theorem or collection of theorems.
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