I understand the definition of real numbers in set theory. We define the term "Dedekind-complete ordered field" and prove that all Dedekind-complete ordered fields are isomorphic. Then it makes sense to say that any of them can be thought of as "the" set of real numbers. We can prove that a Dedekind-complete ordered field exists by explicitly constructing one from the ordered field of of rational numbers. But how do you state the axioms of the real numbers without using set theory? How did mathematicians do it before every useful branch of mathematics was shown to fit inside a set theory? The completeness axiom is the only one that's causing me any trouble: In a set-theoretic approach, the axiom can be stated like this: "Every non-empty subset of ##\mathbb R## that has an upper bound has a least upper bound". Is there a good way to state it without reference to sets? I thought about replacing this axiom with one about metric space completeness instead, but the definitions of "metric space" and "limit" mention real numbers, so this sounds circular. The reason I'm interested in this is that it's relevant to my understanding of what mathematics is. ZFC set theory is much prettier of course, but it doesn't seem wrong to define a much smaller branch where the primitives (undefined concepts) are real numbers, addition and multiplication (and perhaps a few more things), instead of sets and set membership.