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 nonArchimedean 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.
The least-upper-bound property states that every nonempty subset of real numbers having an upper bound must have a least upper bound in the set of real numbers. The rational number lineQ does not have the least upper bound property. An example is the subset of rational numbers This set has an upper bound. However, this set has no least upper bound in : the least upper bound as a subset of the reals would be, but it does not exist in. For any upper bound, there is another upper bound with. For instance, take, then is certainly an upper bound of, since is positive and ; that is, no element of is larger than. However, we can choose a smaller upper bound, say ; this is also an upper bound of for the same reasons, but it is smaller than, so is not a least-upper-bound of. We can proceed similarly to find an upper bound of that is smaller than, say, etc., such that we never find a least-upper-bound of in. The least upper bound property can be generalized to the setting of partially ordered sets. See completeness.
Dedekind completeness
Dedekind completeness is the property that every Dedekind cut of the real numbers is generated by a real number. In a synthetic approach to the real numbers, this is the version of completeness that is most often included as an axiom. The rational number line Q is not Dedekind complete. An example is the Dedekind cut L does not have a maximum and R does not have a minimum, so this cut is not generated by a rational number. There is a construction of the real numbers based on the idea of using Dedekind cuts of rational numbers to name real numbers; e.g. the cut described above would name. If one were to repeat the construction of real numbers with Dedekind cuts, one would obtain no additional numbers because the real numbers are already Dedekind complete.
Cauchy completeness
Cauchy completeness is the statement that every Cauchy sequence of real numbers converges. The rational number line Q is not Cauchy complete. An example is the following sequence of rational numbers: Here the nth term in the sequence is the nth decimal approximation for pi. Though this is a Cauchy sequence of rational numbers, it does not converge to any rational number. Cauchy completeness is related to the construction of the real numbers using Cauchy sequences. Essentially, this method defines a real number to be the limit of a Cauchy sequence of rational numbers. In mathematical analysis, Cauchy completeness can be generalized to a notion of completeness for any metric space. See complete metric space. For an ordered field, Cauchy completeness is weaker than the other forms of completeness on this page. But Cauchy completeness and the Archimedean property taken together are equivalent to the others.
The nested interval theorem is another form of completeness. Let be a sequence of closed intervals, and suppose that these intervals are nested in the sense that Moreover, assume that as. The nested interval theorem states that the intersection of all of the intervals contains exactly one point. The rational number line does not satisfy the nested interval theorem. For example, the sequence is a nested sequence of closed intervals in the rational numbers whose intersection is empty. Nested intervals theorem shares the same logical status as Cauchy completeness in this spectrum of expressions of completeness. In other words, nested intervals theorem by itself is weaker than other forms of completeness, although taken together with Archimedean property, it is equivalent to the others.
The monotone convergence theorem states that every nondecreasing, bounded sequence of real numbers converges. This can be viewed as a special case of the least upper bound property, but it can also be used fairly directly to prove the Cauchy completeness of the real numbers.
Bolzano–Weierstrass theorem
The Bolzano–Weierstrass theorem states that every bounded sequence of real numbers has a convergent subsequence. Again, this theorem is equivalent to the other forms of completeness given above.
The intermediate value theorem states that every continuous function that attains both negative and positive values has a root. This is a consequence of the least upper bound property, but it can also be used to prove the least upper bound property if treated as an axiom.
