Who first used the Completeness Axiom for real numbers?

Question

I was studying calculus and the following question came to my mind: Who was the first person to use or suggest the use of the Completeness Axiom of the Real Numbers?

Answer 1

Nobody. Those who were first did not have a clear idea of real numbers or completeness, and by the time the concepts took shape those who used them were no longer first, see MacTutor, The real numbers: Stevin to Hilbert. The first to state completeness as an axiom, to back up his prior axiomatization of geometry, was Hilbert in Über den Zahlbegriff (1900), see SEP, Formal Axiomatics for commentary. But he sure was not the first to use it.

The intuitive notion of "real numbers" was implicit in Stevin's use of infinite decimals that became increasingly common in 17-18th centuries, and completeness was "used" when assuming that graphs of functions intersect where it looks like they do, etc. But the concepts remained vague and implicit until late 19th century. Cauchy "used" completeness in Cours d'Analyse (1821), but as Grabiner writes in The Origins of Cauchy's Rigorous Calculus:

"...though Cauchy implicitly assumed several forms of the completeness axiom for the real numbers, he did not fully understand the nature of completeness or the related topological properties of sets of real numbers or of points in space... Cauchy did not have explicit formulations for the completeness of the real numbers. Among the forms of the completeness property he implicitly assumed are that a bounded monotone sequence converges to a limit and that the Cauchy criterion is a sufficient condition for the convergence of a series."

Bolzano was more explicit with his definition of real numbers as convergent sequences of rationals from 1817 on, but historians disagree as to whether his development was logically satisfactory. Even if it was, it was ahead of his time and went largely unnoticed.

In any case, in Bolzano's treatment, as well as in Weierstrass's, Cantor's, Dedekind's and Heine's later into the century, completeness appears not as an axiom but as a theorem, a consequence of their constructions of real numbers in terms of rational sequences or cuts, see Snow, Views on the real numbers and the continuum for details. Frege complained that such constructions do not guarantee that the resulting systems are consistent, but, as we now know from Goedel's results, no such guarantees can be given in principle, whether by constructions or by Hilbert style axiomatizations.