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The Axiom of Completeness states that any non-empty set with an upper bound has a least upper bound. When and why was this concept of least upper bound dubbed "completeness"?

It's true, of course, that least upper bound is one way to establish the completeness of the real number system. But there are many other equivalent ways, none of them called "Axiom of Completeness".

In particular, see https://math.stackexchange.com/questions/4870101/is-the-axiom-of-completeness-logically-equivalent-to-there-is-no-proper-superse where it is explored that "completeness" may have originally had a different meaning: that any thing that could be a number is a number, or, more formally, that $\mathbb R$ is a maximal ordered Archimedean field, such that there is no proper superset of $\mathbb R$ that is an ordered Archimedean field.

To me, this suggests that the notion of completeness was first defined this way: that $\mathbb R$ is complete in the sense that nothing could be added to it without taking away essential properties, and only afterwards did people say "this is equivalent to saying that any non-empty set with an upper bound has a least upper bound" and so they called that second property completeness. Are there any sources shedding light on this?

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    $\begingroup$ The earliest statement called the "Axiom of Completeness" that I know is at the end of chapter 1 of Hilbert's Foundations of Geometry. This seems to establish his goal of "a simple and complete set of independent axioms" set out in the Introduction. I don't know much beyond this, unfortunately. I don't know how "completeness," which seems to mean logically complete here, came to mean "no gaps" in the reals. Hilbert says it implies it. Maybe that's how. $\endgroup$
    – Michael E2
    Feb 25 at 23:31
  • $\begingroup$ @MichaelE2 Thanks. I found the source and posted it below. Is there really no prior usage? $\endgroup$ Feb 26 at 2:47
  • $\begingroup$ As @Conifold showed, Dedekind discussed continuity, as a property of the real number system, in terms of completeness and absence of gaps. This is well before Hilbert, obviously. Dedekind does not use the term "axiom of completeness." Hilbert's use, as I said, seems to be about logical completeness of an axiomatic system. Cantor does not seem to use "completeness" in his series of articles in the 1880s about "infinite linear point-manifolds" (undendliche, lineare Punktmannichfaltigkeiten), even though he discusses Dedekind's construction of the real number system. I know little else, though $\endgroup$
    – Michael E2
    Feb 26 at 13:24

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According to Burn, Irrational numbers in English language textbooks, 1890–1915: Constructions and postulates for the completeness of the real numbers, "completeness" was first used by Dedekind in 1872, and then by Hilbert in 1899. This should not be too surprising. Meta-mathematical considerations concerning classes of number systems were rare before axiomatizations, they were studied in concrete realizations, so the term would not have been very useful. Dedekind was a rare exception anticipating later algebraic formalism. Hilbert needed it to distinguish standard geometry from algebraic and non-Archimedean models, and those were not studied until the very end of 19th century.

Both Dedekind's and Hilbert's uses were sporadic, widespread adoption of "completeness" only happened in 1920-s, with the general theory of metric spaces. Dedekind and others did use "continuity" derived from the "continuum" for similar purposes before that. Dedekind seems to prefer "continuity" and talks about "all continuous domains". The least upper bound property and the Archimedean axiom are sometimes called "axioms of continuity" even today.

Here is Burn:

"There is an anachronism in our title, for the widespread use of the term "completeness" to describe the extension of the rational numbers to the real numbers only became conventional from its occurrence in the study of metric spaces, though it was used by Dedekind in [1872] and in his correspondence with Lipschitz [Dedekind 1876], and by Hilbert in his axioms for geometry [1899] (in every translation, and every edition after the first).

In the period with which we are concerned, the issue would generally have been worded differently. How, it would have been asked, could the continuity of the real line be guaranteed or how could the continuum be characterized? Dedekind had used the term "completeness" in [1872] both to describe the closure of a number field under the four arithmetical operations, and as a synonym for "continuity." We will retain the modern term ''completeness'' for the sake of clarity, since the term ''continuity,'' in modern usage, applies only to functions."

Dedekind's Stetigkeit und irrationale Zahlen (1872) is translated into English by Berman, it is well-known for developing (Dedekind) cuts. The 1876 letter to Lipschitz is printed in Dedekind's Gesammelte mathematische Werke, vol. 3, p. 473, 1930. Here is from Berman's translation:

"If now, as is our desire, we try to follow up arithmetically all phenomena in the straight line, the domain of rational numbers is insufficient and it becomes absolutely necessary that the instrument $R$ constructed by the creation of the rational numbers be essentially improved by the creation of new numbers such that the domain of numbers shall gain the same completeness, or as we may say at once, the same continuity, as the straight line.

[...] The above comparison of the domain $R$ of rational numbers with a straight line has led to the recognition of the existence of gaps, of a certain incompleteness or discontinuity of the former, while we ascribe to the straight line completeness, absence of gaps, or continuity. In what then does this continuity consist? Everything must depend on the answer to this question, and only through it shall we obtain a scientific basis for the investigation of all continuous domains."

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Michael E2 suggests the first usage of the Axiom of Completeness was Hilbert (p.21) in 1899 (!!!), who writes:

Remark. To the preceeding five groups of axioms, we may add the following one, which, although not of a purely geometrical nature, merits particular attention from a theoretical point of view. It may be expressed in the following form:

Axiom of Completeness. To a system of points, straight lines, and planes, it is impossible to add other elements in such a manner that the system thus generalized shall form a new geometry obeying all of the five groups of axioms. In other words, the elements of geometry form a system which is not susceptible of extension, if we regard the five groups of axioms as valid.

This axiom gives us nothing directly concerning the existence of limiting points, or of the idea of convergence. Nevertheless, it enables us to demonstrate Bolzano’s theorem by virtue of which, for all sets of points situated upon a straight line between two definite points of the same line, there exists necessarily a point of condensation, that is to say, a limiting point. From a theoretical point of view, the value of this axiom is that it leads indirectly to the introduction of limiting points, and, hence, renders it possible to establish a one-to-one correspondence between the points of a segment and the system of real numbers.

I'm shocked that no such term wasn't used in the 1800s when completeness was being conceived.

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