 The following year, Dedekind published a paper [1872] in which he provided an axiomatic analysis of the structure of the set \(\mathbf{R}\) of real numbers. He defined it as an ordered field that is also complete (in the sense that all Dedekind-cuts on \(\mathbf{R}\) correspond to an element in \(\mathbf{R}\)); completeness in that sense has the Archimedean axiom as a consequence. Cantor too provided a definition of \(\mathbf{R}\) in 1872, employing Cauchy sequences of rational numbers, which was an elegant simplification of the definition offered by Carl Weierstrass in his lectures. The form of completeness axiom that Weierstrass preferred was Bolzano’s principle that a sequence of nested closed intervals in \(\mathbf{R}\) (a sequence such that \([a_{m+1},b_{m+1}] \subset [a_{m},b_{m}]\)) “contains” at least one real number (or, as we would say, has a non-empty intersection).