A perfect world

Mathematicians have axiomatized the real line as a one-dimensional continuum, as a complete ordered Archimedean field, as a real closed field, or as a system of binary decimals on which arithmetical operations are performed in a certain way. Each of these axiomatizations is tacitly understood (...) as an axiomatization of the same real line.

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(...) there is no way of dealing with mathematical items rigorously except through axiom systems. But this is like saying that there is no way of communicating ideas except through words. Although any idea has to be represented in sentences, the same idea may be expressed by completely different sentences. An idea is "independent" (...) of the words used to express the idea. When we assert that a mathematical item is "independent" of any particular axiom system, we mean this "independence" in much the same way as independence of ideas from language.

Gian-Carlo Rota, Indiscrete Thoughts.