Linked article reminded me of a passage in "Topoi - The Categorial Analysis of Logic" by Robert Goldblatt: "It would be somewhat misleading to infer [...] that

Linked article reminded me of a passage in "Topoi - The Categorial Analysis of Logic" by Robert Goldblatt: "It would be somewhat misleading to infer [...] that foundational systems act primarily as a basis out of which mathematics is actually created. The artificiality of that view is evident when one reflects that the essential content of mathematics is already there before the basis is made explicitly, and does not depend on it for its existence. We may for example think of a real number as an infinite decimal expression, or a point on the number line. Alternatively it could be introduced as an element of a complete ordered field, an equivalence class of Cauchy sequences, or a Dedekind cut. None of these could be said to be the correct explanation of what a real number is. Each is an embodiment of an intuitive notion and we evaluate it, not in terms of its correctness, but rather in terms of its effectiveness in explicating the nature of the real number system." Tim Gowers makes a good case for the usefulness of the "infinite decimal expression" view of real numbers.