| Themes > Science > Mathematics > Calculus > Real Number > Getting rid of the pictures |
The "point on a line" answer is not a fully satisfactory answer, because it is not axiomatic or algebraic. It relies on pictures that we don't really understand. For instance, the set of real numbers and the set of rational numbers have essentially the same picture, but their algebraic properties differ in ways that are very important for analysts. Imagine studying that picture of a line under a super microscope. If you could magnify the line at a very high power -- say at a magnification of a googolplex, or better yet a magnification of infinity -- would it still look the same? Or would you see a row of dots separated by spaces, like the dots in a picture in a newspaper? (It turns out that, in some sense, the real numbers would still look like a line under infinite magnification, but the rational numbers would be dots separated by spaces. But that is only a vague and intuitive statement, not anything precise that we can use in proofs.) The only way to get rigorous answers to these questions is to set up a very careful system of axioms about geometry ... but that amounts to the same thing as setting up a careful set of axioms about the algebraic properties of the real numbers. It turns out that the latter is a little easier, so we may as well concentrate on the algebraic aspects of the situation. To answer questions like this, ultimately we have to get away from the pictures; we have to understand the real numbers entirely in terms of formulas. As a preview, here is the definition that we're going to end up with: the real line is a Dedekind-complete ordered field. That's complicated, so we'll work our way up to it in stages. We'll discuss:
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