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PlanetMath: real number (919 words) |
 | The real numbers can also be defined as the unique (up to isomorphism) ordered field satisfying the least upper bound property, after one has proved that such a field exists and is unique up to isomorphism. |
| However, an ordered group (and a field is a group under the operations of addition and subtraction) defines a uniform structure, and uniform structures have a notion of completeness (topology); the description in the Completeness section above is a special case. |
| But the original use of the phrase “complete Archimedean field” was by David Hilbert, who meant still something else by it. |
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