real number system

real number system

  1. (mathematics) The complete Archimedean ordered field (of real numbers).
    • As we have mentioned before, the structure of Euclidean geometry, as formalized through the axioms of Hilbert, produces an archimedean ordered field. To com- plete the story, one can add to these axioms the further requirement that this field is maximal in the sense that it cannot be embedded inside any larger archimedean ordered field. It turns out then that any such ordered field is isomorphic to any other, and thus there is essentially one such ordered field. This ordered field is the real number system R.
      The real number system is a complete, linearly ordered field \mathcal{R} = (\mathbb{R}, +, \cdot, <).
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