Ordered Field of Rational Cuts is Isomorphic to Rational Numbers

Theorem

Let $\struct {\RR, +, \times, \le}$ denote the ordered field of rational cuts.

Let $\struct {\Q, +, \times, \le}$ denote the field of rational numbers.

Then $\struct {\RR, +, \times, \le}$ and $\struct {\Q, +, \times, \le}$ are isomorphic.

Proof

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Sources

• 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Real Numbers