# Real Numbers form only Ordered Field which is Complete

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## Theorem

The set of real numbers $\R$ is the only ordered field which also satisfies the Continuum Property.

## Proof

From Real Numbers form Ordered Field we have that $\R$ forms an ordered field.

From the Continuum Property we have that $\R$ is complete.

It remains to be shown that any ordered field which also satisfies the Continuum Property is isomorphic to $\R$.

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## Sources

- 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**order properties**(of real numbers)