# Real Numbers form only Ordered Field which is Complete

## 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 $\R$.