The geometric definition of an ordered field: * an [[nlab:ordered local ring]] $R$ where for all elements $x \in R$, $x$ is invertible or $x = 0$ * equivalently, an ordered local ring which satisfies [[nlab:trichotomy]]. The [[nlab:limited principle of omniscience]] implies that the discrete (i.e. Cauchy, Escardo-Simpson, decidable Dedekind) real numbers are the terminal such an ordered field. More importantly, it implies that every pointwise continuous function is uniformly continuous via the [[nlab:lesser limited principle of omniscience]]. [[!redirects Sandbox > history]]