# Homotopy Type Theory Dedekind real numbers > history (Rev #10)

## Definitions

The locally $\mathcal{U}$-small Dedekind real numbers for a universe $\mathcal{U}$ is defined as the Archimedean ordered integral domain $\mathbb{R}_\mathcal{U}$ with a strictly monotonic function $i:\mathbb{I}_\mathcal{U} \to \mathbb{R}_\mathcal{U}$ from the locally $\mathcal{U}$-small Dedekind real unit interval $\mathbb{I}_\mathcal{U}$ to $\mathbb{R}_\mathcal{U}$ such that $i(0) = 0$ and $i(1) = 1$.