## Idea ## I am going to define this in terms of Archimedean ordered Q-algebras... ## Definition ## Strict order axioms: $\lt$ * For all terms $a:\mathbb{R}$, $a \lt a$ is false. * For all terms $a:\mathbb{R}$, $b:\mathbb{R}$, $c:\mathbb{R}$, $a \lt c$ implies $a \lt b$ or $b \lt c$ * For all terms $a:\mathbb{R}$, $b:\mathbb{R}$, not $a \lt b$ and not $b \lt a$ implies $a = b$. * For all terms $a:\mathbb{R}$, $b:\mathbb{R}$, $a \lt b$ implies not $b \lt a$ Archimedean property: * For all terms $a:\mathbb{R}$, $b:\mathbb{R}$, and $c:\mathbb{R}$, $a \lt a + b$ and $a \lt a + c$ implies that there exists a natural number $n:\mathbb{N}$ such that $b \lt n c$, where $n c$ is the additive $n$-th power (n-fold addition) One axioms: $1$ * $1 \lt 1 + 1$ ## See also ## * [[real numbers]] category: not redirected to nlab yet