\begin{definition}[Definition:Rank Axioms (Matroid)/Definition 1]
Let $S$ be a finite set.
Let $\rho: \powerset S \to \Z$ be a mapping from the power set of $S$ into the integers
$\rho$ is said to satisfy the '''rank axioms''' {{iff}}:
{{begin-axiom}}
{{axiom | n = \text R 1
        | mr= \map \rho \O = 0
}}
{{axiom | n = \text R 2
        | q = \forall X \in \powerset S \land y \in S
        | mr= \map \rho X \le \map \rho {X \cup \set y} \le \map \rho X + 1
}}
{{axiom | n = \text R 3
        | q = \forall X \in \powerset S \land y, z \in S
        | mr= \map \rho {X \cup \set y} = \map \rho {X \cup \set z} = \map \rho X \implies \map \rho {X \cup \set y \cup \set z} = \map \rho X
}}
{{end-axiom}}
\end{definition}