Axiom:Axiom of Dependent Choice/Right-Total

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Axiom

Let $\mathcal R$ be a binary relation on a non-empty set $S$.

Suppose that:

$\forall a \in S: \exists b \in S: b \mathrel {\mathcal R} a$

that is, that $\mathcal R$ is a right-total relation.


The axiom of dependent choice states that there exists a sequence $\sequence {x_n}_{n \mathop \in \N}$ in $S$ such that:

$\forall n \in \N: x_{n + 1} \mathrel {\mathcal R} x_n$