Induction on time scales

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Let $\mathbb{T}$ be a time scale. Let $t_0 \in \mathbb{T}$ and assume that $\{S(t) \colon t \in [t_0,\infty)\}$ is a family of statements satisfying the following axioms:

  • the statement $S(t_0)$ is true
  • if $t \in [t_0,\infty)$ is right-scattered and $S(t)$ is true, then $S(\sigma(t))$ is true
  • if $t \in [t_0,\infty)$ is right-dense and $S(t)$ is true, then there is a neighborhood $U$ of $t$ such that $S(s)$ is true for all $s \in U \cap (t,\infty)$
  • if $t \in (t_0, \infty)$ is left-dense and $S(s)$ is true for all $s \in [t_0,t)$, then $S(t)$ is true.

Then $S(t)$ is true for all $t \in [t_0,\infty)$.