Definition
For a set of ordinals $\boldsymbol\alpha$ and ordinals $\gamma$, $\beta$, let
$$\boldsymbol\alpha \xrightarrow[\gamma]{}\beta$$
symbolize the proposition that
- $(\boldsymbol\alpha,<)$ has order-type $\gamma$ for the natural ordering $<$
[alternatively: $\boldsymbol\alpha$ is a strictly increasing $\gamma$-indexed set of ordinals]- $\beta \not\in \boldsymbol\alpha$
- $\beta$ is the supremum of $\boldsymbol\alpha$
(read: "$\boldsymbol\alpha$ is a $\gamma\rightarrow\beta$-sequence").
For all $\gamma\rightarrow\beta$-sequences $\boldsymbol\alpha$ it holds that
- $\gamma \leq \beta$
- $\gamma$ and $\beta$ are both limit or both successor ordinals
Examples
$\boldsymbol\alpha \xrightarrow[1]{}\beta$ means $\boldsymbol\alpha = \lbrace \alpha \rbrace$ and $\beta = \alpha + 1$
$\boldsymbol\alpha \xrightarrow[\omega]{}\beta$ means $\boldsymbol\alpha$ is a countable fundamental sequence with limit $\beta$
$\boldsymbol\alpha \xrightarrow[\beta]{}\beta$ means $\boldsymbol\alpha=\beta = \lbrace \alpha : \alpha < \beta \rbrace$
$\boldsymbol\alpha=\beta$ implies $\boldsymbol\alpha \xrightarrow[\beta]{}\beta$
Definition
With each ordinal number $\gamma$ a second-order property $\Pi_\gamma$ of properties $P$ of ordinal numbers is associated:
$$(\forall \beta)\ (\forall \boldsymbol\alpha \xrightarrow[\gamma]{}\beta)\ \Big( \big( (\forall \alpha \in \boldsymbol\alpha)\ P(\alpha)\big) \Rightarrow P(\beta)\Big)$$
This meta-property reads:
When for an arbitrary $\beta$ all members of an arbitrary $\gamma\rightarrow\beta$-sequence $\boldsymbol\alpha$ have property $P$, then also $\beta$ has (= inherits) property $P$.
Compare with the parameter-free second-order property $\Pi$:
$$(\forall \beta)\ (\forall \boldsymbol\alpha \xrightarrow[\beta]{}\beta)\ \Big( \big( (\forall \alpha \in \boldsymbol\alpha)\ P(\alpha)\big) \Rightarrow P(\beta)\Big)$$
which is a blown-up way to say $(\forall \beta) \big( (\forall \alpha < \beta)\ P(\alpha)\big) \Rightarrow P(\beta)$.
Examples of usage
[Simple induction]
If $P(0)$ and $P$ has property $\Pi_1$ then $P(\alpha)$ for all finite ordinals $\alpha$.
[Transfinite induction]
If $P(0)$ and $P$ has property $\Pi$ then $P(\alpha)$ for all ordinals $\alpha$.
Question
How do generalizations with respect to properties $\Pi_\gamma$ look like?
What would especially be the condition $\Phi_\omega$ in:
If $P(0)$ and $P$ has property $\Pi_\omega$ then $P(\alpha)$ for all ordinals $\alpha$ obeying condition $\Phi_\omega$.