Definition:Closed Set under Progressing Mapping

Definition

Let $x$ and $y$ be sets.

Let $g$ be a progressing mapping.

We say that:

$y$ is closed under $g$ relative to $x$

if and only if:

$\forall z \in y \cap \powerset x: \map g z \in y$

That is:

$z \in y \land z \subseteq x \implies \map g z \in y$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text {III}$ -- The existence of minimally superinductive classes: $\S 7$ Cowen's theorem: Definition $7.1$