Definition:Closed Set under Progressing Mapping
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Definition
Let $x$ and $y$ be sets.
Let $g$ be a progressing mapping.
We say that:
- $y$ is closed under $g$ relative to $x$
- $\forall z \in y \cap \powerset x: \map g z \in y$
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 $7.1$