Definition:Superinductive Class
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Definition
Let $A$ be a class.
Let $g: A \to A$ be a mapping on $A$.
Then $A$ is superinductive under $g$ if and only if:
- $A$ is inductive under $g$
- $A$ is closed under chain unions.
That is:
\((1)\) | $:$ | $A$ contains the empty set: | \(\ds \O \in A \) | ||||||
\((2)\) | $:$ | $A$ is closed under $g$: | \(\ds \forall x:\) | \(\ds \paren {x \in A \implies \map g x \in A} \) | |||||
\((3)\) | $:$ | $A$ is closed under chain unions: | \(\ds \forall C:\) | \(\ds \bigcup C \in A \) | where $C$ is a chain of elements of $A$ |
Also see
- Results about superinductive classes can be found here.
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 $\text I$ -- Superinduction and Well Ordering: $\S 2$ Superinduction and double superinduction