Definition:Minimally Inductive Class under General Mapping
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Definition
Let $A$ be a class.
Let $g: A \to A$ be a mapping.
Definition 1
$A$ is minimally inductive under $g$ if and only if:
\((1)\) | $:$ | $A$ is inductive under $g$ | |||||||
\((2)\) | $:$ | No proper subclass of $A$ is inductive under $g$. |
Definition 2
$A$ is minimally inductive under $g$ if and only if:
\((1)\) | $:$ | $A$ is inductive under $g$ | |||||||
\((2)\) | $:$ | Every subclass of $A$ which is inductive under $g$ contains all the elements of $A$. |
Definition 3
$A$ is minimally inductive under $g$ if and only if $A$ is minimally closed under $g$ with respect to $\O$.
Also see
- Results about minimally inductive classes can be found here.
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 4$ A double induction principle and its applications: Definition $4.2$