# 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 $4.2$