Definition:Inductive Class

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Definition

Let $A$ be a class.


Then $A$ is inductive if and only if:

\((1)\)   $:$   $A$ contains the empty set:    \(\displaystyle \quad \O \in A \)             
\((2)\)   $:$   $A$ is closed under the successor mapping:      \(\displaystyle \forall x:\) \(\displaystyle \paren {x \in A \implies x^+ \in A} \)             where $x^+$ is the successor of $x$
         That is, where $x^+ = x \cup \set x$


Inductive Set

The same definition can be applied when $A$ is a set:


Let $S$ be a set of sets.


Then $S$ is inductive if and only if:

\((1)\)   $:$   $S$ contains the empty set:    \(\displaystyle \quad \O \in S \)             
\((2)\)   $:$   $S$ is closed under the successor mapping:      \(\displaystyle \forall x:\) \(\displaystyle \paren {x \in S \implies x^+ \in S} \)             where $x^+$ is the successor of $x$
         That is, where $x^+ = x \cup \set x$


General Definition

Let $A$ be a class.

Let $g: A \to A$ be a mapping on $A$.


Then $A$ is inductive under $g$ if and only if:

\((1)\)   $:$   $A$ contains the empty set:    \(\displaystyle \quad \O \in A \)             
\((2)\)   $:$   $A$ is closed under $g$:      \(\displaystyle \forall x:\) \(\displaystyle \paren {x \in A \implies \map g x \in A} \)             


Also see

  • Results about inductive classes can be found here.


Sources