Equivalence of Definitions of Minimally Inductive Class

Theorem

Let $A$ be a class.

Let $g$ be a mapping on $A$.

The following definitions of the concept of minimally inductive class under $g$ are equivalent:

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 $\O$.

Proof

$(1)$ implies $(2)$

Let it be given that $A$ is inductive under $g$.

Let $A$ be a minimally inductive class under $g$ by definition 1.

Then by definition:

$A$ has no proper subclass $B$ such that $B$ is inductive under $g$.

Let $A$ have a subclass $C$ which is inductive under $g$.

Then by definition, $C$ is not a proper subclass of $A$.

Thus by definition of proper subclass:

$C = A$

By definition of subclass:

$A \subseteq C$

and:

$C \subseteq A$

That is:

$\forall x \in A: x \in C$

and:

$\forall x \in C: x \in A$

That is: $C$ contains all elements of $A$.

Thus $A$ is a minimally inductive class under $g$ by definition 2.

$\Box$

$(2)$ implies $(1)$

Let it be given that $A$ is inductive under $g$.

Let $A$ be a minimally inductive class under $g$ by definition 2.

Then by definition:

Every subclass of $A$ which is inductive under $g$ contains all the elements of $A$.

Let $C$ be a subclass of $A$ which is inductive under $g$.

By definition of subclass:

$C \subseteq A$

But by hypothesis:

$\forall x \in A: x \in C$

That is:

$A \subseteq C$

By definition of equality of classes it follows that:

$A = C$

and so by definition $C$ cannot be a proper subclass of $A$.

Thus $A$ is a minimally inductive class under $g$ by definition 1.

$\Box$

$(1)$ implies $(3)$

Let $A$ be a minimally inductive class under $g$ by definition 1.

Then by definition:

\((1)\)	$:$							$A$ is inductive under $g$
\((2)\)	$:$							No proper subclass of $A$ is inductive under $g$.

By definition of inductive class:

$\O \in A$

$A$ is closed under $g$.

Putting these definitions together, we have that:

No proper subclass of $A$ such that $\O \in A$ is closed under $g$.

That is, $A$ is minimally closed under $g$ with respect to $\O$

Thus $A$ is a minimally inductive class under $g$ by definition 3.

$\Box$

$(3)$ implies $(1)$

Let $A$ be a minimally inductive class under $g$ by definition 3.

That is, $A$ is minimally closed under $g$ with respect to $\O$.

That is:

$(1): \quad \O \in A$

$(2): \quad$ No proper subclass of $A$ such that $\O \in A$ is closed under $g$.

Hence by definition of inductive class:

\((1)\)	$:$							$A$ is inductive under $g$
\((2)\)	$:$							No proper subclass of $A$ is inductive under $g$.

Thus $A$ is a minimally inductive class under $g$ by definition 1.

$\blacksquare$