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.
$\blacksquare$