User:Dfeuer/Double Induction Principle

Theorem
Let $M$ be a class.

Let $\mathcal R$ be a relation.

Let $g$ be a mapping whose domain includes $M$.

Let $M$ be minimally inductive under $g$.

Suppose that for all $x$ and $y$ in $M$:


 * $x \mathrel{\mathcal R} 0$
 * $x \mathrel{\mathcal R} y \land y \mathrel{\mathcal R} x \implies x \mathrel{\mathcal R} g(y)$

Then for all $x, y \in M$, $x \mathrel{\mathcal R} y$

Lemma
Let $y \in M$.

Then:


 * If $x \mathrel{\mathcal R} y$ for all $x \in M$, then $y \mathrel{\mathcal R} x$ for all $x \in M$.

Proof
Suppose that $x \mathrel{\mathcal R} y$ for all $x \in M$.

Let $B$ be the class of all $z \in M$ such that $y \mathrel{\mathcal R} z$.

By the premise, $0 \in B$.

Suppose that $z \in B$.

Then $y \mathrel{\mathcal R} z$.

By assumption, $z \mathrel{\mathcal R} y$.

Thus by the premise: $y \mathrel{\mathcal R} g(z)$, so $g(z) \in B$.

Thus $B$ is inductive under $g$.

As $M$ is minimally inductive under $g$ and $B \subseteq M$, $B = M$.

Now let $C$ be the class of all $y \in M$ such that for all $x \in M$:


 * $x \mathrel{\mathcal R} y$

By the premise, $0 \in C$.

Suppose that $y \in C$.

Let $x \in M$.

Then by the lemma, $y \mathrel{\mathcal R} x$.

Thus by the premise, $x \mathrel{\mathcal R} g(y)$.

Since this holds for all $x \in M$, $g(y) \in C$.

Thus $C$ is inductive under $g$.

Since $M$ is minimally inductive under $g$, $C = M$.