Transfinite Induction/Schema 2

Theorem
Let $\phi \left({x}\right): \operatorname{On} \to \left\{{\mathrm T, \mathrm F}\right\}$ be a propositional function satisfying the following conditions:


 * $(1): \quad \phi \left({\varnothing}\right) = \mathrm T$
 * $(2): \quad \phi \left({x}\right) = \mathrm T \implies \phi \left({x^+}\right) = \mathrm T$
 * $(3): \quad \left({\forall x < y: \phi \left({x}\right) = \mathrm T}\right) \implies \phi \left({y}\right) = \mathrm T$

where $x^+$ denotes the successor of $x$.

Then, $\phi \left({x}\right)$ is true for all ordinals $x$.

Proof
Defint the class:
 * $A := \left\{{x \in \operatorname{On}: \phi \left({x}\right) = \mathrm T}\right\}$.

Then $\phi \left({x}\right) = \mathrm T$ is equivalent to the statement:
 * that $x \in A$

The three conditions in the hypothesis become:


 * $(1a): \quad \varnothing \in A$
 * $(2a): \quad x \in A \implies x^+ \in A$
 * $(3a): \quad \left({\forall x < y: x \in A}\right) \implies y \in A$

These are precisely the conditions for the class $A$ in the second principle of transfinite induction.

Therefore, $\operatorname{On} \subseteq A$.

Thus, $\phi \left({x}\right)$ holds for all $x \in \operatorname{On}$.

In proofs by transfinite induction using this particular schema, the following terms are used.

Basis for the Induction
The proposition $\phi \left({\varnothing}\right)$ is called the basis for the induction.

Induction Step
The proposition $\phi \left({x}\right) = \mathrm T \implies \phi \left({x^+}\right) = \mathrm T$ is called the inductive step.

It says that the propositional function will pass from an ordinal number to its successor.

Limit Case
$\left({\forall x < y: \phi \left({x}\right) = \mathrm T}\right) \implies \phi \left({y}\right) = \mathrm T$ is called the limit case.

It states that if $\phi$ is true for every ordinal strictly less than $y$, then $\phi$ is true for $y$.

It is essentially proving that the proposition will hold for limit ordinals.

Then, this formulation of transfinite induction says that if the basis for the induction, inductive step, and limit case are all satisfied, then the statement holds for all ordinals.