Principle of General Induction

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Theorem

Let $M$ be a class.

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

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


Let $P: M \to \set {\T, \F}$ be a propositional function on $M$.

Suppose that:

$(1): \quad \map P \O = \T$
$(2): \quad \forall x \in M: \map P x = \T \implies \map P {\map g x} = \T$


Then:

$\forall x \in M: \map P x = \T$


Proof

We are given that $M$ is a minimally inductive class under $g$.

That is, $M$ is an inductive class under $g$ with the extra property that $M$ has no proper class which is also an inductive class under $g$.


Let $P$ be a propositional function on $M$ which has the properties specified:

$(1): \quad \map P \O = \T$
$(2): \quad \forall x \in M: \map P x = \T \implies \map P {\map g x} = \T$


Thus by definition, the class $S$ of all elements of $M$ such that $\map P x = \T$ is an inductive class under $g$.

But because $M$ is minimally inductive under $g$, $S$ contains all elements of $M$.

That is:

$\forall x \in M: \map P x = \T$

as we were to show.

$\blacksquare$


Terminology

Basis for the Induction

The step that shows that the propositional function $P$ holds for $\O$ is called the basis for the induction.


Induction Hypothesis

The assumption made that $\map P x$ is true for some $x \in M$ is called the induction hypothesis.


Induction Step

The step which shows that $\map P x = \T \implies \map P {\map g x} = \T$ is called the induction step.


Also see


Sources