Principle of Induction
Theorem
Principle of Finite Induction
Let $n_0 \in \Z$ be given.
Let $\Z_{\ge n_0}$ denote the set:
- $\Z_{\ge n_0} = \set {n \in \Z: n \ge n_0}$
Let $S \subseteq \Z_{\ge n_0}$ be a subset of $\Z_{\ge n_0}$.
Suppose that:
- $(1): \quad n_0 \in S$
- $(2): \quad \forall n \ge n_0: n \in S \implies n + 1 \in S$
Then:
- $\forall n \ge n_0: n \in S$
That is:
- $S = \Z_{\ge n_0}$
Principle of Mathematical Induction
Let $\map P n$ be a propositional function depending on $n \in \Z$.
Let $n_0 \in \Z$ be given.
Suppose that:
- $(1): \quad \map P {n_0}$ is true
- $(2): \quad \forall k \in \Z: k \ge n_0 : \map P k \implies \map P {k + 1}$
Then:
- $\map P n$ is true for all $n \in \Z$ such that $n \ge n_0$.
Principle of Transfinite Induction
Principle 1
Let $\On$ denote the class of all ordinals.
Let $A$ denote a class.
Suppose that:
Then $\On \subseteq A$.
Schema 1
Let $\map P x$ be a property
Suppose that:
- If $\map P x$ holds for all ordinals $x$ less than $y$, then $\map P y$ also holds.
Then $\map P x$ holds for all ordinals $x$.
Principle 2
Let $A$ be a class satisfying the following conditions:
- $\O \in A$
- $\forall x \in A: x^+ \in A$
- If $y$ is a limit ordinal, then $\paren {\forall x < y: x \in A} \implies y \in A$
where $x^+$ denotes the successor of $x$.
Then $\On \subseteq A$.
Schema 2
Let $\map \phi x$ be a property satisfying the following conditions:
- $(1): \quad \map \phi \O$ is true
- $(2): \quad$ If $x$ is an ordinal, then $\map \phi x \implies \map \phi {x^+}$
- $(3): \quad$ If $y$ is a limit ordinal, then $\paren {\forall x < y: \map \phi x} \implies \map \phi y$
where $x^+$ denotes the successor of $x$.
Then, $\map \phi x$ is true for all ordinals $x$.
Principle of General Induction
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:
\((1)\) | $:$ | \(\ds \map P \O \) | \(\ds = \) | \(\ds \T \) | |||||
\((2)\) | $:$ | \(\ds \forall x \in M:\) | \(\ds \map P x \) | \(\ds = \) | \(\ds \T \implies \map P {\map g x} = \T \) |
Then:
- $\forall x \in M: \map P x = \T$
Principle of Structural Induction
Let $\LL$ be a formal language.
Let the formal grammar of $\LL$ be a bottom-up grammar.
Let $\map P \phi$ be a statement (in the metalanguage of $\LL$) about well-formed formulas $\phi$ of $\LL$.
Then $P$ is true for all WFFs of $\LL$ if and only if both:
- $\map P a$ is true for all letters $a$ of $\LL$,
and, for each rule of formation of $\LL$, if $\phi$ is a WFF resulting from WFFs $\phi_1, \ldots, \phi_n$ by applying that rule, then:
- $\map P \phi$ is true only if $\map P {\phi_1}, \ldots, \map P {\phi_n}$ are all true.