Transfinite Induction
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Theorem
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 $x$.
Sources
- 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $3$. The Axiom of Choice and Its Equivalents