Principle of Mathematical Induction for Minimally Inductive Set
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Theorem
Let $\omega$ be the minimally inductive set.
Let $S \subseteq \omega$.
Suppose that:
- $(1): \quad \O \in S$
- $(2): \quad \forall x: x \in S \implies x^+ \in S$
where $x^+$ is the successor set of $x$.
Then:
- $S = \omega$
Proof
The hypotheses state precisely that $S$ is an inductive set.
Then the minimally inductive set $\omega$ being defined as the intersection of all inductive sets, we conclude that:
- $\omega \subseteq S$
by Intersection is Subset: General Result.
Thus, by definition of set equality:
- $S = \omega$
$\blacksquare$
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 12$: The Peano Axioms
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 7.31$