# Principle of Mathematical Induction for Minimal Infinite Successor Set

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## Theorem

Let $\omega$ be the minimal infinite successor set.

Let $S \subseteq \omega$.

Suppose that:

- $(1): \quad \varnothing \in S$
- $(2): \quad \forall x: x \in S \implies x^+ \in S$

where $x^+$ is the successor set of $x$.

Then:

- $S = \omega$

## Proof

The hypotheses state precisely that $S$ is an infinite successor set.

Then the minimal infinite successor set $\omega$ being defined as the intersection of all infinite successor 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*: $\S 7.31$