Principle of Mathematical Induction for Minimally Inductive Set

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:


 * $S = \omega$