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