# Principle of Mathematical Induction for Minimal Infinite Successor 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:

$\omega \subseteq S$

Thus, by definition of set equality:

$S = \omega$

$\blacksquare$