Equivalence of Definitions of Minimally Inductive Set

Theorem
Let $\omega$ denote the ordinary definition of minimal infinite successor set and let $\omega '$ denote the alternative definition of minimal infinite successor set.


 * $\displaystyle \omega = \omega '$

Proof
Let $K_I$ denote the set of all nonlimit ordinals.

First, $\varnothing \in \omega$ because $\varnothing \in A$ for every infinite successor set $A$.

$\varnothing \in \omega '$ by the fact that $\omega '$ is a nonempty ordinal

If $n \in \omega$, then $n^+ \in \omega$ by the second Peano postulate.

Therefore, by induction over the minimal infinite successor set $\omega$, setting $S = \omega \cap \omega '$ we have that $S = \omega$.


 * $\displaystyle \omega \subseteq \omega '$

Similarly, because $\omega '$ is an ordinal by Minimal Infinite Successor Set is Ordinal, it is well-ordered by $\Epsilon$ by Alternative Definition of Ordinal, so we may apply Well-Founded Induction/Special Case. By induction, we have that


 * $\displaystyle \omega ' \subseteq \omega$

Thus, we may conclude that $\omega = \omega '$.