User:Dfeuer/Existence of Inductive Set implies Axiom of Infinity
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Theorem
Suppose that there is an inductive set $a$.
Then the axiom of infinity holds. That is, the class $\omega$ of all natural numbers is a set.
Proof
Let $x \in \omega$.
Then by the definition of natural number, $x$ is in every inductive set, so in particular $x \in a$.
Thus by the definition of subclass, $\omega \subseteq a$.
By the subset axiom, $\omega$ is a set.
$\blacksquare$