Minimally Inductive Set is Limit Ordinal

Theorem

Let $\omega$ denote the minimally inductive set.

Then $\omega$ is a limit ordinal.

Proof

Let $K_I$ denote the class of all nonlimit ordinals.

Aiming for a contradiction, suppose $\omega$ is not a limit ordinal.

Every element of $\omega$ is a nonlimit ordinal.

So, if $\omega$ is also a nonlimit ordinal, $\omega + 1 \subseteq K_I$.

By the definition of $\omega$:

$\omega \in \omega$

which violates No Membership Loops and is thus contradictory.

It follows from Proof by Contradiction that $\omega$ is a limit ordinal.

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 7.33$