# Minimal Infinite Successor Set is Limit Ordinal

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## Theorem

Let $\omega$ denote the minimal infinite successor 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*: $\S 7.33$