Minimally Inductive Set is Limit Ordinal
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Let $\omega$ denote the minimally inductive set.
Then $\omega$ is a limit ordinal.
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.
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 7.33$