Let $\mathbf{Lim}$ denote the class of limit ordinals. Is it true that $\mathbf{Lim}$ is a proper class? How does one prove that?
2026-03-31 11:26:47.1774956407
Is the class of all limit ordinals a proper class?
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The class of limit ordinals cannot be a set. Its union would be a set of all ordinals, and then the Burali-Forti paradox obtains a contradiction.
The class wouldn't qualify as an ordinal even it if were a set, because an ordinal is supposed to be transitive. But, for example, the class of limit ordinals contains $\omega$ but not its element $42$, so it is not transitive.