Is $\varnothing$ a limit ordinal?

Question

This question is from Goldrei's Classic Set Theory:

Let $\lambda$ be an ordinal. If $\cup\lambda=\lambda$, then $\lambda$ is a limit ordinal.

But what if $\lambda=\varnothing$? I think I am doing something wrong by writing $\cup\varnothing=\varnothing$.

Answer 1

One often defines a limit-ordinal as an ordinal which is not $\emptyset$ yet satisfies your mentioned properties. So no, $\emptyset$ is not a limit ordinal, though some authors may use it as a limitordinal since it in some sences behave as one, and it may make some proofs easier.

See also the wikipedia page.

Answer 2

The definition that I know is that all ordinals are successor ordinals, limit ordinals or 0. I believe this answers your question.