$\omega$ is considered a limit ordinal, and an ordinal is considered a limit ordinal if and only if it has no predecessor, however, what about $\omega-1$?
And since limit ordinals cant have any predecessors then how do you extend the following,
$$_\beta N=\bigcup_{\alpha}\;_{\beta-1} N_\alpha$$ Because according to the rules of limit ordinals if $\beta$ is a limit ordinal there shouldn't exist, $\beta-1$. So how would you generalize the formula for $_\beta N$ for all ordinals?