Derived sets of an ordinal
= Derived sets of an ordinal
For an ordinal $\xi$,
$$
\xi'=\{\omega\beta<\xi:\beta>0\},
\qquad
\xi''=\{\omega^2\beta<\xi:\beta>0\}.
$$
Thus the derivative index of $\omega$ is one and that of $\omega^2$ is two.
= Derived sets of an ordinal
For an ordinal $\xi$,
$$
\xi'=\{\omega\beta<\xi:\beta>0\},
\qquad
\xi''=\{\omega^2\beta<\xi:\beta>0\}.
$$
Thus the derivative index of $\omega$ is one and that of $\omega^2$ is two.