User blog comment:Clarrity/List of ordinals (up to omega^omega)/@comment-30118230-20180225131945/@comment-1605058-20180225201806

For any recursive limit ordinal \(\alpha<\omega_1^\mathrm{CK}\) you can give an FS in a matter completely analogous to \(\omega_1^\mathrm{CK}\), except at every step you have to restrict your attention to ordinals smaller than \(\alpha\).

Specifically, let \(\alpha[0]=0\) and for \(n>0\) take \(\alpha[n]\) be the ordinal larger than \(\alpha[n-1]\) and smaller than \(\alpha\) which, out of all ordinals in this interval, can be coded in Kleene's O with the least number.