User blog comment:Scorcher007/Large countable ordinal notation up to Z2 and ZFC/@comment-11227630-20191006161445

There might be more things below zoo 2.22 ordinal.

In the range of β=(1st 2nd-order-gap with length "X-ordinal after β"), there should be a β=(1st 2nd-order-gap with length "next 2nd-order gap ordinal after β"); (but it does not exist, β=(1st 2nd-order-gap with length "next 2nd-order gap ordinal after β"+1) or β=(1st 2nd-order-gap with length at least "next 2nd-order gap ordinal after β") instead, due to consequent gaps.)

Continuing upward, we can have β0 a gap-length-"next β1 that is gap-length-'next gap-length-1 ordinal after β1' ordinal after β0" ordinal, and more layers. We can also have ω layers - then there is \(\alpha=\sup\{\beta<\alpha|(L_\alpha\backslash L_\beta)\cap\mathcal P(\omega)=\varnothing\}\), similar to a nonprojectable ordinal.

Next, when α is П2-reflecting on \(\{\beta<\alpha|(L_\alpha\backslash L_\beta)\cap\mathcal P(\omega)=\varnothing\}\), the α might be the 3rd-order Δ2 gap ordinal, analogous to zoo 2.16.