User blog comment:Scorcher007/Large countable ordinal notation up to Z2 and ZFC/@comment-11227630-20181124020537

The concept of "gap" does not go this far.

For countable α,β, Lβ,Lβ+α∈Vβ+α+1∈Pω1(ω), then Lβ∩Pω1(ω)=Lβ≠Lβ+α=Lβ+α∩Pω1(ω), so there is no ω1-th order gap ordinal.