User blog comment:Hyp cos/Fundamental Sequences in Taranovsky's Notation/@comment-40154718-20190916083849/@comment-11227630-20190916103520

In TON, things with cofinality greater than ω just work like this, but they actually are countable and recursive. For example, Ω_1 seems to work as a diagonalizer, but it is actually (in system n) the limit of C(...C(Ω_n,0)...,0) (with n 0's), C(...C(C(Ω_n,Ω_n),0)...,0), C(...C(C(C(Ω_n,Ω_n),Ω_n),0)...,0), C(...C(C(C(C(Ω_n,Ω_n),Ω_n),Ω_n),0)...,0), etc.

To analysis the strength, Taranovsky set some gaps below some ordinals such as Ω_1. If there is no gap, all ordinals would be recursive.