User blog comment:ArtismScrub/Hierarchical Cardinal Notation (HCN)/@comment-34812936-20180228064722/@comment-30754445-20180228125948

No.

Φ(Ω,0) has a cofinality of Ω.

Φ(Φ(Ω,0),0) has a cofinality of Φ(Ω,0) >> Ω.

So the two cardinals cannot be equal.

For similar reasons, Φ(ɪ,0) must be equal to ɪ.