User blog comment:Testitemqlstudop/FPT, large ordinals, ordinal FGH and a new Fundamental Sequence system/@comment-35470197-20191031122541/@comment-39541634-20191031134819

No.

Γ₀ is much larger than the limiit of the sequence ε₀, ζ₀, η₀, ... .

To reach Γ₀, you need to extend that sequence to ordinal indexes. Then, you will have Γ₀ as the first fixed point of "α↦α-th ordinal in the sequence" (or alternatively: α↦φ(α,0)).

The whole situation is quite similar the way ζ₀ relates to the ε numbers. ζ₀ is what you get after you exhaust the ε's, but ζ₀ is not the limit of the sequence:

ε₀, ε₁ , ε₂ , ε₃ , ...

This latter limit is simply εω. At this point, you can still plug in larger and larger ordinals in the subscript, without any trouble. You only hit the wall once you reach the first fixed-point of "α↦εα", which is exactly what ζ₀ means.