User blog comment:Nayuta Ito/The attempt to make my own ordinals collapse function/@comment-28606698-20170423081909/@comment-30754445-20170423092005

For this to work, you'll also need to add the transformation x→Ωx to the list of allowed functions.

Otherwise, the limit of the system will be:

ρ(Ω×Ω×...) = ρ(Ωω) = φ(ω,0).

Alternatively, if you want to try something really interesting, instead of allowing  x→Ωx, make multiple collapsing functions. So we'll have:

ρ2(0) = Ωω

ρ2(1) = Ωω 2

and so on.

If I'm not mistaken, this will eventually lead to:

ρ(ρ2(Ω)) = Γ₀

and

ρ(Ω2) = SVO

(which seems weird... so I might have missed something in my analysis)