User blog comment:Ubersketch/Homomorphism between Cantor's normal form of finite degree and primitive sequence system/@comment-35470197-20190809223713/@comment-39541634-20190812081340

What? No. I'm afraid you misunderstood Username's explanation.

His f is not a single function. It depends on the ordinal we wish to collapse.

What Username is proposing, is that for a given ordinal X with cofinality Ω, we look for a "natural looking function" f whose desired behavior depends on X: Namely, a function that obeys f(Ω)=X.

So for ψ(Ω), we want a function that gives f(Ω)=Ω. The most natural choice here is f(α)=α, so we have:

ψ(Ω)[0]=ψ(f(0))=ψ(0)

ψ(Ω)[1]=ψ(f(ψ(0)))=ψ(ψ(0))

ψ(Ω)[2]=ψ(f(ψ(ψ(0))))=ψ(ψ(ψ(0)))

And so on.

(the above is the sequence 1, ω, ω^ω, ω^ω^ω... which is the FS for ε0)

On the other hand, for Ω^Ω, we want a function that gives f(Ω)=Ω^Ω. A natural choice here would be f(α)=Ω^α, and so we have:

ψ(Ω^Ω)[0]=ψ(f(0))=ψ(Ω^0)=ψ(1)=ω

ψ(Ω^Ω)[1] = ψ(f(ω)) = ψ(Ω^ω) = φ(ω,0)

ψ(Ω^Ω)[2] = ψ(f(ψ(Ω^ω))) = ψ(Ω^ψ(Ω^ω)) = φ(φ(ω,0),0)

And so on.

(which is a FS of Γ0)

So as an intuitive framework, it works, at least up to the BHO. The only problem with it is that the statement "we look for the most natural function f that..." is highly subjective, hence it is not well-defined.

(with a single function OCF this can easily be remedied by strictly defining the function as one which replaces the rightmost Ω in your expression with an α. Of-course, since this definition is notation-dependent, you'll need to explicitly define your "standard canonical notation" for the ordinals first).