User blog comment:Moooosey/Simple psi-- a fairly simple ocf/@comment-43798125-20191026225036/@comment-39541634-20191027003843

It seems to me that what OP is asking, is whether an OCF can be strengthened by adding ordinals to the initial sets, rather than changing the permissible generator functions. Of-course removing all the "stuck points" is theoretically impossible. But here is the closest thing I could find to what the OP is looking for: A simple (but not very efficient) thing you can do, is to arbitrarily add more uncountables after Ω to your inital sets.

Note that I am not talking about adding another collapse function. We're just adding more uncountable for the use of the same OCF. It's a pretty weak extension, but it looks like

For example, say you add Ω2 to your initial sets. You leave all the other rules (from your 1st version) intact.

Then, we'll have:

ψ(Ω2) = ψ(the limit of the stuff we can write with Ω's) = ψ(Ωxω) = εω

So simply by adding a new symbol, we got unstuck! Now we can continue:

ψ(Ω2+Ω) = εω+1

ψ(Ω2+Ω+Ω) = εω+2

ψ(Ω2+Ω2) = εωx2

ψ(Ω2+Ω2+Ω2) = εωx3

And the limit of this will be:

ψ(Ω2+Ω2+Ω2+...) = εω^2

And we're stuck again. But at least we are now at a higher point then we were previously.

Adding another uncountable (say Ω3) will get the system up to εω^3, and so on. To systemize this, you could use another trick: Allow for the use of any ordinal of the form Ωx where x is a countable ordinal we've already built. This will get you up to ζ0 but no further (can you see why?).

Of-course this is a terribly inefficient extension method, because the real power of OCFs doesn't come from the initial set. It's the way your function generates new ordinals that gives it its power: Replacing "a+b" with "a^b" in your permissible generation rules already gets you from εω to the BHO. That's much much larger than ζ0, and it was achieved by much simpler means.

(As a side note: Really powerful OCFs actually tend to use the opposite approach. Instead of adding more ordinals to the initial set, they start with less. They throw out Ω itself, because it can be generated with 0 and 1 and the permissible functions. For example, you might have Ω=ψ₁(0), which is built using the elements 0,1 and the function ψᵦ(α))