User blog comment:Syst3ms/YAUD/@comment-30754445-20180810132619/@comment-30754445-20180813103329

"Correct me if I'm wrong, but the value of the subscript actually matters in the collapse."

In the context your asking: no, it doesn't matter.

What matters, in the end, is not the exact value of the subscript, but what it does during the collapse.

For example, if I took Deedlit's inaccessible-based notation and changed the values of Ωx from "the xth cardinal" to "the x*2-th cardinal", it would still result in precisely the same countable ordinals when I do the final collapse with ψ0.

The uncountable ordinals will of course be different, including the outputs of the various ψ's whenever they return uncountable ordinals. But this doesn't change the end result.

Of-course you need to be careful when you do that. What we need here, basically, is precisely the kind of restricting-rules you've given. Things like "cof(I)=I"  tell us that there's a limit to our freedom in choosing what actual ordinal is represented by I. But you need these anyway. So you haven't really gained anything by insisting that "these are not ordinals".

Also, as I've already said earlier, when using an OCF, you don't need to "set the value" of anything other than your atomic expressions (I, M and so on). The whole point of OCFs is that they provide you with a general set of rules whose job is to generate these ordinals for you.

What you do need to check, is that these rules indeed do what you wanted them to do. But you need to do that anyway, right? And you also need to check that the intermediate uncountable ordinals (or in your non-OCF: the intermediate symbols that look like uncountable ordinals) indeed do whatever job you intended them to do.

And this, actually, is much easier to do with a proper OCF. It is much easier to analyze things when you can compare them to something familiar, like uncountable ordinals. With your non-OCF, the analysis is just as complicated, but you are forced to do it the dark. Without actual to ordinals to serve as a steady reference, you'll get completely lost very quickly.