User blog comment:Syst3ms/A sketch for an — actually — formal definition of UNOCF/@comment-35470197-20180803231131/@comment-35470197-20180804123134

4) > See my edit, it provides an equality relation with the I function, which is used in other, formal OCFs iirc

Sorry, I do not know it. Is it a certain extension of the two-variable \(I_{\alpha}(\beta)\) used in the definition of Rathjen's OCF? It enumerates the closure of the class of \(\alpha\)-inaccessibles.

> But do you have any idea how I would compare two C function expressions properly ?

Although I do not precisely understand what \(C\) is, I can say that if \(C\) is presented as an enumeration function of some system \(X\) of classes, then we compare two values of \(C\) using properties of \(X\). For example, values of Veblen function (resp. the cardinal analogue of Veblen function, the enumeration function of the closure of higher inaccessibles) are easily compared by the definition of enumeration.

I did a related computation here: https://googology.wikia.com/wiki/User_blog:P%E9%80%B2%E5%A4%A7%E5%A5%BD%E3%81%8Dbot/Ordinal_Notation_Associated_to_a_Proper_Class_of_Ordinals#Iterated_Enumeration_Function

So in general, when we consider an OCF (a function for actual ordinals but not placeholders), we usual compare values in an abstract (set-theoretic) way using variables but not explicit expressions of inputs.

On the other hand, when we consider an ordinal notation system (a system of formal strings of placeholders and function symbols), then the "value" of a function symbol does not make sense, and hence we directly define a recursive relation \(<\) on the system using formal expressions and observing a related OCF.

So the strategy depends on whether you are actually defining an OCF or an ordinal notation system. I guess that mixing pure placeholders into OCFs could become obstruction here by the reason above. But... it is possible that just I lack knowledge. Other specialists might have good idea.

> Don't bother with 3, I fixed it already.

Oops. Sorry.