User blog comment:Syst3ms/A formal definition for UNOCF/@comment-35470197-20180728080043/@comment-35470197-20180729090540

> \(\ddots\) doesn't require nesting, it allows it.

So what are the precise definition of \(\ddots\) and \(\cdots\)? If you allow anything other than \(\psi\)'s, "allowing nesting" does not make sense. For example, you can defined \(\varphi\) as \(\varphi = \psi\), and use a nesting of \(\varphi\) in \(\cdots\). Then \(\cdots\) does not contain a nesting of \(\varphi\), but works in the same way as \(\ddots\).

> And also, what in the world are you talking about with "0,Ω,+,ψ,C" ?

If you want to define something by "allowing" several operations, you need to restrict constants (e.g. \(0, \Omega\)) and functions (e.g. \(+,\psi,C\)) which appear in expression. Otherwise, for example, "allowing a nesting" does not makes sense, as I mentioned above.

> Also also, wouldn't the best idea be to make some standard form in this case ?

Exactly. Also, making an ordinal notation system (with a subset of standard forms) in a recursive way is a good method, which is often used in papers on OCFs.

In order to define standard forms (without an ordinal notation system), you need to determine when inequalities like \(\psi_{\kappa}(\alpha) = \psi_{\lambda}(\beta)\) hold. Of course, you can define standard forms without determining it, but then you will find that the standard forms do not work well when you need to prove the well-definedness of your formal definition.

I am sorry that my explanation is not so sophisticated. Other speciallists such as PsiCubed2, Deedlit, and LittlePeng could give you explanations much easier to understand...

In order to understand the problems in your presenting formal definitions, it is better to read several papers on OCFs. For example, Buchholz's paper is a good starting point to grasp the precise way to construct an OCF and the associated ordinal notation system.