User blog comment:Syst3ms/A formal definition for UNOCF/@comment-35470197-20180728080043/@comment-35470197-20180728221040

>>Sorry, that's far beyond what I can do myself.

But your work is still significant as our first step. It will be fantastic if we finally get a formal definition of the OCF.

>nonono, you misunderstood. I just said that it would be easier to talk if you joined the discord server. "On an unrelated note" is the same thing as "by the way".

Ooh... Sorry, I did not know the idiom. I got new knowledge here :D

Well, since I am not so good at chatting, discord might not suits me well. Sorry for it.

By the way, of course, you can use I and M as constant term symbols. I know that you do not need the large cardinals at all at this step. But when we prove that the ordinal notation system actually work, we often need the large cardinals.

For example, in the paper of the OCF with weakly inaccessibles by Rathjen, the definition of the OCF with formal symbols corresponding to weally inaccessibles is based on \(\textrm{ZFC}\), but in the proof of the well-foundedness, he used a result on the OCF with weakly Mahlo, which is literally based on the existence of a weakly Mahlo cardinal. (He wrote that the large cardinal axiom would be completely removed in the future, but I think that he have not written it yet. I do not think that he needs 30 years.)

We need a formal proof of the well-definedness of the OCF you formalised, when we apply it to a formal proof of BM1 as you intend. At that stage, we understand whether we actually need large cardinals or not.

I list up points which might be necessary to check:
 * The well-definedness of \(\psi\).
 * The uniqueness of the expression of ordinals using \((0,+,\Omega,\psi,C)\) or the existence of a recursive subset of the associated ordinal notation consisting of standard forms.
 * The recursiveness of the order \(<\) restricted to the associated ordinal notation.
 * The recursiveness of the computation of the cofinality of \(\psi_{\kappa}(\alpha)\).
 * The recursiveness of the system of fundamental sequences on the associated ordinal notation.
 * The definition of the relation \(< I\), if you do not actually use large cardinals.

> Then for your question, I should have mentioned that cof(δ)<κ cof(δ)<κ. That should clear up any ambiguity.

It is not sufficient. For example, when \(\kappa = I\) and \(\alpha = 2^I\), then \(\alpha\) has no decomposition satisfying your conditions. Maybe you are implicitly assuming a restriction of the domain of \(\psi_{\kappa}\).