User blog comment:PsiCubed2/How to make Deedlit's Mahlo-level notation more intuitive/@comment-35470197-20180807000338/@comment-35470197-20180808233927

> But you say you're having doubts. So obviously, for this discussion, we can no longer just assume that everything in Deedlit's notation works.

Uh huh, I do not doubt others. Asking something on their statements just means that I, being poor at your theory, could not understand reasons or proofs. If you have ideas to solve them, please help me. At least, I believe that Deedlit wrote his outstandingly poweful equality psi_{Omega_1}(varepsilon_{M+1}) = PTO(KPM) with an explicit proof inside him.

> Can you elaborate a bit more about the problems?

To begin with, since Deedlit's OCF is defined in a way completely different from Rathjen's OCF, I could not compare them.

I list up main non-trivial differences. Please allow me if all of them are well-known and obviously solved.
 * 1) Deedlit used kappa in his C(a,kappa), while Rathjen used kappa^- in his C_{kappa}(a). It looks to work differently when kappa is weakly inaccessible, because kappa^- actually refers to the predescessor of the degree of its weak inaccessibility in order to collapse cardinals with higher weak inaccessibility.
 * 2) Deedlit defined chi(a,b) as the enumeration of the equality gamma = C(a,gamma) cap M, while Rathjen defined chi_a(b) as the enumeration of the closure relative to M of the class {kappa | kappa notin B(a,kappa) wedge a in B(a,kappa)}, where B(a,k) is defined in a similar way to C(a,b) in Deedlit's OCF.
 * 3) So Deedlit directly defined C(a,b) as a counterpart of C_{kappa}(a), and implicitly obtained the closure of the class associated to B(a,b) from C(a,b). It is not so trivial for me that such a C(a,b) actually works as strong as the C(a,b) in Rathjen's OCF.
 * 4) Deedlit defined his OCF on any ordinals, while Rathjen's OCF is limited to the first strongly critical ordinal above M. This restriction is used in order to ensure the behaviour of his normal forms.
 * 5) Deedlit allowed any applications of operators +, phi, and chi, while Rathjen allowed applications obeying the rules on normal forms. Such a technique works when we define a notation system, but I do not know it works as desired when we define an OCF itself because such a restriction is often used in order to ensure that the OCF actually collapse ordinals.
 * 6) Deedlit stated that his psi_{Omega_1}(varepsilon_{M+1}) coincides with PTO(KPM), while Rathjen stated his psi_{Omega_1}(psi_{chi_{varepsilon_{M+1}}(0)}(0)) coincides with it.
 * 7) Therefore at least up to PTO(KPM), his associated ordinal notation would be  assumed to be as strong as (or stronger than) Rathjen's associated ordinal notation, even though Deedlit's OCF has many advantages on the simplicity of the definition.

> Also, what's "Gamma"?

M^Gamma just means the first strongly critical ordinal above M, i.e. min {a > M | varphi(a,0) = a}.

> BTW I don't know what specific paper Rathjen you're refering to. A link would be appreciated.

I am refering to "Ordinal notations based on a weakly Mahlo cardinal", which is open here (last line).

https://www1.maths.leeds.ac.uk/~rathjen/preprints.html