User blog comment:Ubersketch/A proposal for a standard/@comment-11227630-20190813030043/@comment-35470197-20190813050540

> 2 and 3 simply remove the corresponding part of comparison algorithm

No. They will change standard forms, and hence the comparison algorithm will change if you do not have an algorithm to determine standard forms. For example, the inequality ψ_{Ω_1}(ε_0) < ψ_{Ω_1}(Ω_1) breaks when you remove 2.

> But I have no idea about specific comparison algorithms currently.

I am certain that creating those algorithms are much more difficult than you guess now. You are right in the sense that Rathjen used assumptions which are unnecesary for the strength in set theory in order to create algorithms, which are very long. The existence of the algorithms themselves are the main topic in ordinal analysis, and hence it means that Rathjen used the assumptions for the purpose on proofs. On the other hand, if you remove the assumptions, you will lose algorithms, which are very important in actual analysis. Therefore I disagree that the assumptions are unnecessary for googology.