User blog comment:C7X/CNF Comparison Algorithm/@comment-32213734-20200106184503/@comment-35470197-20200107064525

@C7X

What I said is that you need to define the recursive set T of normal form expressions of ordinals and a comparison algorithm on T in order to define what you call "a comparison algorithm for ordinals". Then usually you need to define a comparison algorithm for a wider recursive set than T, because the definition of T often mentions to the comparison itself.

> if it references the comparison algorithm itself

No, it is just a circular logic, because the comparison algorithm on T is not defined when you define T. I meants, if you use the comparison of ordinals, which is uncomputable, in the definition of T, then it is not computable in the sense that you do not have specified a way to compute it.

@Rgetar

Oh, are you considering the comparison for ordinals which can be greater than ε_0? Anyway, as I wrote above, it is not an algorithm unless you specify the set T of valid expressions.