User blog comment:P進大好きbot/New Issue on Traditional Analyses/@comment-34422464-20190828003038/@comment-35470197-20190828005714

It might be confusing, but OCFs are recursively defined in set theory, which essentially refers to infinity. On the other hand, when we talk about algorithms, then we need to work recursively in arithmetic, which refers only to finitely many finite numbers.

Therefore although \(\in\)-relation is well-defined in set theory by definition, it does not ensure that the \(\in\)-relation for given two expressions of ordinals can be interpreted into a relation which can be completely determined by the expressions.

Also, it is good to know when we work in ordinals beyond \(\psi_0(\Omega_{\omega})\) with respect to extended Buchholz's OCF, we usually need a comparison algorithm in order to define an algorithm to compute FSes. Therefore if we lack a comparison algorithm, it usually means that we lack an algorithm for fundamental sequences.