User blog comment:P進大好きbot/New Issue on Traditional Analyses/@comment-31663462-20190825021820/@comment-39541634-20190826061056

As an expansion on what P-bot said:

SBA is right that for any two ordinals "writable in n symbols" (for a given n) there is a computable alogrithm of comparison. A simple table with all the possible pairings will do the trick.

The problem is that:

(1) The existence of an algorithm for a fixed value of n does not imply the existence of a general algorithm for all n.

(2) Knowing that "there exists an algorithm to do X" is very different from actually constructing and implementing a workable algorithm for doing X. It doesn't even guarantee that there's a reasonably short algorithm for doing X (by "reasonably short" I mean that it can fit the physical universe).