User blog comment:C7X/CNF Comparison Algorithm/@comment-32213734-20200106184503/@comment-35470197-20200106234909

To be more precise, yours and the OP's are not algorithms. An algorithm only accepts inputs in a fixed countable set with a fixed enumeration, e.g. the set of natural numbers and a recursive set of formal strings consisting of fixed finitely many letters. The set of ordinals below ε_0 is countable, but you have not specified an enumeration or a coding into formal strings in a fixed recursive set.

Usually, in order to define the recursive set of CNFs, we need to use the comparison algorithm. Therefore you need to extend the algorithm to a wider set of formal strings in order to avoid circular logic. In particular, specifying the recursive set of formal strings (and the algorithm to check whether given formal strings are CNFs or not) is non-trivial and hence is not ignorable.