User blog comment:Fejfo/bitstring lexographical notation/@comment-35470197-20190913005559/@comment-35470197-20190913235932

Right, I agree with the issue. I also considered a way to formalise your system and simultaneously fundamental sequences without using ordinals, but unfortunately I could not do so. Of course, I can create fundamental sequences by just immitating fundamental sequences of ordinals below a fixed ordinal and hiding them. Then the definition looks independent of ordinals, but the well-foundedness of the ontation is doubtful beyod the fixed ordinal. Therefore it is not a solution to remove a given bound by the fixed ordinal. There might be more brilliant idea, and hence I am looking forward to seeing your formalisation in the future.