User blog comment:PsiCubed2/Something to Ponder Regarding Sequences and Ordinal Notations/@comment-33713741-20181121153145/@comment-30754445-20181121162110

(3,1) is the first element in the fundamental sequence of (4) that is bigger than (3).

And technically, since the first few elements of a FS have no effect on it's limit, you can set (3,1) to be whatever ordinal you wish (as long as it is between (3) and (4)).The system would work perfectly even if you set (3,1) to (3)+1 (the ordinal that comes immediately after (3)).

If you also want the system to look nice and intuitive, however, you'd probably want (3,1) to be much much larger than (3). And if you don't kepnow how to do that (I certainly don't) then you'll probably prefer to keep (3,1) undecided for the time being.

There's a reason why nobody has given a value to (0,0,0)(1,1,1)(2,2,2)(3,3,3)(4,4,3) in any version of BMS... :-)