User blog comment:Boboris02/Large number combinatorics III:Sigma ordinal notation/@comment-1605058-20170212163020

The function \(\sigma(0,\alpha)\) enumerates the successor ordinals, so \(\sigma(1,\alpha)\) enumerates the limit ordinals (and \(0\)), then \(\sigma(2,\alpha)\) again enumerates all successor ordinals and so on, switching between successors and limits. Are you sure this is what you mean? Also, nothing indicates how to define it for limit ordinals, so this is only well-defined for finite ordinals.

In the definition of \(L(\alpha)\) you refer to \(L(\delta)\) for \(\delta\) greater than \(\alpha\). This isn't a valid definition - a (transfinite) recursive definitions refer only to smaller ordinals, and if you refer to larger ordinals, it stops making any sense.

If \(\alpha\) is a limit ordinal, then its fundamental sequences don't have last terms. Also note that no ordinal has "the" fundamental sequence - each ordinal has many, so you have to specify some system of FSes, or mention that you assume there is one fixed.