User blog comment:QuasarBooster/Higher order worms?/@comment-34422464-20190606232043/@comment-35470197-20190607084951

I imagined another formulation of "\(\omega\)-th order worm" in the following way: For example, the following are \(\omega\)-th order worm by the definition above: Then the lexicographic ordering on the set of finite order worms can be extended to a well-ordering by a similar way to the ordinal notation system associated to extended Buchholz's OCF if I am correct, and the expansion rule of finite order worms can be extended to that of (\omega\)-th order worms in a similar way to the canonical recursive system of fundamental sequences below \(\psi_0(\Omega_{\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}}})\).
 * A list of natural numbers and \(\omega\)-th order worms is again an \(\omega\)-th order worm.
 * 1) The empty list.
 * 2) Any list of natural numbers, i.e. \(1\)-st order worm.
 * 3) Any list of \(1\)-st order worms, i.e. \(2\)-nd order worm.