User blog comment:Edwin Shade/A Small Question/@comment-30754445-20171029013412/@comment-26428969-20171029104836

A mcuh better definition for "recursive ordinal":

α is recursive iff there exists a well-ordering of the natural numbers of order-type α such that there exists a computable algorithm that, given two natural numbers, correctly decides which natural number is greater by that well-ordering.

Note that you can recursively do "arithmetic" on any ordinal smaller than ω_1^CK so your definition would make  ω_1^CK recursive, which is incorrect.