An ordinal is always recursive iff it is less than another recursive ordinal, so the Church-Kleene ordinal is also the supremum of all recursive ordinals. It is a limit ordinal, since the successor of a recursive ordinal is also recursive. Since every computable well-ordering can be identified by a distinct Turing machine, of which there are countably many, \(\omega_1^ ext{CK}\) is also countable.