User blog comment:Ikosarakt1/Uncountable function cannot exists/@comment-1605058-20130707061148/@comment-1605058-20130708191554

Because there might be no largest ordinal of given complexity, choice may be arbitrary. Suppose we have nonrecursive function with following property: BB(n) has complexity n, and this arbitrary choice always chooses BB(n). What is the order type? It is limit of BB(0), BB(1), ... BB(n), ... Because it is unbounded sequence of natural numbers, limit is w. So order type of some nonrecursive function is recursive. Now what?