The countable ordinals are themselves either countable or uncountable. They cannot be countable since that would involve a set with itself as an element, so they are uncountable.
If they are uncountable, then either they form a consistent totality or they are a proper class. If they form a consistent totality, then a least uncountable ordinal exists, otherwise not.
So how is it determined that the sequence of countable ordinals is or is not a consistent totality? My problem is that it cannot be said they form a consistent totality because there is an uncountable limit ordinal since that would be circular. Therefore, how exactly is this resolved?
Edit, as requested: My question is not equivalent to the question, "What axioms are you using? ZFC?" Nor is it answered by (I haven't figured out how to format the symbols on this site so I'll just spell it out): "The set of all countable ordinals is the set of those elements x of the cardinal number two to the aleph-null such that x is a countable ordinal", which seems to be a way of saying x is a countable ordinal if it is a member of the set of countable ordinals, except two to the aleph-null is a number: it is not a set but the cardinality of a set, so it doesn't actually have elements. I suppose the intention of that answer could be taken as that all countable sets are elements of the power set of N, except the assertion isn't true. Nor is my question addressed by comments to that answer. So far as I can tell, the answerer did not define omega plus one, nor have I ever seen two to the aleph-null described as an ordinal. Besides which, the relation of the cardinal number two to the aleph-one to the limit ordinal omega-one is outside ZFC. So I honestly don't see what any of this has to do with my question or why I was asked to edit it accordingly.