I am little confused with this. In any model of set theory (or is only in ZFC?), we start with an initial set, and then by transfinite recursion we build the universe of sets, which is a proper class. The universe of sets can differ, for instance, we don't know if $V=L$, or we at least can assume it either true or false and get different models? (or is it different theories?). I guess there are many other different universes beyond $V$ and $L$, either by transfinite recursion or by some other (unknown to me) method.
The specific question is: Is the set of ordinals in any of those models the same?. I know that there are properties of ordinals (or cardinals) that are not absolute. For instance, there are no ordinals that are measurable cardinals in $L$, but an ordinal that is a measurable cardinal in some other model can still belong to $L$, only that in $L$ it doesn't have that property. So, finally my question: do all the universe of sets of set theory have the same ordinals on every model? In other words, is the universe of sets of the same "size" on all models, so that the proper class or ordinals is well defined, and its size the same? (if that is not the case, do $V$ and $L$ have the same number of ordinals?
So in the end, what I want to know is if the proper class of all ordinals is well defined and always the same, or if it is dependent of the particular set theory or model we are considering in a specific case.