of a big deal, in this sense, because much like in a completion of a metric space the original space is dense, we expect that the ordinals of $M$ are the ordinals of the larger model. Otherwise we may end up changing the model too much.It should be remarked, though, that given any family of non-empty sets $\cal F$, we can always force a choice function from that particular family. Often in the cost of making $\bigcup\cal F$ countable (or otherwise well-orderable), which may or may not collapse cardinals.It should be remarked, regardless to the above, that sometimes there is an extension which does add ordinals:Jon Barwise, Admissible Sets and Structures: An Approach to Definability Theory (Berlin: Springer-Verlag, 1975), 365-379.Every countable transitive model of