 There is a marked similarity in the way \(\mn_2\) arises from \(\mn_1\) and in the way \(\mn_3\) arises from \(\mn_2\). In both cases we simply add a power set construction on top of what the model already has. Second-order logic alone is very good with power sets as we saw in  §5.3.  A consequence of this is that all the higher order logics (of any finite order and up to a point also of infinite order) from second-order on have Turing-equivalent decision problems and the same Hanf- and Löwenheim numbers (see  §4).  Thus, even though higher than second-order logics are strictly speaking stronger than second-order logic no difference can be seen by the usual criteria of strength of expressive power.