Two properties (or concepts) $F$ and $G$ are said to be equinumerous if they have the same cardinality, i.e. if they can be put in one-to-one correspondence with each other. This can be very easily defined in polyadic second-order logic: we just need to say that there exists a two-place relation $R$ such that $Rxy$ and $Rxz$ implies $y=z$, $Rxz$ and $Ryz$ implies $x=y$, $Rxy$ implies $Fx$ and $Gy$, $Fx$ implies that $Rxy$ for some $y$, and $Gy$ implies $Rxy$ for some $x$. (I think that's right.) My question is can we define equinumerousity in monadic second-order logic, i.e. without the use of relations?
If so, can we prove Frege's theorem in monadic second-order logic? Also, are there other important notions which can only be defined in polyadic second-order logic?
Any help would be greatly appreciated.
Thank You in Advaance.