 One should note that  carrier of the  final coalgebra may be taken to be    certain theories in this language.   These may be described extrinsically as the theories of all points in all coalgebra.   It is more informative, however, to  set down a logical system and then consider the maximal consistent sets in the system. With the right definition, the maximal consistent sets do turn out to be the carrier of a final coalgebra for the functor. 