Optimality Theory assumes the candidate set generated for any given input is of infinite cardinality. If all of the candidates in the candidate set were potential winners (optimal candidates under some ranking), then OT would have predicted an infinite typology—there would be infinitely many possible languages. However, Samek-Lodovici and Prince (1999) have shown that in standard OT (with only markedness and IO Correspondence constraints), only a finite number of candidates from the infinite candidate set can actually be winners—the (infinite) majority of the candidates in the candidate set is harmonically bounded, and will therefore never be selected as winners under any ranking.
 

Their argument for the finite cardinality of the set of potential winners rests on the assumption that cardinality of CON is finite. If every possible ranking between the n constraints in CON were to select a unique winner, then there can be maximally n! different winners for any given input.
 

The addition of non-IO Correspondence constraints to CON threatens this general result. Both OO Correspondence constraints and Sympathy constraints can result in an otherwise harmonically bounded candidate being selected as winner. Non-IO Correspondence constraints therefore decrease the size of the set of harmonically bounded candidates and increase the size the set of potential winners. A question that therefore needs answering: Is the set of potential winners still of finite cardinality in an OT grammar with Sympathy constraints and OO Correspondence constraints? If this question can be answered in the affirmative, then OT predicts a finite typology even with non-IO Correspondence constraints added to CON. However, if the addition of non-IO Correspondence constraints increases the set of potential winners to an infinite size, then an OT grammar with these constraints added to CON will predict an infinite typology.
 

In this paper I argue that under reasonable assumptions it can be shown that the cardinality of the set of potential winners is finite even with the addition of Sympathy and OO Correspondence constraints. I argue that each of Sympathy Theory OO Correspondence Theory adds only finitely many constraints to CON. I then use the same argument that Samek-Lodovici and Prince use for standard OT. There are only finitely many rankings (n!) between finitely many constraints (n). Even if each of these rankings were to select a unique winner, only a finite number of winners can be selected.