Infinite regress lies within every democratic procedural choice. If society members try to select an appropriate rule [social choice correspondence (SCC)] entirely endogenously, they will need an appropriate rule to choose such a rule. However, this should also be selected by an appropriate rule to choose a rule to choose a rule, and so on. This paper explores how to solve this infinite regress. A preference profile over the set of alternatives is said to converge if, at a sufficiently high level, every feasible SCC in the menu ultimately results in the same alternative, and hence, further regress has no effective meaning. A menu is said to be convergent if all preference profiles converge under the menu (i.e., infinite regress can "always" be resolved). First, we characterize the convergent menus under a special case. Then, we prove two general possibility theorems: (1) there exists a menu of SCCs that is strongly convergent (i.e., the outcome is uniquely determined); (2) any set of scoring rules can be extended to a superset that is asymptotically convergent for a large society (i.e., the probability of a convergent profile occurring goes to one as the population goes to infinity). Therefore, such a large society can "almost always" resolve the infinite regress by adding multiple SCCs. These theorems are expected to build new ground for SCCs in a distinct way from the axiomatic characterizations of standard social choice theory.
