Selection Principles in Metalogic

In metalogic we take logic itself as the object of study. For this we do three things: define our logical formulae (Grammar), define what those logical formulae mean (Semantics), and define the rules for proofs (proof theory). In a crude sense the semantics of a logic tell us which logical formulae are true, and the proof-theory of a logic tells us which logical formulae can be deduced by proof from a set of axioms. These axioms are the objects of study in this project. The properties studied in most depth are soundness and completeness. Basically put, soundness is the following property: Every logical formula that is provable is guaranteed to be true. Completeness is this: Every logical formula that is guaranteed to be true is provable. The selection principle that I explore is this: Given a sequence of sound and complete sets of logical formulae, can we pick one formula from each, such that the set of all of our selections is complete? As it turns out, the answer is: no.