Let $A$ be a list of axioms which we assume to be sound (for example, PA or ZFC). Godel's incompleteness theorems imply that if we add only finitely many (true) axioms to $A$, the new list $B$ will yield an incomplete theory.
One may relativize and quantify this idea as follows. Let $\Phi$ be any set of sentences. Say that a list $A$ settles $\phi$ if for any $\phi \in \Phi$ we have $A \vdash \phi$ or $A \vdash \lnot\phi$. Let $\Phi_n$ denote the set of all sentences with at most $n$ characters, and let $g(n)$ denote the size of the shortest list of true sentences (shortest in relation to the number of characters) we must add to $A$ to settle $\Phi_n$. Then $g$ is nondecreasing, and we have an upper bound of the form $g(n) \leq CK^n$.
By a pigeon-hole argument, $g$ is unbounded. If for example $A$ is the set of axioms of $ZFC$ and $n_1$ is the number of characters in a (suitable) formulation of the continuum hypothesis, we have $g(n_1)>0$. Are effective lower bounds known for $g$ ?