 Define a set \(\Gamma\) of sentences of the language \(\LKe\) to be maximally consistent if \(\Gamma\) is consistent and for every sentence \(\theta\) of \(\LKe\), if \(\theta\) is not in \(\Gamma\), then \(\Gamma,\theta\) is inconsistent. In other words, \(\Gamma\) is maximally consistent if \(\Gamma\) is consistent, and adding any sentence in the language not already in \(\Gamma\) renders it inconsistent. Notice that if \(\Gamma\) is maximally consistent then \(\Gamma \vdash \theta\) if and only if \(\theta\) is in \(\Gamma\).