 Solution

A set of sentences is maximal consistent if it is consistent and no proper extension by further sentences remains consistent. Equivalently it is consistent, deductively closed, and decides every sentence: for each $\theta$, exactly one of $\theta$ and $\neg \theta$ belongs to it.

For example, if $\Sigma ⊢\theta$ but $\theta \notin \Sigma$, adding $\theta$ does not create a new contradiction, contrary to maximality. Thus maximality includes deductive closure, not only having as many independent axioms as possible.