A set $\Gamma$ of propositional formulas is closed under derivations if, for any $\varphi$, we have $\Gamma \vdash
\varphi \Rightarrow \varphi \in \Gamma$.
A set is maximally consistent if it is
consistent and for any $\Gamma'$ s.t. $\Gamma \subseteq \Gamma'$ and $\Gamma'$
consistent, then $\Gamma ' = \Gamma$ (i.e. there is no larger set of
propositions that is consistent.)
Every maximally consistent set is closed under derivations (or closed under consequence). I am trying to determine if the reverse is true. My intuition is that it is not; i.e., that there are sets cloesd under derivations that are not maximally consistent. However, I haven't been able to find an example of such set.
A particular problem is that the counterexample I'm looking for must be infinite. Any finite set $\{\varphi_1, \ldots, \varphi_n \}$ of arbitrary size $n$ is not closed under derivation. (It is easy to imagine infinite combinations, using the connectives $\land, \lor,$ etc, of the elements of the set, which a finite set could never contain).
Any guidance is appreciated.
Best Answer
HINT: Let the set of sentence symbols be a singleton set $S=\{p\}$, and let $\Gamma$ be the set of tautologies, i.e. the "closure under derivations" of the empty set.