Ive encountered the following question (question of true \ false):
For every consistent set (but not maximally consistent) of propositions $\Phi$ exists consistent set (but not maximally consistent) of propositions $\Phi'$ such that: $\Phi \subseteq \Phi'$
This claim is false, but i really don't understand why, it seems reasonable that if there is some consistent set $\Phi$ but not maximal consistent, so there is a claim that says that there exists a module for $\Phi$, and because the condition of not being maximum consistent i can conclude there is at least 2 modules, lets assume $\varphi \in \Phi$,
then if we define $\Phi' = \Phi\cup \left\{ \neg\neg\varphi\right\}$, so $\Phi\subseteq \Phi'$, and we can imply this 'trick' for every consistent set.
where am i wrong?