We say that a set $\Gamma$ of formulas in a language $L$ is complete if it is consistent and for each formula $\varphi$ in $L$, exactly one of $\varphi$ and $\neg\varphi$ is in $\Gamma$. Show that if $\Gamma$ is complete, then it is maximal consistent (i.e. every set properly containing $\Gamma$ is inconsistent)
Can anyone help me out? I am totally lost and my professor never thought us this.
Would this work?
Assume that $\Gamma$ is complete, then by definition it is also consistent. This fulfills the first condition of being max. consistent.
For the 2nd condition, I will prove its equivalent. So suppose $\varphi\not\in\Gamma$, I need to prove that $\Gamma\cup\{\varphi\}$ is inconsistent.
But by the def. of completeness, either $\varphi\in\Gamma$ or $\neg\varphi\in\Gamma$ - we have $\varphi\not\in\Gamma$, so $\neg\varphi\in\Gamma$. Thus $\Gamma\cup\{\varphi\}\vdash\neg\varphi$.
On the other hand we $\varphi\in\Gamma\cup\{\varphi\}$, so $\Gamma\cup\{\varphi\}\vdash\varphi$ -but this means $\Gamma\cup\{\varphi\}$ is inconsistent.
Yup, you're exactly right!
(Well, you're mathematically right. Your claim "I am totally lost" is quite wrong, however. :P)
A technical aside:
It's worth noting that the only rule of $\vdash$ you're using here is that $\psi\in\Theta$ implies $\Theta\vdash\psi$: you use that to conclude that $\Gamma\cup\{\varphi\}\vdash\neg\varphi$ (since $\neg\varphi\in\Gamma$) and that $\Gamma\cup\{\varphi\}\vdash\varphi$ (since $\varphi\in\{\varphi\}$). So this result is really quite general and applies to a wide range of "deduction systems." In full generality, what you've proved is the following:
Right now this isn't too interesting. Down the road, though, the general question of what sorts of deductive systems have what properties becomes quite an interesting one, with two relevant terms being abstract model theory (for the semantic side of things, primarily focusing on logics with quantification) and algebraic logic (for the syntactic side of things, with more emphasis on propositional logics).