I am studying some model theory. At the moment I am considering countably saturated models. Here my book (Chang & Keisler) needs the concept of types (maximal consistent sets of formulas). The authors state:
"The set $T$ of all sentences which belong to a type $\Gamma(x_1...x_n)$ is a maximal consistent theory."
I can easily see that $T$ must be a consistent theory because every model $\mathfrak{A}$ in which $\Gamma$ is realized, will be a model of $T$.
Now my question is, why $T$ is maximal consistent. Given a sentence $\sigma$ that is consistent with $T$, we have to show that $\sigma$ is in $T$. For this it would suffice to prove that $\sigma$ is in $\Gamma$, but I cannot see how to do this. does a model of $T \cup \{ \sigma \}$ already realize $\Gamma \cup \{\sigma \}$? I guess not...
Thank you in advance for any help!
For consistent sets of formulas "maximal" is equivalent to "contains $\varphi$ or $\lnot\varphi$ for every formula $\varphi$".
For consistent theories "maximal" is equivalent to "contains $\varphi$ or $\lnot\varphi$ for every sentence $\varphi$".
A type, being a maximal consistent set of formulas, in particular contains $\varphi$ or $\lnot\varphi$ for every sentence $\varphi$. So the set of sentences in the type is a maximal consistent theory.