Is every model of $\Gamma$ a model of $Cn(\Gamma)$?

Question

Is every model of $\Gamma$ a model of $Cn(\Gamma)$ ?

$Cn(\Gamma)=\{\sigma:\Gamma \models \sigma\}$ This is the set of all sentences logically implied by $\Gamma$ .

This could help me to understand some facts of elementary first order logic.

Answer 1

The answer is yes. Here's why:

by definition given a set of formulas $\Gamma$ and a formula $\varphi$ then $$\Gamma \models \varphi$$ if for every $M$ which is a model of $\Gamma$ ($M \models \Gamma$) then $M$ is also a model for $\varphi$ ($M\models \varphi$).

By definition $\sigma \in Cn(\Gamma)$ iff for every structure $M$ such that $M\models \Gamma$ we have that $M \models \sigma$.

So for every $\sigma \in Cn(\Gamma)$ and every structure $M$ such that $M \models \Gamma$ we have $M \models \sigma$, i.e. every model of $\Gamma$ is a model of every formula in $Cn(\Gamma)$ (i.e. $M \models Cn(\Gamma)$).