The definition is:
"A set of formula's $\Sigma$ axiomizes $Th(M)$ if for all sentences $\phi$ the following applies: $\phi \in Th(M) \Leftrightarrow \Sigma \models \phi$"
Where $Th(M) = \{ \phi | \phi$ is a sentence and $M \models \phi \}$ and $M$ is a model.
What I'm reading than says the following:
According to the definition, a set of sentences $\Sigma$ axiomizes the theory $Th(M)$ if for all sentences $\phi$ the following applies: $\phi \in Th(M) \Leftrightarrow \Sigma \models \phi$
With a given set $\Sigma$ en model $M$, the implication from left to right is usually hard to proof.
My question is: why is this hard to proof? Don't you check if the given sentence is true in the model et voila; you know that it follows from $\Sigma$?
Let $M$, for example, be the complex numbers under addition and multiplication. Let $\Sigma$ be the theory of algebraically closed fields of characteristic $0$.
For this example, the implication from left to right says that the theory of algebraically closed fields of characteristic $0$ is complete. This is a non-trivial result.