Reading Hodges, I encountered two definitions of types of a theory. I think I really grasped the concept, so that either definition he uses I can uderstand proofs. But nonetheless I cannot fully see the equivalence between these two definitions and that is disturbing to me.
In section 2.3 we have the following:
Definition 1 A n-type of a theory $T$ is a collection of formulas $\Phi(\bar{x})$ such that there is a model $A$ of $T$ and a n-tuple $\bar{a}$ from $A$ such that $A\models\phi(\bar{a})\quad\forall\phi\in\Phi$. We then say that another model $B$ of the same theory $T$ realizes the type if the same olds in $B$, otherwise we say it omits the type.
In section 5.2 things are handled differently, starting from structures instaed of theories:
Let $A$ be a structure and $\bar{b}$ an n-tuple of elements from $A$. The complete n-type of $\bar{b}$ over the set of parameters $X\subset A$ with respect to $A$ is collection $\Phi(\bar{x},\bar{y})$ of formulas such that $A\models\phi(\bar{b},\bar{a})\quad\forall\phi\in\Phi$ for some $\bar{a}\subset X$.
Then we say that a complete n-type is such a collection of formulas but allowing for some $\bar{b}$ which lies in some elementary extension $B$ of $A$. In the parricular case where $\bar{b}$ is actually in $A$ we say $A$ realizes the type otherwise we say it omits the type. A type of $A$ is simply a subset of some complete type of $A$.
Thanks to the compactness theorem, we can show that a collection of formulas $\Phi(\bar{x})$ is a type of $A$ if and only if $A\models\exists\bar{x}\bigwedge\Psi(\bar{x})$ where $\Psi(\bar{x})$ varies in finite subsets of $\Phi(\bar{x})$. Similarly a maximal collection of formulas with this property is the same thing as a complete type of $A$.
With this background we say that
Definition 2 A type of a theory T is a collection $\Phi(\bar{x})$ of formulas such that $T'=T\cup\{\exists\bar{x}\bigwedge\Psi(\bar{x})\}$ is consistent for any finite subset $\Psi(\bar{x})$ of $\Phi(\bar{x})$
I cannot see why these two are equivalent. In parricular 1 seems stronger to me. I would try to show that 2 implies 1 by the compactness theorem and the equivaent definition of type in a structure given above. But no one assures me that the different structures witnessing the consistency of $T'$ can be put together in a single one.
Thanks in advance for help
P.s. Notice that Equivalent Definitions of Types is a different question.