On p.8 of <http://www.msri.org/publications/books/Book39/files/marker.pdf>, the author writes $\Gamma(\bar{d})$, when $\Gamma$ is, first of all, a set of formulas (not a single one), and it is a formula which has variables, not constants. This doesn't make sense. And what does he mean by $T+\Gamma(\bar{d})$? This would have to mean that we are working in a different language. And why does it imply facts about $\psi\_i(\bar{v})$ when we have $\bar{d}$, i.e. a set of constants, rather than variables, when $\bar{v}$ is a set of variables, not a set of constants?


Similarly, in the proof of the "CLAIM," in the sentence that begins "If $\Sigma$ is inconsistent...", how can we go from $\psi\_1(\bar{d})$ to $\psi\_1(\bar{v})$? One takes constants, the other takes variables.



Confusion about model theory notes
The notation $\Gamma(\bar d)$ means that for every formula $\psi(\bar v)\in\Gamma(\bar v)$ we substitute $\bar d$ for $\bar v$. The set $T+\Gamma(\bar d)$ is just the union $T\cup\Gamma(\bar d)$.


The interchanging of variables $\bar v$ and constants $\bar d$ in the proof comes from the fact that the constants $\bar d$ are new constants that do not appear in $T$. Thus, for example, if $T\models \phi(\bar d)$, then also $T\models\forall \bar v (\phi(\bar v))$.


