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.