I am trying to prove the consistency of $\Gamma = \{\neg \forall v_{1}Pv_{1}, Pv_{2}, Pv_{3}, \dots \}$. What I'm thinking is the following:
Suppose $\Gamma$ is inconsistent. $\{\neg \forall v_{1}Pv_{1}, Pv_{2}\}$ is consistent, so there must be a minimal $k \geq 2$ such that $\{\neg \forall v_{1}Pv_{1}, Pv_{2},\dots, Pv_{k+1} \}$ is inconsistent. By Reductio Ad Absurdum, this means $\{\neg \forall v_{1}Pv_{1}, Pv_{2},\dots, Pv_{k}\} \vdash \neg Pv_{k+1}$.
I want to show that $\{\neg \forall v_{1}Pv_{1}, Pv_{2},\dots, Pv_{k}\}$ is also inconsistent, contradicting the minimality of $k$, but am not sure how to do that -- it seems like I would do this by showing $\{\neg \forall v_{1}Pv_{1}, Pv_{2},\dots, Pv_{k}\} \vdash Pv_{k+1}$, but I don't have any insight right now as to how to show that.
Any thoughts?
I assume the logic is classical first-order. Are you required to solve this problem within some proof theory? If not the following thoughts may help.
Any satisfiable set is consistent.So it suffices to show that $\Gamma$ is satisfied by some model. This can be shown via compactness: Let $\Gamma_0 \subseteq \Gamma$ be finite and so for some $n \in \mathbb{N}, \Gamma_0 \subseteq \Delta :=\lbrace \neg \forall v_1 Pv_1 \rbrace \cup \lbrace Pv_i : 1<i < n \rbrace$. Obviously $\Delta$ is satisfiable and consequently the same holds for $\Gamma_0$. `