Definition:Consistent (Logic)/Set of Formulas

Definition
Let $\mathcal L$ be a logical language.

Let $\mathscr P$ be a proof system for $\mathcal L$.

A collection $\mathcal F$ of logical formulas is consistent for $\mathscr P$ :


 * There exists a logical formula $\phi$ such that $\mathcal F \nvdash_{\mathscr P} \phi$.

That is, some logical formula $\phi$ is not a provable consequence of $\mathcal F$.