Equivalence of Definitions of Consistent Set of Formulas

Theorem
Let $\LL_0$ be the language of propositional logic.

Let $\mathscr P$ be a proof system for $\LL_0$.

Let $\FF$ be a collection of logical formulas.

Definition 1 implies Definition 2
Suppose that $\FF \not \vdash_{\mathscr P} \phi$.

that there exists $\psi$ such that $\FF \vdash_{\mathscr P} \psi$ and $\FF \vdash_{\mathscr P} \neg \psi$.

Then by the Rule of Explosion (Variant 3):


 * $\psi, \neg \psi \vdash_{\mathscr P} \phi$

and therefore, combining the two above lines:


 * $\FF \vdash_{\mathscr P} \phi$

which is a contradiction.

Definition 2 implies Definition 1
According to definition 2, for every logical formula $\phi$ either $\phi$ or $\neg \phi$ is not a $\mathscr P$-provable consequence of $\FF$.

In particular, then, there exists a logical formula $\psi$ such that $\FF \not \vdash_{\mathscr P} \psi$.