State and prove the Completeness Theorem for Propositional Logic.

[You do not need to give definitions of the various terms involved. You may assume the Deduction Theorem, provided that you state it precisely.]

State the Compactness Theorem and the Decidability Theorem, and deduce them from the Completeness Theorem.

Let $S$ consist of the propositions $p_{n+1} \Rightarrow p_{n}$ for $n=1,2,3, \ldots$. Does $S$ prove $p_{1}$ ? Justify your answer. [Here $p_{1}, p_{2}, p_{3}, \ldots$ are primitive propositions.]