Compactness Theorem/Proof using Consistency Principle

Proof
By definition, $T$ is finitely satisfiable means that every finite subset of $T$ is satisfiable.

Because the direction:
 * $T$ satisfiable implies $T$ finitely satisfiable

is trivial, the proof below justifies the converse:
 * $T$ finitely satisfiable implies $T$ satisfiable.

By Extend Theory to Satisfy Witness Property, there exist a language $\LL^*$ and a set of $\LL^*$-sentences $T^*$ satisfying:
 * $T^*$ is finitely satisfiable
 * If $T^*$ is satisfiable, then $T$ is satisfiable
 * For every $\LL^*$-WFF of $1$ free variable $\map \phi x$, there exists some constant $c_\phi$ such that:
 * $T^* \models \paren {\exists x: \map \phi x} \implies \map \phi {x := c_\phi}$

Thus, it suffices to show that $T^*$ is satisfiable.

By Finitely Satisfiable Set of Sentences has Maximal Finitely Satisfiable Extension, there exists a set of $\LL^*$-sentences $T' \supseteq T^*$ such that:
 * For all $\LL^*$-sentences $\phi$, either $\phi \in T'$ or $\sqbrk {\neg \phi} \in T'$

As $T^* \subseteq T'$, it follows that the third property above continues to hold for $T'$.

Thus, by the lemma to Extend Theory to Satisfy Witness Property:
 * $T'$ satisfies the witness property

Additionally, any $\mathrm{PL}$-structure that models $T'$ will also model $T^*$.

But, by Maximal Finitely Satisfiable Theory with Witness Property is Satisfiable, there is such a $\mathrm{PL}$-structure.