# Proof of Completeness Theorem in Enderton's Logic, satisfiability of $\Gamma \cup \Theta \cup \Lambda$

I'm reading the proof of the Completeness Theorem from Enderton's "A Mathematical Introduction to Logic". I'm having issues seeing how the following highlighted sentence actually holds (excerpt from page 137).

Let $$\Lambda$$ be the set of logical axioms for the expanded language. Since $$\Gamma \cup \Theta$$ is consistent, there is no formula $$\beta$$ such that $$\Gamma \cup \Theta \cup \Lambda$$ tautologically implies both $$\beta$$ and $$\neg \beta$$. (This is by Theorem 24B; the compactness theorem of sentential logic is used here.) Hence there is a truth assignment $$v$$ for the set of all prime formulas that satisfies $$\Gamma \cup \Theta \cup \Lambda$$.

I have tried to reason "by contrapositive". That is, suppose a set of (sentential) formulas $$\Sigma$$ is unsatisfiable. Then, vacuously, every truth assignment that satisfies $$\Sigma$$ will also satisfy any formula at all. Hence $$\Sigma$$ tautologically implies any formula. In particular, for any given formula $$\beta$$, $$\Sigma$$ tautologically implies both $$\beta$$ and $$\neg\beta$$.

Is my reasoning correct?

• Looks right to me. Remember it well, cause "consistent implies satisfiable" is used just as often, if not more than "valid implies provable". – spaceisdarkgreen Mar 28 at 1:39