2.6.8: The Completeness Theorem
Let’s combine our results: we arrive at the completeness theorem.
Let \(\Gamma\) be a set of sentences. If \(\Gamma\) is consistent, it is satisfiable.
Proof. Suppose \(\Gamma\) is consistent. By Lemma 10.4.1 , there is a saturated consistent set \(\Gamma' \supseteq \Gamma\) . By Lemma 10.5.1 , there is a \(\Gamma^* \supseteq {\Gamma'}\) which is consistent and complete. Since \(\Gamma' \subseteq \Gamma^*\) , for each formula \(A(x)\) , \(\Gamma^*\) contains a sentence of the form \(\lexists{x}{A(x)} \lif A(c)\) and so \(\Gamma^*\) is saturated. If \(\Gamma\) does not contain \(\eq[][]\) , then by Lemma 10.6.1 , \(\Sat{M(\Gamma^*)}{A}\) iff \(A \in \Gamma^*\) . From this it follows in particular that for all \(A \in \Gamma\) , \(\Sat{M(\Gamma^*)}{A}\) , so \(\Gamma\) is satisfiable. If \(\Gamma\) does contain \(\eq[][]\) , then by Lemma 10.7.1 , for all sentences \(A\) , \(\Sat{\equivclass{M}{\approx}}{A}\) iff \(A \in \Gamma^*\) . In particular, \(\Sat{\equivclass{M}{\approx}}{A}\) for all \(A \in \Gamma\) , so \(\Gamma\) is satisfiable. ◻
For all \(\Gamma\) and sentences \(A\) : if \(\Gamma \Entails A\) then \(\Gamma \Proves A\) .
Proof. Note that the \(\Gamma\) ’s in Corollary \(\PageIndex{1}\) and Theorem \(\PageIndex{1}\) are universally quantified. To make sure we do not confuse ourselves, let us restate Theorem \(\PageIndex{1}\) using a different variable: for any set of sentences \(\Delta\) , if \(\Delta\) is consistent, it is satisfiable. By contraposition, if \(\Delta\) is not satisfiable, then \(\Delta\) is inconsistent. We will use this to prove the corollary.
Suppose that \(\Gamma \Entails A\) . Then \(\Gamma \cup \{\lnot A\}\) is unsatisfiable by Proposition 5.14.2 . Taking \(\Gamma \cup \{\lnot A\}\) as our \(\Delta\) , the previous version of Theorem \(\PageIndex{1}\) gives us that \(\Gamma \cup \{\lnot A\}\) is inconsistent. By Propositions 8.9.2 and 9.8.2 , \(\Gamma \Proves A\) . ◻
Use Corollary \(\PageIndex{1}\) to prove Theorem \(\PageIndex{1}\) , thus showing that the two formulations of the completeness theorem are equivalent.
In order for a derivation system to be complete, its rules must be strong enough to prove every unsatisfiable set inconsistent. Which of the rules of derivation were necessary to prove completeness? Are any of these rules not used anywhere in the proof? In order to answer these questions, make a list or diagram that shows which of the rules of derivation were used in which results that lead up to the proof of Theorem \(\PageIndex{1}\) . Be sure to note any tacit uses of rules in these proofs.