2.6: The Completeness Theorem
- Page ID
- 121609
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 2.6.1: Introduction
- The completeness theorem is one of the most fundamental results about logic.
- 2.6.2: Outline of the Proof
- The proof of the completeness theorem is a bit complex, and upon first reading it, it is easy to get lost. So let us outline the proof.
- 2.6.3: Complete Consistent Sets of Sentences
- Complete sets of sentences leave no questions unanswered. For any sentence \(A\), \(\Gamma\) “says” if \(A\) is true or false.
- 2.6.4: Henkin Expansion
- Part of the challenge in proving the completeness theorem is that the model we construct from a complete consistent set \(\Gamma\) must make all the quantified formulas in \(\Gamma\) true. In order to guarantee this, we use a trick due to Leon Henkin.
- 2.6.5: Lindenbaum’s Lemma
- We now prove a lemma that shows that any consistent set of sentences is contained in some set of sentences which is not just consistent, but also complete.
- 2.6.6: Construction of a Model
- Right now we are not concerned about \(=\), i.e., we only want to show that a consistent set \(\Gamma\) of sentences not containing \(=\) is satisfiable. We first extend \(\Gamma\) to a consistent, complete, and saturated set \(\Gamma^*\). In this case, the definition of a model \(M(\Gamma^*)\) is simple.
- 2.6.7: Identity
- The construction of the term model given in the preceding section is enough to establish completeness for first-order logic for sets \(\Gamma\) that do not contain \(=\). It does not work, however, if \(=\) is present. We can fix this using a construction known as “factoring.”
- 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.
- 2.6.9: The Compactness Theorem
- One important consequence of the completeness theorem is the compactness theorem. The compactness theorem states that if each finite subset of a set of sentences is satisfiable, the entire set is satisfiable—even if the set itself is infinite.
- 2.6.10: A Direct Proof of the Compactness Theorem
- We can prove the Compactness Theorem directly, without appealing to the Completeness Theorem, using the same ideas as in the proof of the completeness theorem.
- 2.6.11: The Löwenheim-Skolem Theorem
- The Löwenheim-Skolem Theorem says that if a theory has an infinite model, then it also has a model that is at most countably infinite.