Math 557 Sep 22

The Model Existence Theorem

Key Concepts

• Extend \(T\) to a Henkin theory \(T_H\), then complete it to a theory \(T'\).

• \(T'\) is a complete Henkin theory for the extended language \(\mathcal{L}_H\).

• For \(T'\), its term model \(\mathcal{A}_{T'}\) satisfies \[ \mathcal{A}_{T'} \models \sigma \; \iff \; T' \vdash_{\mathcal{L}_H} \sigma \] for all \(\mathcal{L}_H\)-sentences \(\sigma\).

Problems

Exercise 1
How do we obtain a model for \(T\) from a model for \(T'\)?

Exercise 2
Why is \(\mathcal{A}_{T'}\) countable if \(\mathcal{L}\) is countable?

Exercise 3
Let \(X\) be the set of all maximally consistent \(\mathcal{L}\)-theories. Recall that the sets \[\langle \sigma \rangle = \{ T \in X \colon \sigma \in T\} \quad (\sigma \text{ $\mathcal{L}$-sentence})\] generate a Hausdorff topology on \(X\).

Show that the topology is compact.