I am trying to understand why the compactness theorem does not apply in infinite logic and I wonder if anyone has a good example and explanation for this?
Edit: By infinite logic I mean logic that allows infinitely many conjunctions and disjunctions. More exactly:
- $M \models \bigvee \Gamma$ iff $M \models \varphi$ for some set of sentences $\varphi \in \Gamma$.
- $M \models \bigwedge \Gamma$ iff $M \models \varphi$ for some set of sentences $\varphi \in \Gamma$.
The compactness theorem:
The compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model.
Thanks in advance!