# proof of compactness theorem for first order logic

The theorem states that if a set of sentences of a first-order language $L$ is inconsistent, then some finite subset of it is inconsistent. Suppose $\Delta\subseteq L$ is inconsistent. Then by definition $\Delta\vdash\perp$, i.e. there is a formal proof of “false” using only assumptions from $\Delta$. Formal proofs are finite objects, so let $\Gamma$ collect all the formulas of $\Delta$ that are used in the proof.

Title proof of compactness theorem for first order logic ProofOfCompactnessTheoremForFirstOrderLogic 2013-03-22 12:44:02 2013-03-22 12:44:02 CWoo (3771) CWoo (3771) 4 CWoo (3771) Proof msc 03B10 msc 03C07