Consistent Theory (First-Order Logic)

first-order-logic

Definition

Consistent Theory (First-Order Logic)

A theory $\Gamma$ is consistent if there exists a structure $\mathcal{M}$ such that $\mathcal{M}\models \phi$ for all $\phi \in \Gamma$.

Satisfaction

Theorem

For finite $\Gamma$, consistency is equivalent to satisfiability of the conjunction:

$$\mathcal{M}\models \phi \text{ for all }\phi \in \Gamma ⟺\mathcal{M}\models \underset{\phi \in \Gamma }{\bigwedge }\phi$$