Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 16H / a

A sentence is a formula with no free variables. Thus its truth in an $L$-structure does not depend on a variable assignment.

A theory in $L$ is a set of $L$-sentences. Here a theory need not already contain every sentence derivable from it.

An $L$-structure $M$ is a model of $T$ when

A theory $T$ is consistent when no contradiction is derivable from it, equivalently when $T⊬\perp$. In a classical proof system this is also equivalent to there being no sentence $\sigma$ for which both $T⊢\sigma$ and $T⊢\neg \sigma$.