Notes - Logic and Proof MT24, Logical theories

Suppose $\sigma$ is a signature. @Define a theory $\pmb T$.

A set of $\sigma$-sentences that is closed under semantic entailment, i.e. if $\pmb T \models F$, then $F \in \pmb T$.

Suppose $\sigma$ is a signature and $\mathcal A$ is a $\sigma$-structure. @Define $\text{Th}(\mathcal A)$.

The set of sentences that are satisfied in $\mathcal A$.

Suppose $\pmb S$ is a set of sentences. How can you construct a theory, and what is $S$ called in this context?

\[\pmb T = \{F : \pmb S \models F\}\]

$\pmb S$ is called the axioms of $\pmb T$.

@Define what it means for a theory $\pmb T$ to be complete.

For any sentence $F$, either $F \in \pmb T$ or $\lnot F \in \pmb T$.