Notes - Logic MT24, Axiomatisations
Flashcards
Suppose:
- $\mathcal L$ is a first-order language
- $\mathcal A$ is an $\mathcal L$-structure
@Define the first-order theory of $\mathcal A$, written $\text{Th}(\mathcal A)$ or $\text{Th}^{\mathcal L}(\mathcal A)$.
\[\text{Th}(\mathcal A) := \\{\sigma \in \text{Sent}(\mathcal L) \mid A \models \sigma\\}\]
i.e. the set of all $\mathcal L$-sentence true in $\mathcal A$.
Suppose:
- $\mathcal L$ is a first-order language
- $\mathcal A$ is an $\mathcal L$-structure
- $\text{Th}(\mathcal A)$ is an $\mathcal L$-theory
@Define an axiomatisation of $\text{Th}(\mathcal A)$.
A complete subset of $\text{Th}(\mathcal A)$, i.e. a set of sentences which hold of $\mathcal A$ which suffice to deduce any sentence which holds of $\mathcal A$.