What's the difference between a model and a $\sigma$ structure?

Question

In model theory, I haven't actually seen the word "model" defined. The only thing I've seen defined is a $\sigma$ structure for some signature $\sigma$.

I read phrases like $A$ is a model of some theory, where $A$ is a $\sigma$ structure. What is the difference? Is a model just a $\sigma$ structure?

The first time I see model defined is in "canonical model", which just gives a specific $\sigma$ structure.

I am reading Hodge's A Shorter Model Theory.

I think a model is just a $\sigma$ structure that satisfies a certain set of formulae (a theory).

Answer 1

Maybe this will help:

• A model of a theory $\mathcal{T}$ over a signature $\sigma$ is a $\sigma$-structure $A$ such that $A \vDash \mathcal{T}$, i.e. every statement in $\mathcal{T}$ is true in $A$.

• Therefore, the following are equivalent statements: "$A$ is a $\sigma$-structure", and "$A$ is a model of $\varnothing$".

See also this related post.

Answer 2

Structure is the official term. A structure is a set equipped with certain functions and relations that correspond to the symbols of a signature (also called language, depending on the author).

The term model is officially only defined for speaking about the relation between a structure and a theory: A structure is a model of a theory if it satisfies all sentences in the theory.

But in reality, outside lectures for beginners almost nobody says structure. We tend to call the things model, even when there is no theory around. This phenomenon is similar to how most people call the Netherlands Holland, or say America for the United States of America, except in contexts where this can lead to confusion.