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

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).

• A model is a $\sigma$ structure which satisfies some axioms (the "theory"). – Qiaochu Yuan Oct 12 '15 at 21:46
• @QiaochuYuan So I am right that a model is just a $\sigma$ structure that models a set of formulae/axioms? – user223391 Oct 12 '15 at 21:47
• @QiaochuYuan If you confirm this in an answer you have a shiny 25 reputation points waiting for you, as I'm sure is just so important. ;) – user223391 Oct 12 '15 at 21:48
• I honestly don't get why I got two downvotes, and one very recently. Would they care to explain? – user223391 Dec 24 '15 at 23:18
• I thought that model was two things, 1) a set of propositions $\Sigma$ and 2) a truth assignment on the atoms $A$ such that all proposition $\Sigma \subset Prop(A)$ are true. – Pinocchio Sep 12 '18 at 14:35

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 $\varnothing$".

• Isn't $\sigma$ usually a signature, not a language? – user223391 Oct 12 '15 at 21:50