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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).

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  • $\begingroup$ A model is a $\sigma$ structure which satisfies some axioms (the "theory"). $\endgroup$ – Qiaochu Yuan Oct 12 '15 at 21:46
  • $\begingroup$ @QiaochuYuan So I am right that a model is just a $\sigma$ structure that models a set of formulae/axioms? $\endgroup$ – user223391 Oct 12 '15 at 21:47
  • $\begingroup$ @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. ;) $\endgroup$ – user223391 Oct 12 '15 at 21:48
  • $\begingroup$ I honestly don't get why I got two downvotes, and one very recently. Would they care to explain? $\endgroup$ – user223391 Dec 24 '15 at 23:18
  • $\begingroup$ 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. $\endgroup$ – Pinocchio Sep 12 '18 at 14:35
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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.

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  • $\begingroup$ Isn't $\sigma$ usually a signature, not a language? $\endgroup$ – user223391 Oct 12 '15 at 21:50
  • $\begingroup$ @avid19 I am used to texts where "language" is used for your "signature", however, according to your definition of signature I believe you are right. $\endgroup$ – 6005 Oct 12 '15 at 21:52
  • $\begingroup$ In Hodge's A Shorter Model Theory, he defines a signature to be all the non logical symbols, and a language is the signature, including the logical symbols. $\endgroup$ – user223391 Oct 12 '15 at 21:54
  • $\begingroup$ @avid19 OK, I have updated the answer to follow that convention. The convention I am used to is that the language does not include the logical symbols either. $\endgroup$ – 6005 Oct 12 '15 at 21:56
  • $\begingroup$ Thanks for the help, enjoy those 25 reputation points :) $\endgroup$ – user223391 Oct 12 '15 at 21:58
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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.

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