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I'm newbie to Model Theory.

I'm a little bit confused about the idea of Theory and the idea of Model. I understand that a structure is a model of a theory if it satisfies every sentence of the theory.

I wonder if it is correct to think like this:

Model is an independent thing : it has its own rules and objects ("model_axioms", definitions, inference rules, ...). And then, what we want to do is to see if a predefined sets of rules (i.e. the theory) can fit the given model.

For example, if we talk about the language of ordered field, and we have the theory of ordered field which gives the very general ideas about the interactions between objects. I know that the set ($\mathbb{R}$, $+$, $.$) together with it standard ordering is a model of ordered field theory. As for me, given the set of real numbers, its operations, its ordering, in fact, there exist already the rules associated. For example, "sum of 2 real numbers is a real number", "there is always a real number between 2 real numbers", "1 + 1 = 2". Here what we want to do is to see if the ideas of ordered field theory can be applied to a thing (and its rules) which already exists.

Could you please tell me if it is a correct way to think ? If not, could you please tell me how should I think about model, theory and their connection ?

Many thanks for your help!

Asaf Karagila
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VDT-QHH
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  • I don't really understand what you've written, but it reminds me of this question, and you might find my answer to that question useful. – Alex Kruckman Apr 25 '24 at 16:14
  • @AlexKruckman: Hi, your answer is very useful. It mentions what I meant but it didn't answer precisely what I want to ask. I'm agree with you where you said about the structure of real numbers which is something precise, defined using Dedekind cut/Cauchy sequence. I think it actually means that in that structure, the objects and the rules (i.e. statements about real numbers) have already existed, independently of the "theory of real closed field". What I meant is that the point of model theory is to see if the "theory of RCF" is aligned with the "things" (rules, objects,..) which already exist – VDT-QHH Apr 26 '24 at 06:03
  • Do you think it is a right way to think about "theory", "model" and the connection between them ? – VDT-QHH Apr 26 '24 at 06:04
  • Ok, I converted my comments to an answer. – Alex Kruckman Apr 26 '24 at 18:28

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You're correct that structures exist independently of theories. But I don't quite agree with your characterization.

You wrote that a structure "has its own rules and objects ('model_axioms', definitions, inference rules, ...)." But the definition of a structure has nothing to do with axioms, definitions, and inference rules. Structures are "ordinary mathematical objects" (or, if you like, set-theoretic objects): sets equipped with certain distinguished elements, operations, and relations. The fact that $1+1 = 2$ is not a "rule" about $\mathbb{R}$, it's just a feature of how the elements called $1$ and $2$ and the function called $+$ behave.

Now, given a structure $M$, there is a complete theory $\newcommand{\Th}{\mathrm{Th}}\Th(M)$ canonically associated to $M$, namely the set of all first-order sentences true in $M$. Sometimes, we want to axiomatize this theory: write down explicit axioms, which generate a theory $T$, and show that $\Th(M) = T$. This could be described as "seeing if the axioms of $T$ align with the structure $M$". But I would not describe this as the "point of model theory".

For one thing, axiomatizing complete theories of explicit structures is just one kind of problem model theory can help us solve - model theory is about much much more than that. Second, when model theorists do try to axiomatize the complete theory of a structure $M$, the goal is not to understand the relationship between the axioms and the truths of $M$. Instead, the goal is usually to understand exactly which structures $N$ are elementarily equivalent to $M$. If $N\models T = \Th(M)$, then $N$ and $M$ are elementarily equivalent: they satisfy exactly the same sentences of first-order logic. For example, since $\mathrm{RCF} = \Th(\mathbb{R})$, every real closed field is elementarily equivalent to $\mathbb{R}$.

Alex Kruckman
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  • Hello, thanks for your answer. However, I don't really agree when you say that the definition of a structure has nothing to do with axioms, definition or inference rule. For example, in the structure of $\mathbb{R}$, I define the concept of functions, then I have some statements about them, for example the Fundamental theorem of calculus. Actually, I must use inference rule on the model to come to this theorem. Otherwise, I don't think we can say much about a model if we rely only on its defined features, without creating new objects and without using inference rules from the predefined set. – VDT-QHH Apr 26 '24 at 21:58
  • @VDT-QHH The fundamental theorem of calculus is proved using the general inference rules of mathematics (i.e., of ZFC set theory). We don't need special inference rules about $\mathbb{R}$ to prove this theorem. Let me also point out that the fundamental theorem of calculus is not even expressible as a first-order sentence in the language of rings, so it is not a theorem of the theory of real closed fields. – Alex Kruckman Apr 26 '24 at 22:17
  • I am surprised. So it means that when working with real numbers in RCF, I lose all the knowledge of real analysis ? Or maybe we need to "enlarge" the language to include at the same time the language of ZFC set theory + language of ring (i.e. a language made by 2 languages) when we want to conserve the calculus's theorem ? – VDT-QHH Apr 27 '24 at 05:44
  • @VDT-QHH I think there is a misunderstanding here about the purpose of studying theories like RCF. The goal of studying RCF is not to have an axiomatic basis for proving theorems about the real numbers. We already have much more flexible and powerful axiomatic bases for proving theorems about the real numbers, namely foundational theories like ZFC. See my answer here. Instead, we primarily study RCF to understand the properties of definable sets and functions relative to this theory (see quantifier elimination and o-minimality). – Alex Kruckman Apr 28 '24 at 01:04
  • From this point of view, model theorists actually prefer theories which are relatively weak in their expressive power, since we can then obtain stronger regularity results about definability in these theories. – Alex Kruckman Apr 28 '24 at 01:07
  • That said, it turns out that many concepts of calculus actually are definable in RCF, just working one function at a time. I.e., we can express that the mean value theorem holds for the definable function $f$, but we can't write down a sentence which says "for all functions $f$, the mean value theorem holds". But thanks to o-minimality, it turns out that the class of definable functions behaves in a much tamer way than the general functions of real analysis. For example, they are always differentiable at all but finitely many points. This is what I mean by "regularity results". – Alex Kruckman Apr 28 '24 at 01:14
  • Hello, many thanks for your answer. I have asked a separate that relates to your comments, could you please have a look at it if you have time ? https://math.stackexchange.com/questions/4906947/how-to-work-with-many-theories-at-the-same-time – VDT-QHH Apr 28 '24 at 06:37