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As a beginner, I'm overwhelmed by the usage of terminology , such as theory, model, interpretation, structure et al, which are omnipresent in Mathematical logic.

Here's my understanding about them:

Creating a theory is synonymous to axiomatization, which is selecting a finite number of sentences, which is expected to be satisfiable,which means these sentences are true in at least one model.

The motivation of distinguishing between theory and model is that a theory always suffers from the indecidability of some meaningful sentences e.g. continuum hypothesis, which can't be avoided once and for all by adding more axioms in the theory. However, as expected in a model, there is no such thing indecidable as continuum hypothesis, or we don't talk about them in the context of a model. Usually, a model is defined as consisting of an underlying set, a set of relations, and a set of functiosn, which are defined intentionally to address certain unsolved problems in the theory. It seems to me it only make sense to talk about a model in the context of a particular sentence of interest, because no model can be fully axiomatized.

I'm also confused with the usage of interpretation, structure, and model. Are they interchangeable?

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    Aximoatizations aren't always finite. Recursively enumerable is usually very useful, and sometimes even more generous axiomatizations are useful -- e.g. the axiomatization formed by taking every theorem as an axiom is useful in the foundations of non-standard analysis.2012-12-30

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A theory and axiomatization are not entirely synonymous. It makes sense to talk about axiomatization of a theory, but there's no such thing as a theory of an axiomatization. The difference is, perhaps, a bit subtle, but it is there. Some authors also consider theory to be by definition closed under taking consequences, which makes the distinction more clear.

Similarly for the differences between model and structure: they mean the same thing on their own, so “Take a model $M$”, and “Take a structure $M$” mean the same, but “model” has a more specific meaning: you can say, for example, that “$M$ is a model of $T$” and you can't really replace “model” with “structure” in this statement. On the other hand, it is more usual to say that something is a structure of a given language than a model.

The word “interpretation”, I've most often encountered to mean the interpretation of a term or a symbol of a language in a given structure (or model ;) ) than anything else, and I don't think it has any synonyms to that effect.

Your paragraph about theories and models seems completely off to me. A model and a theory are two very distinct concepts. A theory is just a set of formulas, while a model is a concrete set with concrete interpretations of all symbols of a given language. They're defined the way they are defined, not necessarily to address any unsolved problems, whatever that's supposed to mean, and it certainly makes sense to talk about a model in full generality, along with its entire theory. I'm not sure what you mean by the statement that „no model can be fully axiomatized”.

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    There is the theory generated by an axiomatization (by which I just mean a set of statements).2012-12-30
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    @Hurkyl: Depending on your meaning of theory, there might be many theories „generated” by axiomatization, the most important ones being the axiomatization itself and its closure under taking consequences. :)2012-12-30
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    Re: "A theory is just a *set of formulas*, while a model is a *concrete set* with concrete interpretations of all symbols of a given language." [my italics]. Can you amplify on the distinction? And what "concrete set" and "concrete interpretations" mean in this context? (eg, the category Set?)2012-12-30
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    @alancalvitti: It means that you have a given set of elements, and for each function symbol you know what it does with any tuple of elements, and for each relation symbol you know whether or not the relation is satisfied by a tuple. I don't think category theory is relevant at this point .2012-12-30
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    @tomasz, for example, take the Collatz (semi)dynamics. It's an open question whether every natural number is in the basin of attraction of the cycle {4,2,1}. Are you saying it's a theory but not a model because we don't know whether the relation is satisfied? (Just trying to make things concrete).2012-12-30
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    @alancalvitti: It's neither. It's just a sentence (or, rather „every natural number is in the basin of attraction of the cycle $\{4,2,1\}$” is a sentence). It may or may not be true in a given model, just as it may or may not be true in true arithmetic (probably one or the other, we just don't know which). I didn't mean “know” literally in my last comment. Perhaps it would be more accurate to say that, rather, the model itself *knows* these things, not us.2012-12-30
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    @tomasz, ok, by the way, Wikipedia: http://en.wikipedia.org/wiki/Model_theory says it's the study of classes of mathematical structures. So why wouldn't category theory be relevant here?2012-12-30
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    I think that the OP's paragraph about theories and models was written to address an unstated question that one might have upon first encountering these notions, namely "why isn't there a one-to-one correspondence between theories (_e.g._ Peano Arithmetic) and 'intended models' of those theories" (_e.g._ $\mathbb{N}$.)2012-12-30
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    @alancalvitti: Much of modern model theory is interacting with algebra. Not necessarily with categories, but with algebra. The reason categories are less useful here is that he tools used in model theory are often logical rather than categorical.2012-12-30
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    @alancalvitti: I meant that we're talking about the very basic notions. It makes sense to talk about e.g. category of structures of a given signature with homomorphisms, or category of models of a given theory (such as the category of groups, abelian groups, rings...), but here we're talking about the very basic notions and none of this is needed. That's what I meant by not relevant *at this point*.2012-12-30
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    A paragraph On page 80, A Mathematical Introduction to Logic, Herbert B. Enderton(2ed) indicates that interpretation might be interchangeable with stucture.2013-01-06
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    "In sentential logic we had truth assignments to tell us which sentence symbols were to be interpreted as being true and which as false. In firstorder logic the analogous role is played by structures, which can be thought of as providing the dictionary for translations from the formal language into English. (Structures are sometimes called interpretations, but we prefer to reserve thatword for another concept, to be encountered in Section 2.7.) "2013-01-06
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    @MettaWorldPeace: well, I was referring to what I've personally encountered in practice. I think I've also heard that the word is sometimes used to that effect (and I haven't said that it is never the case), it's just that I don't recall having ever heard it *actually* used in that meaning during a lecture or a conference talk, hence the impression.2013-01-06