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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. – Hurkyl Dec 30 '12 at 3:28
up vote 2 down vote accepted

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). – Hurkyl Dec 30 '12 at 3:26
@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. :) – tomasz Dec 30 '12 at 3:28
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?) – alancalvitti Dec 30 '12 at 3:28
@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 . – tomasz Dec 30 '12 at 3:30
@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. – tomasz Dec 30 '12 at 3:56

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