How do model theorists define structures?
Let $L$ be a first order language and $A$ be an L-structure. Let $\sigma$ be an L-sentence.
In most of the mathematics I have seen a structure is defined by the axioms it satisfies. These may or may not be expressible as first order sentences. For instance groups have a first order axiomatisation where as the axioms of a topological space are not entirely first order. So I would say $A \models \sigma$ just when there is a proof (not necessarily a formal first order proof) of $\sigma$ from the axioms of $A$. So to me a structure always comes with a set of axioms it satisfies.
Somehow in the model theory I have read the situation seems to be inverted. We are given a structure and are asking whether we can find a (first order) axiomatisation for it? But how then is a structure defined, if not via some kind of axiomatisation? In order to even talk about a structure it seems to me one has to have some kind of axiomatisation already in mind. Concretely, I believe, Gödel showed $Th(\mathbb{N})$ isn't (first order) axiomatizable. But how, if not by some kind of axioms (e.g. PA), is $\mathbb{N}$ defined?
The only way I see out of this is that structures are defined by axioms ( maybe not first order expressible) and model theorists ask when a first order axiomatisation exists. Is this the full story?
Does anyone see what I am missing? Many thanks!
Solution 1:
In most of the mathematics I have seen a structure is defined by the axioms it satisfies.
I'm going to go ahead and disagree with this, at least to a certain extent. Consider the following two approaches to "defining the reals:"
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As the unique complete ordered field.
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Via equivalence classes of Cauchy sequences.
Each of these is done with decent frequency; however, only the former is actually isomorphism invariant, and so only the former should be thought of in terms of axioms (first-order or otherwise). Under the hood, both approaches amount to proving an existence-and-appropriate-uniqueness result in an appropriate background theory, although they demand different amounts of uniqueness.
Even before we get to "first-order vs. other," model theory raises the distinction between isomorphism-invariant properties and constructions which require auxiliary work. This doesn't have to be a normative distinction - I find "auxiliary work" quite interesting - but it is a valuable one in many ways.
OK, so what about $\mathbb{N}$? (Let's work in $\mathsf{ZFC}$ for simplicity.)
Well, first of all note that we again have two very different ways of "defining $\mathbb{N}$:" we could prove (say) that there is a unique-up-to-isomorphism prime model of $\mathsf{PA}$, or we could explicitly define additive and multiplicative structure on $\omega$. In either case, we get a theorem which says "$\mathbb{N}$ exists." Moreover, we can prove auxiliary results connecting the different versions of this theorem, so that things are reasonably copacetic.
Godel's incompleteness theorem does put limits on how we could hope to "characterize $\mathbb{N}$," but it doesn't in any way stand in tension with the foregoing.